Exact constructions of square-root Helmholtz operator

Exact constructionsof square-root Helmholtz operatorsymbols:The focusingquadratic profile

LouisFishman, MaartenV. deHoop

andMattheusJ.N.vanStralen

NavalResearch Laboratory, Code7181,StennisSpaceCenter, MS39529Centerfor WavePhenomena,ColoradoSchoolof Mines,Golden,CO80401PlasmaOpticalFibreB.V., Zwaanstraat 1, 5651CA Eindhoven,TheNetherlands

ABSTRACT

Operatorsymbolsplay a pivotal role in boththeexact,well-posed,one-way reformu-lation of solving the (elliptic) Helmholtzequationandthe constructionof thegener-alizedBremmercouplingseries.The inversesquare-rootandsquare-rootHelmholtzoperatorsymbolsaretheinitial quantitiesof interestin bothformulations,in additionto providing the theoreticalframework for the developmentand implementationofthe ‘parabolicequation’(PE)methodin wave propagationmodeling.Exact,standard(left) andWeyl symbolconstructionsarepresentedfor both the inversesquare-rootandsquare-rootHelmholtzoperatorsin the caseof the focusingquadraticprofile inonetransversespatialdimension,extending(and,ultimately, unifying) thepreviouslypublishedcorrespondingresultsfor thedefocusingquadraticcase[J. Math. Phys.33(5), 1887-1914(1992)].Both (i) spectral(modal)summationrepresentationsand(ii)contour-integral representations,exploiting the underlyingperiodicity of the associ-ated,quantummechanical,harmonicoscillatorproblem,arederived,and,ultimately,relatedthroughthepropagatingandnonpropagatingcontributionsto theoperatorsym-bol. High- andlow-frequency, asymptoticoperatorsymbolexpansionsaregivenalongwith theexactsymbolrepresentationsfor thecorrespondingoperatorrationalapprox-imationswhich provide thebasisfor thepracticalcomputationalrealizationof thePEmethod.Moreover, while thefocusingquadraticprofile is, in somerespects,nonphys-ical, thecorrespondingHelmholtzoperatorsymbols,nevertheless,establishcanonicalsymbolfeaturesfor moregeneralprofilescontaininglocally-quadraticwells.

Keywords: one-waywave equations,pseudodifferentialoperators,normalmodes

1. Intr oduction

The global natureof wave propagationproblems,asmodeledby theelliptic scalarHelmholtzequation,rendersthecomputa-tionalsolutionquitedifficult in extendedinhomogeneousenvi-ronments(1, 2). Thedevelopmentandapplicationof the ‘par-abolic equation’(PE) method(3, 4, 5, 6, 7) hassuccessfullyaddressedthis issuefor appropriately, weaklyrange-dependentenvironments,where, for the most part, one-way (forward)wave fields are computedand back-scatteredenergy is ne-glected.In recentyears,the PE methodhasbeenextendedtofully-coupled,two-way, elliptic wave propagationthroughtwocomplementaryapproaches:(i) theexact,well-posed,one-wayreformulationof elliptic wave propagationproblems(1, 2, 8)and (ii) the constructionand applicationof the generalized

Bremmercouplingseries(9, 10, 11). Both methodsof exten-sionarebasedon ideasandconstructionsfrom wave field de-composition,invariantimbedding(reflectionandtransmissionoperators),andthecloselyrelatedDirichlet-to-Neumann(DtN)operators,andmake useof micro-localanalysis(pseudodiffer-entialandFourierintegral operatorsandpathintegrals).Whilethecurrentfocusin thedirectandinverseanalysisof theabovetwo approachesis primarily on the propertiesand singular-ity structureof thescattering(reflectionandtransmission)andDtN operatorsymbols(1, 12, 13), the inversesquare-rootandsquare-rootHelmholtzoperatorsymbolsarethe initial quanti-tiesof

152 L. Fishman,M.V. deHoop& M.J.N.vanStralen

interestin bothformulations.In many respects,however, despitethedetailedtreatmentof thefully-coupled,two-way, elliptic formu-lationsin theabove-referencedliterature,theoriginal PEmethodprovidesthemoststraightforwardintroductionto theHelmholtzoperatorsymbols.

Startingfrom thetwo-dimensional,space-frequency domain,scalarHelmholtzequation,theformally exactwaveequationforpropagationin a transverselyinhomogeneoushalf-spacesupplementedwith appropriateright-traveling-wave radiationandinitial-valueconditionsis givenby (1, 2)i (1)

where

is theone-way wave function, is a referenceor averagewavenumber(proportionalto frequency), and "! # $%&(' )+* "! , .- /#*0 ,*$%1"243 * (2)

is the square-rootHelmholtzoperator, where) "!

is the refractive index field. The rangecoordinate,5 , is associatedwith the‘one-way’ direction;

!is thetransversecoordinate.In thesubsequentformalanalysis,thetransversecoordinateswill alsobefreely

denotedby 6 .For apointsourceof thevolumeinjectiontype,locatedat 5 and

! !%7, theappropriateinitial-valueconditionor (source)

decompositionis givenby (1)8 ! 9 i : <; 2>= "!8?!%7 @ (3)

where ; 2

denotestheinverseor parametrixof

(i.e., theinversesquare-rootHelmholtzoperator).TheHelmholtzoperatorsymbolsaredefinedin apseudodifferentialoperator(operator-ordering)calculus(14,15,16, 17). LetA A "! !CB

denotethekernelassociatedwith operator

, i.e., D8E 5 ! 9GFIH JLK ! B A "! ! B 8 5 ! B M (4)

andA ; 2 A ; 2 "! ! B

denotethekernelassociatedwith theinverse ; 2

. Thenthesymbol, for theinversesquare-rootHelmholtzoperator, is definedasN 7 OQPR TS 6 9GF H J K/UDVXW/Y i S UI A ; 2 6 6 ZUI (5)

in thestandard(left) pseudodifferentialoperatorcalculus,andas[ O PR TS 6 9F H J K/UDV0WY i S UI A ; 2]\ 6 ? 2* U^ 6 2* UI_ (6)

in the Weyl pseudodifferential operatorcalculus(14, 15, 16, 17). Likewise, definitionsofN 7 O

and[ O

hold for the square-rootHelmholtzoperatoritself.

In termsof thesquare-rootHelmholtzoperatorsymbols,theone-way waveequation(1) canbewrittenas `i aFbH J :dc .K S N 7 O TS ! eV0WY i Sb! gf 5 S 9 (7)

ori aF H J K ! B F H J :dc .K S [ O \ S 2* "! ! B #_hV0WYi' i Sh"!8?j! B 1 5 ! B 9k (8)

in thestandardandWeyl calculi(14, 15, 16, 17), respectively. Here,f 5 S is theFouriertransformof theone-waywavefunction 5 ! with respectto thetransversecoordinate,f 5 S 9GF H J K ! B V0WY .? i Sl! B e 5 ! B @m (9)

For a fixed rangepoint, 5 5bn say, Eqs.(7)and(8) provide the basisfor the nonreflectingboundaryconditionsubiquitousinnumericalwavefield computations(1, 4, 10).o

In VanStralenetal. (10), interchangep and q , andreplace rs (theverticalslownessoperator)by t , ru by v 7 O , and rw 2 by x (and rw * by x ; ).

Exactsquare-rootoperator symbols 153

The fundamentalsolution(propagator),y , associatedwith Eq.(1)canbe expressedasa lattice multivariateintegral, withHamiltonianequalto thesquare-rootHelmholtzoperatorsymbol(1, 2, 11, 18, 19, 20, 21),y 5 !/z 5 B ! B g 5 ? 5 B |~k FbH J. PR 2 C:%c .K S^

; 2 2 K !/

V0WYZ i 2X S "! ?<! ; 2 ,N 7O TS ! # ; 2X 5I (10)

with! ~ ! B ! !

, 5 5 ? 5 B , andwhere

. is theHeavisidefunction.All theintegrationsaretakenover theinterval.?

, ; 2 5 is thestepsizein theparameter alongtherangedirection,and

"! S 0 arethecoordinatesof a path(in

transversephasespace)at thediscretevalues of as - 0

. A similarphase-spacepath-integral representationis foundin theWeyl calculus(1, 2,18, 19, 20,21). Thestructureof thepath-integralrepresentationin Eq.(10)straightforwardlyresultsin thephasespace,marchingalgorithm,which generalizestheTappert/Hardin(6, 7) split-stepFFT algorithmfor thestandardparabolicapproximationto theone-way Helmholtzequation(1) andhasbeendiscussedandillustratedin detailin theliterature(19,20,21).

Moreover, in thecontext of theinversescatteringproblem,thecompositionequationin thepseudodifferentialoperatorcalculusconnectsthesquare-rootHelmholtzoperatorsymbolwith thesquareof therefractive index field through(1, 14, 15, 16, 17)) * 6 ? S * [ O TS 6 (11) dc #*iFIH J¡EK¢K/UbK£#K/¤ [ O ¢g S #U 6 [ O £i S .¤8 6 @V0WY,' : i U/£ ? ¢¤ 1in the Weyl calculus,with a similar expressionin the standardcalculus(14, 15, 16) (seealso AppendixC). The compositionequation(11)alsoservesasthestartingpoint for theexact,approximate,andnumericalconstructionsof thesquare-rootHelmholtzoperatorsymbols(1, 17,22, 23).

Finally, theoperatorsymbolsin theWeyl andstandardpseudodifferentialoperatorcalculi arerelatedto oneanother(14,15,16,17, 19), viz.,N 7 O TS 6 9 dc lFbH JEKl¢K/U [ O ¢¥.UIV0WY,' ? : i 6 ? UI TS? ¢% 1 (12)

while[ O TS 6 9 %c FbH J/ Kl¢K/UQN 7O ¢¥.UIV0WY,' : i 6 ? UI TS? ¢% 1 m (13)

Recastingtheformaloperatorequations(1)and(3), in achosenpseudodifferentialoperatorcalculus,in termsof theappropriateoperatorsymbolsprovidestheexplicit meansto extendFourieranalysisof wavepropagationin (transversely)homogeneousmediato inhomogeneousenvironments:the analysisis carriedout in the transverseFourier (

S) domainwithout leaving the transverse

space( 6 ) domain.The transversespace-wave numberdomainconstitutesthe transversephasespace.The transversevariableS

togetherwith theprincipalpartofN 7 O

form thecomponentsof thecotangentvectorattachedto thewave front.Ratherthanfocusingon theindividual operatoreigenvaluesandeigenfunctionsasis donein thetraditionalspectralanalysis,

the focushereis on the operatorsymbols:they containthe completespectralinformationin just the appropriatemannerto leadimmediatelyto theinfinitesimalpropagatorandinitial wavefield.

Operatorsymbolsarea naturalquantity to consider. For example,they provide the ‘generalizedslownesssurface’ in geo-physics(11), the natural multidimensionalextension of the scattering(reflection and transmission)coefficients in the one-dimensionalformulation(2, 13), andthe framework to quantize(semi-)classicaltheoriesin quantumphysics(24). While eithertheWeyl or thestandardpseudodifferentialoperatorcalculusprovidesacompletedescriptionof thepropagationproblem,they are,in many respects,complementary, andcanbeusedin conjunctionto advantage.For example,thesymmetryinherentin theWeylcalculuscanoften be exploited in operatorsymbolconstructionsandanalysis(particularlythat involving integratedenergy-fluxconservation calculationsandthat separatingthe effectsdueto anisotropy from thosedueto heterogeneity),while the standardcalculusnaturallyresultsin morecomputationallyefficientalgorithms(19,20).

The brief outline presentedfrom Eq.(5) to Eq.(13)indicatesthe pivotal role that both the inversesquare-rootandsquare-root Helmholtzoperatorsymbolsplay in the PE methodfor wave propagation.The explicit constructionof the one-way waveequationandinitial wavefield, path-integral solutionrepresentationandsubsequentnumericalalgorithm,computationalboundaryconditions,andfundamentalinverserelationshipall dependcrucially upontheanalysisandsubsequentpropertiesof the relevantHelmholtzoperatorsymbols.The extensionto the scatteringand DtN operatorsymbolsin the fully-coupled,two-way, elliptic


formulationsonly reinforcesthe importanceof their role (1, 2, 8, 9, 10, 11). From this perspective, the constructionof exactsquare-rootHelmholtzoperatorsymbolsis of greatvaluein illuminating thegeneralmathematicalpropagationtheory, in additionto providing benchmarksolutionsfor both asymptoticand numericaloperatorsymbol constructions(1, 22, 23). Furthermore,the explicit constructionof thesenontrivial symbols,correspondingto a fractionalpower of the indefinite,transverseHelmholtzoperator, is of mathematicalinterestin its own right. Sincethe relevant (frequency-domain)operatorslie outsideof the well-developedtheoryof elliptic pseudodifferentialoperators(1, 15), anew asymptotic,operatorsymbolcharacterizationis required(1,9, 12). The constructionis accomplishedby incorporatingcomplex and spectralanalysesin the calculusof pseudodifferentialoperators(seealsoShubin(25)).

In view of thepotentiallyilluminating role of exactoperatorsymbolconstructions,this paperpresentstheexactsymbolcon-structionsfor boththeinversesquare-rootandsquare-rootHelmholtzoperatorsin thecaseof thefocusingquadraticprofile in onetransversespatialdimension,extending(and,ultimately, unifying) thepreviously publishedcorrespondingresultsfor thedefocus-ing quadraticcase(23).Thefocusingcaseis particularlyinterestingin thecontext of theformationof caustics.Theresultsaregivenfor boththeWeyl andstandard(left) operatorsymbols.While thefocusingquadraticprofile is, in somerespects,nonphysical,thecorrespondingHelmholtzoperatorsymbols,nevertheless,establishcanonicalfeaturesfor moregeneralprofilescontaininglocally-quadraticwells.Section2 derivesHelmholtzoperatorsymbolrepresentationsin theform of spectral(modal)summationsfollowingfrom Eqs.(5)and(6) andthestandardspectraltheoryof thecorrespondingoperatorkernels.In Section3, contour-integral represen-tations,exploiting theunderlyingperiodicityof theassociated,quantummechanical,harmonicoscillatorproblem,arederivedfortheHelmholtzoperatorsymbols.Thesecontour-integral representationsunify thefocusinganddefocusingquadraticprofilecases,providing theappropriateanalyticcontinuationresultsanda convenientform for thesubsequentasymptoticanalysis.Theanalyticcontinuationresultsarecombinedwith the contour-integral representationsconstructedin the time-Fourierdomainto derive thesquare-root‘Helmholtz’ operatorsymbolsin thetime-Laplacedomainin Section4. Theapplicationof standardasymptoticmeth-odsto thesecontour-integral representationsthenresultsin boththehigh-andlow-frequency expansionsof theHelmholtzoperatorsymbolsin Section5.Thehigh-frequency asymptoticresultsareof particularinterestin view of thefractional,transverseHelmholtzoperatorsfalling outsidethescopeof elliptic pseudodifferentialoperatortheory. In Section6, thespectral(modal)summationandcontour-integral representationsarerelatedthroughanexplicit considerationof thepropagatingandnonpropagatingcontributionsto theHelmholtzoperatorsymbols,andthemultiresolutionandgeneralizedscreenpropertiesof theone-way propagationprocessarerevealedthroughthestructureof theWeyl andstandardoperatorsymbols,respectively. Theexactsymbolrepresentationsfor thewell-known, operatorrationalapproximationsof thesquare-rootHelmholtzoperator(1, 3, 4, 5, 6, 10, 23), whichprovide thebasisfor thepracticalcomputationalrealizationof thePEmethod,arepresentedin Section7 in boththespectral(modal)summationandcontour-integral forms for the focusingquadraticprofile. Theseconstructionsextend(and,ultimately, unify) the previously pub-lishedcorrespondingresultsfor thedefocusingquadraticcase(23).Numericalrealizationsof theexactandapproximateHelmholtzoperatorsymbolsarepresentedin Section8, while the resultsderived in the precedingsectionsareappliedto illustrateseveralpointspertinentto directandinversewave propagationmodelingin multidimensional,extendedinhomogeneousenvironmentsintheconcludingdiscussionpresentedin Section9. AppendicesA-C provide thenecessarymathematicaldetail.

2. Spectral (modal) summationoperator symbol representations

A spectral(modal) summationsymbolrepresentationfor the inversesquare-rootHelmholtzoperatorfollows, in principle, fromEqs.(5)and(6) andthe standardspectraltheory for the correspondingoperatorkernel

A ; 2 "! !CB . In the usualmanner(4, 11),A ; 2 "! ! B

is representedby a convergent,complex Dunford integral in termsof theassociated *

operatorresolvent– which issimplyproportionalto theGreen’s functionfor thecorresponding,effectiveone-dimensional(range-transformed)Helmholtzequa-tion. The Green’s function is constructedin termsof the usualbilinearproductandcorrespondingWronskianof theappropriatesolutionsof thehomogeneousequation.Whenever – for a specificprofile – theeffective one-dimensionalHelmholtzequationcanbesolvedin closedform, theassociatedoperatorresolventcanbeexplicitly constructed,theoperatorspectrumidentified,andtheDunfordintegralrepresentationsubsequentlyevaluatedto yield amodalsummationrepresentationfor thekernel

A ; 2 "! ! B . Appli-

cationof Eq.(5)and/orEq.(6)thenresultsin thespectral(modal)summationrepresentationfor theinversesquare-rootHelmholtzoperatorsymbol.Thecorrespondingsymbolrepresentationfor thesquare-rootHelmholtzoperatorthenfollows from compositionwith thesymbolfor the

*operator(4, 11, 23).

Thefocusingquadraticprofile in onetransversespatialdimensionis definedthrough)+* "! 9)+* ?+¦ * ! * (14)

with) ¦¨§<© ª . Theprofile in Eq.(14)is (i) a modelfor waveguidingstructuresin classicalphysicsapplications(26), (ii) the


well-known harmonicoscillatormodelin quantummechanics(27), and(iii) a much-studied,exactly solubleproblemin operatorspectraltheory(28).

A. The Schwartzkernel

Theabove-outlined,general,spectral(modal)summationrepresentationconstructionfor theinversesquare-rootHelmholtzoperatorkernelhasbeenpresentedbefore.For a concisereview of thegeneralspectraltheoryandthedetailedcalculationsfor thefocusingquadraticprofilecase,seeVanStralen(4); thefinal result

is givenbyA ; 2 "! ! B 9 c 243 * «C -¬E -' ? : ¬ -]?<® 1 243 * : ; «l¯ « "! ! B @

whichexpression,for laterconvenience,is rewritten in theformA ; 2 "! ! B 9 ? V0WY \C°± c i _ c 243 * «¥ -¬9 -'i : ¬ -]?<® 1 243 * : ; « ¯ « "! ! B M (15)

with¯ « "! ! B ² ³ « \ ´¦ 243 * ! _E³ « \ ´¦ 243 * ! B _Z (16)³ « "µ ² VXW/Y \ ? 2* µ * _E « "µ M (17)

where «

is theHermitepolynomial(29),while® ) * %¶ (18)¶ ¦ ·m (19)

With theeffectiverefractiveindex associatedwith mode ¬ givenby' ? ¶ : ¬ -Q?j® 1 243 * , thenumberof propagatingmodes, ,

is obtainedfrom theestimate:¸ ?¹-Lº¹®»º :¸ - mIn Eq.(15),the principal valueof the square-rootfunction is taken,consistentwith the right-traveling wave condition,enforcingthe radiationconditionat infinity to be satisfiedin the usualmanner(1, 4, 24). The seriesrepresentationis understoodin thedistributionalsense(4, 30).

B. The standard (left) symbol

For thespectral(modal)summationrepresentations,it is technicallyeasierto first constructthestandardHelmholtzoperatorsym-bolsandsubsequentlydeducethecorrespondingWeyl formsusingEq.(13).Thus,substitutingEq.(15)into Eq.(5),interchangingtheorderof integrationandsummation,andapplyingtheHermite,Fouriertransform(29),FlH J¼K µ V0WY

iµ½ .³ « "µ 9 :dc 243 * i « ³ « 4½ M (20)

yield thedesiredexpressionfor thestandard,inversesquare-rootHelmholtzoperatorsymbolN 7 OQPR TS 6 9 ? V0WY \ °± c i _j¾ :¶i¿ 243 * «C -¬E -'i : ¬ -]?® 1 243 * ¾ i:^¿

«QÀ « TS 6 @ (21)

where

À « TS 6 9ÁV0WY .? i S 6 ³ « \ ´¦ 243 * 6 _E³ « \ ¦ 243 * S _Z (22)

seealsoVanStralen(4, (4.118)).In the transformationfrom thekernelto thestandardsymbol,notetheoccurrenceof theGaborwavelet, c ; 243 ± V0WY i µ½ ³ "µ , for ¬ Â . Applicationof the ‘large ¬ ’ asymptoticexpansionsfor boththeHermitepolynomialsandthegammafunction(29), ¬E , establishestheconditionalconvergenceof theseriesin Eq.(21).

In VanStralen(4, (4.114)),replaceÃ by ÄÃ , Å by Æ * , Ç by ÈÉ * , 2* Ã ; 2 Ç 243 * by Ê , while Ë 0Ì ; 243 * shouldbeidentifiedwith Í ; 2 ; in thenotation

of thispaper, thespectrumof theoperator(cf. Eq.(2)) Æ *dÎ qdÏ/Ð ÎÒÑXÓ ÄÃ * Ï"Ô *$ follows as Ë *Õ «¥iÖa× Ê Î´ØÙ Ð Ñ × Ú Ï , Ù ÖÛCÜ Ñ Ü Ø Ü>ÝXÝ>Ý .


Thestandardsymbolfor thesquare-rootHelmholtzoperatorfollows, in principle,on composingN 7 O TS 6 with

N 7 OQPR TS 6 with theaidof thestandardintegral compositionequation(14,15,16), i.e.,N 7 O TS 6 9 :dc F H J/hK/UbK¢MN 7O PR TS? ¢¥ 6 .N 7 O TS 6 ? UI@V0WY .? i Þ¢UIMm (23)

Applying thiscompositionequationin its differentialcounterpartform, withN 7 O ) * ?ß¦ * 6 * ?·S * (cf. Eq.(14)),yieldsN 7 O TS 6 9 )+* ?à¦ * 6 * ?S *0/N 7OQPR TS 6 , ¶ *' : i 6 ¥áN 7OQPR TS 6 i ,*á N 7 OPR TS 6 1 m (24)

SubstitutingEq.(21)into Eq.(24)thenresultsin theformalseriesN 7 O TS 6 9 :d¶ 243 * «C -¬E ' ? : ¬ -]?<® 1 243 * ¾ i: ¿«QÀ « TS 6 @

which, by the sameabove-referencedasymptoticexpansions,is divergent, reflectingthe nonvalidity of the interchangeof thesummationanddifferentiationoperationsin thecompositioncalculation(4, 31).

A properexpressionforN 7 O

, essentiallymaintainingthemodaldecompositionform in Eq.(21),can,however, bederivedin thefollowing manner. Partitioningtheseriesin Eq.(21)givesN 7 OQPR TS 6 9 ? V0WY \C°± c i _j¾ :¶^¿ 243 *jâ8ã ; 2«¥ -¬E -'

i : ¬ -]?® 1 243 * ¾ i:^¿

« À « TS 6 «¥ ã -¬E -'i : ¬ -Þ?à® 1 243 * ¾ i: ¿

«ÞÀ « TS 6 Cäå (25)

whereæ is apositive integerwithæèçé¸ - mTheinfiniteseriesin Eq.(25)containsonly nonpropagatingmodes.In thesecondtermin (25),for ¬àê æ , applicationof theidentity(Magnuset al. (29, p.6))c 243 * : ¬ -Þ?à® 243 * GF K£e£0; 243 *IV0WYi' ? : ¬ -Þ?® 4£ 1 (26)

followedby theinterchangeof theorderof integrationandsummation,resultin thecontribution from theinfinite sum,?i¾ :c,¶ ¿ 243 * F K£e£X; 243 *IVXW/Yi' ë®?¹- 4£ 1 «¥ ã -¬9 ¾ i: ¿

«ÞÀ « TS 6 8V0WY .? : ¬ £# mThe interchangeof the operationsis justified in Appendix A through the uniform convergenceof power series(31) and theRiesz/Youngtheorem(32). Writing theinfinite seriesin theabove termin theform«¥ ã -¬9 ¾ i:i¿

«ÞÀ « TS 6 8V0WY .? : ¬ £#² «C -¬E ¾ i:i¿«QÀ « TS 6 LV0WY .? : ¬ £>? ã ; 2«¥ -¬9 ¾ i: ¿«QÀ « TS 6 LV0WY .? : ¬ £# (27)

andapplicationof theMehlerformula

thenyield thereplacementof Eq.(25),N 7 OQPR TS 6 9 ? V0WY \ °± c i _j¾ :¶ ¿ 243 * ã ; 2 «¥ -¬E -'i : ¬ -]?<® 1 243 * ¾ i: ¿

« À « TS 6 ?i¾ -c,¶I¿ 243 * F K£e£0; 243 *bV0WY ë® £>ì' í £> ?Zîk £ï æ 1 (28)

TheMehlerformulais givenbyØ 243 *lð#ñò Î ×9ó ÏÞô «¥ Î´Ù,õ Ï ; 2 Î i ÓØ Ï «lö « Î ÷ Ü.ø Ïbù ð#ñCò Î × Ø ó Ïûú « Öjü Î ó Ï in thispaper’s notation,seeRainville (33,

p.198);i.e.,

üis agenerator of ý Ø 243 *Î´Ù,õ Ï ; 2 Î i ÓØ Ï « ö «þ .


whereí £>ÿ V0WY ? 2* : £>, i V : £> ?a- V : £># 243 * (29)îk £ï æ ÿ : 243 *,VXW/Y .? £> ã ; 2«¥ -¬E ¾ i: ¿« À « TS 6 V0WY .? : ¬ £>M (30)

and ¶ ; 2 ´¦ * 6 *g S *M (31) ¶ ; 2 ¦ 6 S m (32)

TheconstructionofN 7 O TS 6 via thestandardcompositionequationfollowsagainfrom

Eq.(24).SubstitutingtherepresentationforN 7 OQPR TS 6 givenin Eq.(28)into Eq.(24)allowsfor theinterchangeof thedifferentiation

operationswith thesummationandintegrationoperations.This follows from thefactthat(i) theseriesappearingin theexpressionarefinite and(ii) the integrandis an integrablefunction with exponentialdecayfor large

£andadmitsuniform boundsfor the

functionandits subsequentderivatives(31). Indeed,the point of the constructionprocedurewasto replacethe infinite seriesinEq.(25)with anappropriatelydifferentiable(equivalent)integral representation.Theresultof thecalculationcanbewritten in theform N 7 O TS 6 9 :d¶ 243 * ã ; 2«¥ -¬E ' ? : ¬ -]?® 1Ò243 *a¾ i:i¿

« À « TS 6 ?i¾ ¶c ¿ 243 * F K£e£0; 243 *IV0WY ë® £>ì' £> ?M £ï æ 1 (33)

which is thecounterpartof Eq.(28),where £>² V0WY ? 2* : £#i i V : £# ?¹- V : £># 243 * ®? V : £>##* ? : i V : £# : £> ? : £> (34)M £ï æ ² : 243 *,V0WY .? £> ã ; 2 «C -¬E ' ? : ¬ -]?<® 1 ¾ i: ¿« À « TS 6 8V0WY .? : ¬ £>M (35)

andtheprincipalvalueof thesquare-rootfunctionappearingin Eq.(33)is taken.Theresultin Eq.(33),in essence,supplementstheapproximate,truncatedseriesrepresentationfor

N 7 O TS 6 derivedby VanStralen(4), with an integral which resultsin the exactexpression.As such,theexpressionin Eq.(33)is well-definedin the limitæ , thenonconvergentbehavior of the(first) seriesbeingexactly cancelledby theappropriatecontribution from theintegralterm in the neighborhoodof

£ . The detailsof this cancellationwill result in computationallyconvenientoperatorsymbol

representations,thus,focusingon theneighborhoodof£g

, partitioningtheintegralon'

in Eq.(33)into thecontiguoussets' 1and' M

with ç , andnotingthatF K£e£0; 243 *bV0WY ë® £# M £ï æ 9 (36): ° 3 * 243 * ã ; 2 «C -¬E ' ? : ¬ -]?® 1 ¾ i:^¿

« À « TS 6 2 í 2 .- C: z : z? : ¬ -Þ?<® #Mallow for

N 7 O TS 6 to bewritten in theformN 7 O TS 6 9 :d¶ 243 *"! ; 2 «¥ -¬E ' ? : ¬ -Q?à® 1"243 *a¾ i: ¿«ÞÀ « TS 6 $# - : i ¾ c ¿ 243 * ' ? : ¬ -Þ?à® 1"243 * 2 í 2 .- C: z% C: z? : ¬ -Þ?à® #&

i :%¶ 243 * ã ; 2«C ! -¬E : ¬ -]?® 243 *¹¾ i:^¿

« À « TS 6 (37)$# -Q? : ¾ c ¿ 243 * : ¬ -Q?® 243 * 2 í 2 .- : z : z%? : ¬ -]?® #&

158 L. Fishman,M.V. deHoop& M.J.N.vanStralen?i¾ ¶c ¿ 243 * F' K£e£0; 243 *,VXW/Y ë® £>( £> ? i

¾ ¶c ¿ 243 * F K£e£0; 243 *bV0WY ë® £#' £# ?M £ï æ 1 where2 í 2 .- C: z C: z? : ¬ -]?® #g 2* M; 243 *^F K£e£X; 243 *,V0WY,' ? : ¬ -Þ?<® 4£ 1 (38)

is theincompletegammafunctionexpressedasaconfluenthypergeometricfunctionasgivenin Magnuset al. (29, p.337).With the following estimates,it will be establishedthat the limit æ) in Eq.(37),termwise, exists.The first and third

termsin Eq.(37)areindependentof æ andhencewell-definedin thelimit æ* . Theseriesin thesecondtermof Eq.(37)canbeboundedby+++++ ã ; 2«¥ ! -¬E : ¬ -]?® 243 *¹¾ i:i¿

«ÞÀ « TS 6 ,# -Þ? : ¾ c8¿ 243 * : ¬ -Þ?à® 243 * 2 í 2 .- C: z C: z%? : ¬ -]?<® #& ++++- ã ; 2«¥ ! -¬E : ¬ -Q?® 243 * ¾ -:,¿« ï À « TS 6 ï ++++ -Þ? : ¾ c8¿ 243 * : ¬ -Þ?à® 243 * 2 í 2 .- C: z : z? : ¬ -]?<® # ++++ m (39)

Theoscillatorfunctionproductadmitstheboundstatedin Abramowitz andStegun(34, p.787),ï À « TS 6 ï º : « ¬E. * (40)

with .$/ - m 0213 4 forS 6 §·© ª . Equations(26) and(38)establishtheequalityc 243 * : ¬ -Þ?à® 243 * # -Þ? : ¬ -]?<® 243 *c 243 * : 243 * 2 í 2 .- : z C: z? : ¬ -]?® #& F K£e£ ; 243 * VXW/Yi' ? : ¬ -Þ?<® 4£ 1 (41)

which,uponapplyingstandardboundingtechniques5 , yield theestimate++++ -Þ? : ¾ c ¿ 243 * : ¬ -Þ?® 243 * 2 í 2 .- C: z C: z%? : ¬ -]?<® # ++++- ¾ - c¿ 243 * - : ¬ -Q?® 243 * V0WYi' ? : ¬ -Þ?® 1 m (42)

CombiningEqs.(40)and(42)with Eq.(39)thengives+++++ ã ; 2«¥ ! -¬E : ¬ -]?® 243 *¹¾ i: ¿«ÞÀ « TS 6 $# -Q? : ¾ c ¿ 243 * : ¬ -Q?® 243 * 2 í 2 .- : z : z%? : ¬ -]?® #& ++++º . *8¾ - c¿ 243 * V0WYi' ? .-]?® 1 ã ; 2«C ! V0WY .? : ¬ 9]m (43)

The absolutevaluesof the termsin the seriesare boundedby the termsin a convergent geometricseriesforS 6 §»© ª , thus

establishingabsoluteanduniformconvergencein thelimit æ6 andthebound+++++ «C ! -¬E : ¬ -Þ?à® 243 * ¾ i:i¿« À « TS 6

5 Theestimate,7 98 óCó ; 243 * ð>ñò ù × Î´ØÙ Ð Ñ × Ú Ï ó ú;:=< ; 243 * 7 98 ó ð#ñò ù × ÎûØÙ Ð Ñ ×·Ú Ï ó úÖ < ; 243 *ÎûØÙ Ð Ñ ×·Ú Ï ; 2 ð#ñò ù × < Î´ØÙ Ð Ñ ×·Ú Ïûú .

Exactsquare-rootoperator symbols 159$# -Q? : ¾ c ¿ 243 * : ¬ -Q?® 243 * 2 í 2 .- : z : z%? : ¬ -]?® #& ++++º . *8¾ - c¿ 243 * V0WYi' ? :¸ -]?® 1 --]? V0WY .? : 9 m (44)

The integral in the fourth andfinal termin Eq.(37),noting that£ § ' &

, canbewritten in theform (cf. Eqs.(34)-(35)andtheimplicit useof theMehlerformula)F K£e£0; 243 *bV0WY ë® £>ì' £# ?M £ï æ 1 GF K£e£X; 243 *IVXW/Y ë® £> : 243 *,V0WY .? £> «¥ ã -¬E ' ? : ¬ -]?<® 1 ¾ i: ¿

«À « TS 6 LV0WY .? : ¬ £> (45)

and– bringingtheintegrationinsidethesummation– boundedby++++ F K££X; 243 *IV0WY ë® £#ì'> £> ?M £ï æ 1 ++++º . * ¾ : ¿ 243 * V0WY,' ? :dæ -]?<® 1 --]? V0WY .? : 9 (46)

applyingthepreviousresultsin Eqs.(40)and(41)-(42).Takingthe limit æ? in Eq.(46)thenestablishesthat thefourth termgivesazerocontribution.

Combiningthe previous results,andactuallytaking the limit æ@ in Eq.(37),the expressionforN 7 O TS 6 cannow be

writtenasanabsolutelyanduniformly convergentinfinite seriesandawell-definedintegral:N 7 O TS 6 9 :%¶ 243 * «¥ -¬E ¾ i:^¿«QÀ « TS 6 BA ' ? : ¬ -Þ?® 1 243 * : i ¾ c ¿ 243 * ' ? : ¬ -Þ?<® 1 2 í 2 .- C: z C: z? : ¬ -Þ?<® #C?

i¾ ¶c¿ 243 * F K£e£0; 243 *bV0WY ë® £#( £#@m (47)

BacksubstitutingEq.(36)for æ ¸ in Eq.(47),theresultin Eq.(47)canbeexpressedin thealternateformN 7 O TS 6 9 :d¶ 243 *"! ; 2 «¥ -¬E ' ? : ¬ -Q?à® 1 243 * ¾ i: ¿«ÞÀ « TS 6

i :%¶ 243 * «¥ ! -¬E : ¬ -]?® 243 * ¾ i:i¿

« À « TS 6 (48)$# -Q? : ¾ c ¿ 243 * : ¬ -Q?® 243 * 2 í 2 .- : z : z%? : ¬ -]?® #&?i¾ ¶c ¿ 243 * F K£e£0; 243 *IV0WY ë® £>ì' £# ?DM £ï ¸ 1

which, in conjunctionwith the (arbitrary)choiceof ç , canbe usedto balanceandcontrol the magnitudeof the termsin a

numericalcalculationof theoperatorsymbol.AsFE

butHG¨

, thefirst two termsin Eq.(48)representan increasinglybetterapproximatemodaldecompositionof theoperatorsymbolfor anappropriatelydefined,finite regionof thephasespace(4).

Applying theanalysisusedto transformEq.(33)into Eq.(48)to Eq.(28)convertstheconditionallyconvergentinfinite seriesrepresentationfor

N 7 OÞPRgivenin Eq.(21)into anabsolutelyanduniformly convergentinfinite seriesandawell-definedintegral.

C. The Weyl symbol

Exactexpressions,in thedesiredmodalform, for boththeinversesquare-rootandsquare-rootHelmholtzoperatorsymbolsin theWeyl calculuscanbededucedfrom theprecedingstandardcalculusresultsin adirectfashionrequiringrelatively little calculation.From the expressionfor

N 7 OQPR TS 6 given in Eq.(28),it is evident that the transformationmappingN 7 OQPR TS 6 [ O PR TS 6


expressedin Eq.(13)actsonly on thefunctions

À « TS 6 andí £>

. Further, by construction,í £>

in Eq.(29)is the æ9 limitofîk £ï æ in Eq.(30),making

í £>a generatorof the set

À « TS 6 . Consequently, the analogousfunction Ií £> in the Weylcalculuswill generatetheappropriatetransformationof theset

À « TS 6 . Thus,it is only necessaryto identify Ií £> to ultimatelywrite all of theexactexpressionsin theWeyl representation.

It is apparentby theconstructionin Eq.(28)that,for thecase®»º -

(wheretherearenopropagatingmodes),?i¾ -c,¶ ¿ 243 * F K£e£ ; 243 * VXW/Y ë® £>ií £#

isN 7 OPR TS 6 . Since Ií £# mustplay the analogousrole for

[ O PR TS 6 for® º -

, carryingout transformationEq.(13)withEq.(29),it follows thatIí £>9ÁV0WYL' ? £> 1 V £# (49)

(notethattheTS 6 dependenceis containedsolelyin thevariable ). Applicationof thegeneratingfunction(Rainville (33,p.213))V0WYKJ ? 5 ¤-]? ¤ML-]? ¤ «¥ ¸ T « 5 e¤ « ï ¤Iï º - (50)

immediatelyidentifiesthetransformedorthogonalpolynomialsetthroughtheexpansionV0WYL' ? N £> 1 V £>9 : V0WY .? £# «C PO « MV0WY .? : ¬ £> (51)

asO « gÁV0WY .? .? « ¸ ~ « : M (52)

orO « gÁV0WY .? -¬E : « * í .? ¬ ? ¬ z?LzX?.- C: #@ (53)

where ¸ T « . is the simpleLaguerrepolynomial (33), * í .? ¬ ? ¬ z?Lz?.- : # is a generalizedhypergeometricfunction asgiven in Rainville (33, chap.5),andtheequivalenceexpressedin Eq.(52)andEq.(53)follows directly from thedefinitionsof therespective functions(33).

It is now conjecturedwith Eq.(49)andEqs.(51)-(53)thattheeffectof thetransformationfromthestandardto theWeyl calculusfor theinversesquare-rootHelmholtzoperatorsymbolis summarizedby thecorrespondencesí £> ? QIí £# (54)

andÀ « TS 6 ? : 243 * ¬E .? : i « O « Mm (55)

Applicationof Eqs.(54)and(55) to theexpressions(21)and(28) in thestandardcalculusthenprovidestheresults[ OQPR TS 6 9 ? V0WY \ °± c i _ :¶ 243 * «¥ -'i : ¬ -]?<® 1 243 * O « (56)

and [ OQPR TS 6 g ? V0WY \ °± c i _ :¶ 243 * ã ; 2 «C -'i : ¬ -]?® 1 243 * O « ?

i¾ -c,¶I¿ 243 * F K£e£0; 243 *bV0WY ë® £>J Ií £# ? Iîì £ï æ L (57)

where,with Eqs.(30)and(55),Iî £ï æ g : VXW/Y .? £> ã ; 2«¥ RO « LV0WY .? : ¬ £>Mm (58)


For thecaseof thesquare-rootHelmholtzoperatorsymbol,Eq.(55)is supplementedwith thecorrespondence £> ? I £#& (59)

where £>

is givenby Eq.(34)andI £#&ÁV0WYL' ? £> 1 V £>S ®? V £>##* ? £> (60)

following from Eqs.(13)and(34). Applicationof Eqs.(55)and(59) to theexpression(33) in thestandardcalculusthengivestheresult[ O TS 6 9 :d¶ 243 * ã ; 2 «C ' ? : ¬ -]?<® 1"243 * O « ?

i¾ ¶c ¿ 243 * F K£e£0; 243 *IV0WY ë® £>J I £# ? I M £ï æ L (61)

whereI & £ï æ & : V0WY .? £> ã ; 2 «C ' ? : ¬ -Þ?<® 1 O « LVXW/Y .? : ¬ £> (62)

(cf. Eq.(35)).Thededucedconstructionsgivenin Eqs.(56)-(58)and(61)-(62)canbeconfirmedthroughthedirectverificationthat

À « TS 6 and : 243 * ¬E .? : i « O « form astandard-Weyl ‘transformpair’ with respectto Eqs.(12)-(13).The analysisleadingto the expressionsgiven in Eqs.(47)and(48) for

N 7 O TS 6 asthe sumof an absolutelyanduniformlyconvergentinfinite seriesanda well-definedintegral thenfollows in essentiallythesamefashionfor theWeyl symbol

[ O TS 6 ;thefinal results,following directly from theapplicationof thecorrespondencesgivenin Eqs.(55)and(59) to thestandardcalculusexpressions,aregivenby[ O TS 6 g :%¶ 243 * «C O « BA ' ? : ¬ -Þ?® 1"243 *g : i ¾ cL¿ 243 * ' ? : ¬ -Þ?<® 1 2 í 2 .- C: z C: z? : ¬ -Þ?<® #C?

i¾ ¶c¿ 243 * F K£e£0; 243 *bV0WY ë® £# I £# (63)

and [ O TS 6 9 :d¶ 243 *"! ; 2«¥ ' ? : ¬ -]?® 1"243 * O « i :%¶ 243 * «¥ ! : ¬ -Þ?® 243 * O

« (64)$# -Q? : ¾ cL¿ 243 * : ¬ -Q?® 243 * 2 í 2 .- : z : z%? : ¬ -]?® #&?i¾ ¶cQ¿ 243 * F K£e£0; 243 *IV0WY ë® £>J I £> ? I M £ï ¸ L m

Theprincipalresultsof Section2 aretheHelmholtzoperatorsymbolrepresentationsgiven in Eqs.(21),(47), (48), (56), (63)and(64).

3. Contour-integral operator symbol representations

It follows from theconstructionin Eq.(57)thattheinversesquare-rootHelmholtzoperatorsymbolcanbewritten in theform[STVUWOQPR TS 6 9 ? i¾ -ci¶,¿ 243 * F K££0; 243 *bV0WYL' ® £ ? £> 1 V £#M ®»º - (65)


wheretherearenopropagatingmodes.For thecorrespondingdefocusingquadraticprofile (compareEq.(14))definedby)+* "! 9)+* ¦ * ! *with) ¦é§·© ª , theoperatorsymbolintegral representation(23)[YXZ TOQPR TS 6 9 ? V0WY \C°± c i _Z¾ -c,¶I¿ 243 * F K£e£ ; 243 * V0WYL' i ë® £i\[ £## 1 V £>M (66)

where[ ¶ ; 2 ´¦ * 6 * ?·S * (67)

(compareEq.(31)),in conjunctionwith Eq.(65),leadto therelation[STVUWOQPR TS 6 9 |~~] i ] [ XZ TOPR TS 6 M ® ºG- (68)

adoptinganobvioussuperscriptnotationin thissectionfor clarity. Theanalyticcontinuationresultin Eq.(68)canbeextendedto thesquare-rootHelmholtzoperatorsymbolfollowing theconstructionin Eq.(61)andnotingthecorrespondingintegral representationfor thedefocusingquadraticprofilegivenin Fishman(23).

Theprincipalgoal in this sectionis to extendthe integral representationin Eq.(65)andthesubsequentanalyticcontinuationresultin Eq.(68)connectingthefocusinganddefocusingquadraticprofiles(andtheanalogousresultsat thelevel of thesquare-rootHelmholtzoperatorsymbol)from

®»º-to® § '

, therebyexplicitly accountingfor thepresenceof thepropagatingmodes.Thebasicideais to modify an

¦-rotation/contour-integrationconstruction,whichunderliestheEq.(68)analyticcontinuationresult

for® º -

, by explicitly incorporatingthe periodicity of the associated,parabolic(Schrodinger)equationfundamentalsolution(propagator)into theHelmholtzoperatorsymbolconstructionprocedureinherentin Eq.(66).

A. The complex¦

-rotation and contour integration: analytic continuation for® º -

First, the¦

-rotation/contour-integrationconstructionfor the®åº -

casewill be outlined,followedby the explicit incorporationof theunderlyingperiodicityto producethemodifiedconstructionleadingto thedesiredintegral representationsfor theHelmholtzoperatorsymbolsfor

® § ' . Theextendedanalyticcontinuationresults,following directly from theintegral representations,

will thenbeestablished.For the

¦-rotation/contour-integrationconstructionprocedure,the startingpoint is theexact,closed-formexpressionfor the

Weyl symbolfor theinversesquare-rootHelmholtzoperatorin Eq.(66),with theaid of Eqs.(18)-(19),written as(notethechangeof integrationvariable,

£ £ ¦ )[ XZ TOQPR TS 6 9 ? V0WY \C°± c i __^ ca` 243 * F K£e£0; 243 *IV0WY i )+* £i\[ ´¦ £## V ´¦ £>Xm (69)

In Eq.(69), let¦ ¦ ¦ V0WY

i b , - b º ch: , with the correspondingtransformation[ [ ¶ ; 2 V0WY .? i b I' VXW/Y : i b ¦ * 6 * ?·S * 1 , sothatin thelimit bcLc^C: , [ i . Considerthecontourintegral[ed ? V0WY \ °± c i _e^ ca` 243 *gfh KiYii; 243 *bV0WY i )+* i¼D[ ´¦ il# V ´¦ ibM (70)

where j 6k 2 'k * 'k ° 'k ± asillustratedin Fig. 1. For the integrandin Eq.(70),in the complexi-plane,the branchpoint

(associatedwithi ; 243 *

) is at theorigin with thebranchline chosento lie alongthenegative real-axis,andtheisolatedsingularities(associatedwith thezerosof

l ) arelocatedat thepointsi -¦ V0WYKJ i ¾ c : ? b ¿ L : ¬ - c : ¬ ? : ?L- - : 0 m (71)

The¦

-rotationandsubsequentcontourintegrationhave beenspecificallyconstructedso that the contributions fromk 2 and

k °will ultimately result in the operatorsymbol for the focusingcaseandthe analyticcontinuationof the operatorsymbol for thedefocusingcase,respectively. Applicationof the Cauchyintegral theorem(35) followedby standardarguments(35) to establishthatthecontributionsfrom

k * andk ±

vanish,respectively, in theª and m_ limits for

- b º ch: (Fig. 1) and® ºG-

resultin theequality? VXW/Y \ °± c i _F^ c ` 243 * F K£e£ ; 243 * V0WY i /) * £i[ ´¦ £## V ´¦ £>9 (72)


- planeτ

branch line1Γ

2Γ3Γ

4Γ Reτ

Imτ

isolated singularities

Rρ

−ϕ

branch pointπ2 −ϕ

Figure1. Thecontourof integration n Ö s 2 Ð s * Ð s ° Ð s ± andtheintegrandsingularitystructurein thecomplexo -planefor theHelmholtzoperatorsymbolconstructionin Eq.(70).? VXW/Y \ °± c i _<¾ -ci¶I¿ 243 * VXW/Y \ ? 2* i b _&F K£e£0; 243 *,VXW/YL' i VXW/Y .? i b ® £iD[ £## 1 V £#& - b º 2* c ®»º - mWhile thelhsof Eq.(72)is notwell definedin thelimit bpc8c^C: , therhsof Eq.(72)is continuousat b ch: , suggestingthat[STVUWOQPR TS 6 9 ? i

¾ -ci¶ ¿ 243 * F K££0; 243 *bV0WYL' ® £ ? £> 1 V £#M ®»º - whichcanbeverifiedby establishingtheequivalenceof thisequalityandthecorrespondingspectral(modal)summationrepresen-tation in Eq.(56)by proceedingfrom the identity in Eq.(51).The

¦-rotation/contour-integrationconstructionprocedureresultsin

thedesiredoperatorsymbolintegral representation,while explicitly illustratingtheanalyticcontinuationrelationshipbetweenthefocusinganddefocusingquadraticprofilecases,at leastfor

® ºG-, i.e., in theabsenceof propagatingmodes.

B. Periodicity

TheWeyl symbolconstructionprocedurefor theinversesquare-rootHelmholtzoperatorpresentedin (4, 23) is basedon Eq.(6)inconjunctionwith (DeHoopandGautesen(9, (7.1)))A ; 2 "! ! B 9 ? : i rq !/z ! B M (73)

which is justa restatementof Eq.(3),whereq

is theHelmholtzGreen’s function,satisfying' * j *$ * ) * "! 1 q 5 !/z 5 B ! B 9 ? = 5 ? 5 B = "!?<! B M (74)

supplementedwith anoutgoing-wave radiationcondition(1, 23). Theparabolicequationfundamentalsolution,s £0 !/z ! B say, isrelatedto

q 5 !/z 5 B ! B through(23)q 5 !/z ! B g : ;b° 3 *8¾ i c ¿ 243 * F K£e£X; 243 *IV0WYp 2* i £i 5 *£0; 2 s £0 !/z ! B M (75)

andsatisfies i utI .- C: * *$ 2* ) * "! ?¹- s £X !/z ! B 9 (76)

supplementedbys !/z ! B g = "!?<! B ]m (77)


For thefocusingquadraticprofile,theparabolicequationfundamentalsolutiontakestheform implied in Fishman(23,(A1)) andgivenin Schulman(36,p.38),s £0 !z ! B 9v^ ¦ :%c i ´¦ £> ` 243 * V0WY A 2* i # )+* ?¹- £i ¦ ´¦ £> ' ! *& "! B #* 1 l %´¦ £# ? : !¥! B &RC (78)

in accordancewith thecorrespondingquantummechanicalharmonicoscillatorformulation(27, 36). This functionis, to within anexponentialphasefactor, periodicin

£with period

:%ch ¦ . SubstitutingEq.(78)into Eq.(75),thesubsequentresultinto Eq.(73),andapplyingthetransform(6) leadto thelhsof Eq.(72)‘evaluated’at b c^C: :[STVUWOQPR TS 6 xw ? V0WY \C°± c i _e^ ca` 243 * F K£e£X; 243 *IVXW/Y .?ry £>uz ´¦ £#& (79)

wherey ?i )+* ? i

¦&®and

z ´¦ £>EÁV0WYL' ?i ´¦ £> 1 V ´¦ £#@m

Thekey to theextensionof Eqs.(65)and(68) is thereductionof thissemi-infiniteintegral to anintegraloverasingleperiodthroughtheapplicationof thefollowing theorem.

Theorem. Letz ´¦ £>

bea :%ch ¦ -periodicfunctionfor

£ §<.? and¦a§j

, and,further, let thefollowing integral,F K£e£0; 243 *bV0WY .?ry £#z ´¦ £#@ ª V y Lç (80)

exist. Thenthesemi-infiniteintegral in Eq.(80)canbereducedto anintegraloveroneperiodin theformF K£e£0; 243 *bV0WY .?ry £>|z ´¦ £>9¾ ¦:%c ¿ 243 * F *~ 3 ] K£ µ/.- C: ´¦ £ C:%c X V0WY .? :%c y ¦ #&VXW/Y .?ry £>|z ´¦ £>M (81)

whereµ is theLerchtranscendentalfunctiondefinedby (29, 37)µ 9 «¥ ¬ ; « G ?L- ? : ï ï º - m (82)

TheLerchtranscendentalfunctioncanbeanalyticallycontinuedinto thecutcomplex -planevia theintegralrepresentation(Erdelyietal. (37,p.27))µ g -k F K£e£~; 2 V0WY .? £#-Þ? V0WY .? £> ª V ¼ç k ª V ¼ç k not on therealaxisbetween

-and

, (83)

andwherek .

is thegammafunction(29,37). It is continuedin

via therelationshipgivenin Erdelyi et al. (37, p.27),µ 9 2 µ i ; 2 «C ¬ X; « - : m (84)

(A moredetailedtreatmentof theLerchtranscendentalfunctioncanbefoundin thereferences(29, 37).)

Proof. Startingwith theintegralon therhsin Eq.(81)andapplyingEq.(84)yield¾ ¦:dc¿ 243 * F *~ 3 ] K£ µ.- C: ´¦ £ C:dc X.VXW/Y .? :dc y ¦ #&V0WY .?ry £#z ´¦ £#g¾ ¦:%c¿ 243 * F 2 *~ 3 ] *~ 3 ] K£ µ/.- C: ´¦ £ C:%c X V0WY .? :%c y ¦ #&VXW/Y .?ry £>|z ´¦ £>jF *~ 3 ] K£e£0; 243 *bV0WY .?ry £>uz ´¦ £>& ê - (85)


- planeτ

Reτ

Im τ

3Γ’

ρ

branch lines

branch points

isolated singularities

2π/ω

π2

−ϕ

−ϕ2Γ

1Γ4Γ’

’

’

Figure2.Thecontourof integration n B Ö s B 2 Ð s B * Ð s B ° Ð s B ± andtheintegrandsingularitystructurein thecomplexo -planefor theHelmholtzoperatorsymbolconstructionin Eq.(88).

exploiting theperiodicityofz. Thesecondintegralon therhsof Eq.(85)canbewrittenasF *~ 3 ] K£e£0; 243 *bV0WY .?ry £>|z ´¦ £>9F K£e£X; 243 *IV0WY .?ry £#|z ´¦ £# ? F *~ 3 ] K£e£0; 243 *bV0WY .?ry £>uz ´¦ £>Mm (86)

Ontheotherhand,considerthefirst integralontherhsof Eq.(85).SubstitutingEq.(82)into this integral, interchangingtheorderofintegrationandsummation,andexploiting theperiodicityof

zresultin¾ ¦:dc¿ 243 * F 2

*~ 3 ] *~ 3 ] K£ µ.- C: ´¦ £ C:dc X.VXW/Y .? :dc y ¦ #&V0WY .?ry £#uz ´¦ £# «C F « 2 *~ 3 ] «¥ *~ 3 ] K£e£X; 243 *IVXW/Y .?ry £>uz ´¦ £>9GF *~ 3 ] K£e£X; 243 *IV0WY .?ry £#|z ´¦ £#Mm (87)

CombiningEqs.(87),(86)and(85) resultsin thelhsof Eq.(81).

C. Symbolcontour-integral representationsfor the focusingcase

While thesemi-infiniteintegral in theoperatorsymbolconstructionfor thefocusingquadraticprofile is notwell definedatpresent,the above theoremandthe previous

¦-rotation/contour-integrationconstructionfor the case

® º-motivatethe following con-

structionprocedurefor[ TVUWOPR TS 6 . Considerthecontourintegral (compareEq.(70))[ B d ? V0WY \C°± c i _^ ca` 243 * ¾ ¦:dc¿ 243 *¹ fh Ki µ/.- C: ´¦ i C:%c X V0WY :%c i ® #eV0WY i ) * i¼\[ ´¦ ib# V ´¦ il (88)

for® GG - : , wherethecontourj B k B 2 \k B * k B ° \k B ± is definedin Fig. 2. For theintegrandin Eq.(88),in thecomplexi

-plane,thebranchpointsassociatedwithµ/.- C: ´¦ i C:%c X V0WY :%c i ® # arelocatedat thepointsi ? :dc¦ ¬ ¬ - : (89)


- planeτ

Imτ

Reτ

’

2π

2π

Figure3. Thecontourof integration B in thecomplex o -planefor theHelmholtzoperatorsymbolrepresentationinEq.(91).

with the associatedbranchlines chosento lie alongthe negative reali-axis,andthe isolatedsingularitiesare locatedas in the

previousconstructionfor thecase® ºG-

. Theseriesrepresentationin Eq.(82)forµ.- C: ´¦ i :dc X.V0WY :dc i ® # is uniformly and

absolutelyconvergentforï ï º - , conditionallyconvergentfor

ï ïC - and G - , anddivergentfor - , where ÁV0WY :dc i ® .Thus,for thefocusingquadraticprofile,theLerchtranscendentalfunctionin Eq.(88)is definedby theseriesin Eq.(82),residingonthecircleof convergence,away from - , i.e.,

® G - : .Application of the Cauchyintegral theorem(35) to Eq.(88)followed by an evaluationof the contribution along

k B ±in them limit resultin? V0WY \ °± c i _e^ ¦ :` 243 * -c F *~ 3 ] K£ µ.- : ´¦ £ :dc X.V0WY :%c i ® #eV0WY i ) * £i[ ´¦ £># V ´¦ £# ? V0WY \C°± c i _ - :d¶ 243 * c V0WY .? i b F *~ K£ µ.- C: £ C:%c V0WY .? i b X#VXW/Y :dc i ® #VXW/YL' i VXW/Y .? i b ® £iD[ £## 1 V £>+V0WY \ 2± c i _ ¾ :¶ ¿ 243 * F KV0WY .? i

¥ µ/.- C: V0WY .? iCX#V0WY :%c i ® # V0WYL' i :dc ® V0WY .? i

¥,\[ :dc V0WY^' i b ? ¥ 1 # 1 V :dc V0WY,' i b ? ¥ 1 M (90) - b º ch: , ® G - : . While thelhs in Eq.(90)is not well defined,at present,in thelimit bBcch: , therhsin Eq.(90)iscontinuousat b c^C: , suggestingthat

[ TVU%WOPR TS 6 is givenby therhsof Eq.(90)evaluatedat b ch: , i.e.,[STVU%WOQPR TS 6 g ? V0WY \ 2± c i _ - :d¶ 243 * c (91) F Ki µ/.- C: .? i C:dc i/.V0WY :dc i ® #eV0WY¼' ® i ? ib 1 V il@ ® G - : wherethecontour B in thecomplex

i-planeconsistsof the

ª V i -axis fromi

toi :%c andthecirculararc,

ï ihïb :dc ,from

iD :%c toiß

i :%c , asillustratedin Fig. 3. Applicationof theCauchyintegral theorem(35) to Eq.(91)thenyieldsthefinalintegral representation[ TVU%WOQPR TS 6 g ? V0WY \ 2± c i _ - :d¶ 243 * c (92) F Kui µ.- : .? i :dc i/>V0WY :dc i ® #V0WYL' ® i ? il 1 V ib& ® G - :


- planeτ

Reτ

Im τ

singularitiesisolated branch point

branch line−π

π

2π

3π

Figure 4. The contourof integration and the integrand singularity structurein the complex o -plane for theHelmholtzoperatorsymbolrepresentationin Eq.(92).

wherethe contour in Eq.(92)startsatiZ

andendsatiZ

i :%c , keepingthe integrandsingularities,which includebranchpointsat (cf. Eq.(89))i ?

i :%c ¬ ¬ - : 0 (93)

with theassociatedbranchlineschosento lie on thenegative imaginaryi-axis,andisolatedsingularitiesat (cf. Eq.(71))i

i : ¬ - c : ¬ ? : ?L- > - : 0 (94)

‘outside’ thecontourwith respectto thehalf-planeª V i ìç , asillustratedin Fig. 4. In Eq.(92),theoccurrenceof Ií definedin

Eq.(49)is recognized.Now, thespecificchoice, , for thecontour , shown in Fig. 5, providestheappropriatedefinitionfor thelhs integral in Eq.(90)in thelimit bcLch: , viz.,? V0WY \ °± c i _e^ ¦ :` 243 * -c F *~ 3 ] K£ µ.- : ´¦ £ :dc X.V0WY :%c i ® #eV0WY i )+* £i[ ´¦ £># V ´¦ £# ? V0WY \°± c i _ - :d¶ 243 * c F *~ K£ µ.- C: £ C:%c X.V0WY :dc i ® #eV0WYL' i ë® £ ? £># 1 V £>Mwhere the path of integration along

' :dc 1 is understoodto pass‘below’ the integrand singularitiesat£a c^C: and

ch:(cf. Eq.(79)).

Verification of Eq.(92)– Theoperatorsymbolrepresentationgiven in Eq.(92)canbeverifiedin thefollowing manner. Choosingthecontour to lie entirelyin thehalf-plane

ª V i ¼ç andavoid theintegrandsingularities,andapplyingEq.(84)for -

toEq.(92)resultin? V0WY \ 2± c i _ - :%¶ 243 * c F Kui µ.- : .? i C:%c i#V0WY :%c i ® #eV0WYL' ® i ? il 1 V ib


- planeτ

Reτ

singularitiesisolated

Im τ

τ

τ’

branch point

π

π/2

3π/23π/2+δ

3π/2−δ

π/2−δ

π/2+δ

−π/2branch line

2π

Figure 5. Thecontoursof integration and B in thecomplex o -planeusedfor theproperdefinitionof the left-handsideintegral in Eq.(90)andthehigh-frequency asymptoticconstructionsin Eqs.(129)and(147);

Û=aK ÓØin generaland

Û= Ñfor thespecificasymptoticevaluations. ?

i¾ -c,¶I¿ 243 * F LKii ; 243 * V0WY¼' ® i ? ib 1 V il ? V0WY \ 2± c i _ V0WY :dc i ® :d¶ 243 * c F Ki µ.- C: -Q?¹ i C:%c i>V0WY :%c i ® #V0WYL' ® i ? il 1 V ibM ® G - : m (95)

Examiningthefirst termon therhs in Eq.(95)andapplyingthegeneratingfunctionresultin Eqs.(51)and(52), interchangingtheorderof integrationandsummation,whichis justifiedby theuniformconvergenceof powerserieswithin theirradiusof convergenceandtheapplicationof theRiesz/Youngtheorem(31, 32), andcarryingout theremaining

i-integrationresultin?

i¾ -c,¶ ¿ 243 * F KiYi,; 243 *IVXW/YL' ® i ? ib 1 V ib ? VXW/Y \°± c i _ : 3 *¶ 243 * «¥ O « 2 í 2 .- C: z C: zd? i :dc : ¬ -Q?à® #M (96)

utilizing thecomplex form of theintegralrepresentationof theincompletegammafunctionexpressedasaconfluenthypergeometricfunction(29) givenin Eq.(38).Applying standardconfluenthypergeometricfunctionidentities(29,37) thenyieldstheexpression? V0WY \ °± c i _ :¶ 243 * «¥ -'

i : ¬ -]?® 1 243 * O « (97)ßV0WY \°± c i _ : VXW/Y :dc i ® ci¶ 243 * «C -'

i : ¬ -]?® 1 243 * O « .- C: z- : z i :%c : ¬ -]?<® #

for thefirst termontherhsin Eq.(95),where .~z~z

is thesecondsolutionof theconfluenthypergeometricdifferentialequationasgivenin Magnuset al. (29,chap.vi)andErdelyi et al. (37, chap.vi).Thelarge ¬ asymptoticexpansions(29,37) for both ¸ ~ « : and

.- : z%- : z i :%c : ¬ -?Z® # establishtheconvergenceof thetwo seriesin Eq.(97).Following from thespectral(modal)summationrepresentationof the inversesquare-rootHelmholtzoperatorsymbolgiven in Eq.(56),thefirst termin Eq.(97)is seento beexactly

[ TVUWOPR TS 6 .Examiningthe secondtermon the rhs in Eq.(95)andapplyingthegeneratingfunction resultin Eqs.(51)and(52), the inte-

gral representationin Eq.(83),interchangingtheorderof integrationandsummation,andcarryingout anelementaryexponential


integrationover thecontour in thecomplexi-planeresultin theexpression? V0WY \ °± c i _ : ° 3 * V0WY :dc i ® c,¶ 243 * «¥ O « ÞF K£e£0; 243 *E' £I i :%c : ¬ -]?® 1 ; 2 V0WY .? £>&m

(98)

Utilizing theintegral representation(Magnuset al. (29, p.277))F K£e£ ; 243 * ' £I i :%c : ¬ -]?® 1 ; 2 V0WY .? £> -' : i : ¬ -]?® 1 243 * .- C: zd- C: z i :dc : ¬ -]?<® # (99)

in Eq.(98)thenresultsin theexpression? V0WY \°± c i _ : V0WY :dc i ® ci¶ 243 * «¥ -'i : ¬ -Þ?® 1 243 * O « .- C: z- : z i :%c : ¬ -Þ?à® # (100)

for the secondterm on the rhs in Eq.(95),which exactly cancelsthe secondterm of expression(97). Hence,addingthe termsin expressions(97) and(100), in view of Eq.(56),thenestablishesthe contour-integral representationfor

[ TVUWOQPR TS 6 given byEq.(92).

The expressionfor[ TVUWO TS 6 follows from the equationcomposing

[ TVUWO PRwith[ TVUWO ) * ?j¦ * 6 * ?àS * (cf. Eq.(11))in

accordancewith theWeyl calculus(23) (compareEq.(24)),i.e.,[STVU%WO TS 6 g (101) )+* ?à¦ * 6 * ?S *0i .- :l#* ,* ´¦ C:/#*X ,*á i S / ?é i 6 ¦ * / ¥á h[STVUWOQPR TS 6 @mSubstitutionof Eq.(92)into Eq.(101)allows for the subsequentinterchangeof the differentiationandintegrationoperationsfol-lowing from theuniformconvergencepropertiesof theappropriateintegralsinvolved(31). Theresultingcalculationtakesthefinalform [STVU%WO TS 6 g ? V0WY \ 2± c i _Z¾ ¶:i¿ 243 * -c F Ki µ.- : .? i C:%c i#V0WY :%c i ® # V0WYL' ® i ? ib 1 V il ® ? V ib##* ? ib ® G - : (102)

in which theoccurrenceof I £> definedin Eq.(60)is recognized.

Standard symbols– Thecorrespondingresultsfor thestandardoperatorsymbolsfollow directly from therelationshipin Eq.(12)andtake theformN 7% T U%WOQPR TS 6 9 ? V0WY \ 2± c i _ - :d¶ 243 * c F Ki µ.- C: .? i C:dc i/>VXW/Y :dc i ® # V0WY ® i ? 2* : ibI i V : ib ?¹- V : il# 243 *h ® G - : (103)

(equivalentto replacingin Eq.(92) Ií byí

, cf. Eq.(29)),andN 7% T U%WO TS 6 9 ? V0WY \ 2± c i _j¾ ¶:,¿ 243 * -c FLKi µ/.- C: .? i C:%c i>V0WY :%c i ® # V0WY ® i ? 2* : ibI i V : ib ?¹- V : il# 243 * ® ? V : il##* ? : i V : ib : il ? : ib ® G - : (104)

(equivalentto replacingin Eq.(102)I by, cf. Eq.(34)).In Eqs.(103)and(104),thecontour is now confinedto thehalf-planeª V i ê , avoidingtheintegrandsingularities,andthefunction

V : ib# 243 * is analyticin theneighborhoodof thecontourandis givenby thepower serieswhichresultsfrom thereductionof thegenerator

íwhen

S 6 G .


- planeτ

Imτ

1Γ

2Γ

3Γ

Reτ

’’

’’

’’

2π

R

Figure 6. Thecontourof integration Ö n B B Ö s B B2 Ð s B B* Ð s B B° in thecomplex o -planechosenfor theevaluationof theHelmholtzoperatorsymbolrepresentationin Eq.(105).

D. Symbolcontour-integral representationsfor the defocusingcase

ThecorrespondingHelmholtzoperatorsymbolsfor thedefocusingquadraticprofilecanbeexpressedby contour-integral represen-tationsanalogousto thosegivenin Eq.(92)andEqs.(102)-(104)for thefocusingcase.TheWeyl symbolfor theinversesquare-rootHelmholtzoperatoris givenby[ XZ TO PR TS 6 g ? i

- :%¶ 243 * c (105) F Ki µ.- C: .? i :dc i/#VXW/Y .? :dc ® #eV0WYL' i ë® i[ ib# 1 V ilM ® GkmVerification of Eq.(105)– Following from theCauchyintegral theorem(35),thecontour in thecomplex

i-planecanbespecifi-

cally chosenas j B B ,k B B2 \k B B* Fk B B° , asillustratedin Fig. 6, to beevaluatedin thelimitª . Applying Eq.(84)for

-to

thecontribution to Eq.(105)alongk B B2 gives?

i- :d¶ 243 * c F K£ µ/.- C: .? i C:%c 4£X>V0WY .? :%c ® #eV0WY' i ë® £i\[ £## 1 V £# ? V0WY \C°± c i _¾ -c,¶ ¿ 243 * F K£e£0; 243 *,VXW/YL' i ë® £i[ £># 1 V £> ? i

V0WY .? :dc ® :%¶ 243 * c F K£ µ.- C: -Þ?¹ i C:%c 4£X#V0WY .? :%c ® #eV0WYL' i ë® £,\[ £## 1 V £>M ® Gk (106)

whenthelimitª is taken.Following from theexactoperatorsymbolrepresentationsfor thedefocusingcaseestablishedby

Fishman(23) andgiven in Eq.(66),thefirst termon therhsin Eq.(106)is seento beexactly[ XZ TO PR TS 6 . It is readilyestablished

thatthecontribution to Eq.(105)alongk B B* vanishesin thelimit

ª . Utilizing theperiodicityof thehyperbolicfunctions, "! i :dc 9 "! and V "! i :dc 9 V "! M (107)

in theevaluationof thecontribution to Eq.(105)alongk B B° thenresultsin theterm

iV0WY .? :%c ® :d¶ 243 * c F K£ µ/.- C: -]?é i :dc 4£0>V0WY .? :dc ® #V0WYL' i ë® £i[ £># 1 V £>M ® Gk (108)

in the limitª . Adding thetermsin Eqs.(106)and(108),in view of the integral representationin Eq.(66),thenestablishes

theinversesquare-rootHelmholtzoperatorsymbolrepresentationgivenin Eq.(105).(Notethatspecifyingthecontour j B B inEq.(92)for thefocusingcaserecoversthecontour-integral representationin Eq.(65)for

®»º -.)

Theconstructionof[ XZ TO TS 6 via theWeyl compositionequation(23), in a manneranalogousto Eqs.(101)-(102),thenpro-

videstherepresentation[ XZ TO TS 6 9 ? ¾ ¶:^¿ 243 * -c F Ki µ/.- C: .? i C:dc i/>VXW/Y .? :dc ® #

Exactsquare-rootoperator symbols 171 V0WYL' i ë® i¼[ il# 1 V ilY i ® i[ V ib##* ? ib9 ® Gkm (109)

Standard symbols– Thecorrespondingintegral representationsfor the standardoperatorsymbolsagainfollow from Eq.(12)intheformN 7% XZ TO PR TS 6 9 ? i

- :d¶ 243 * c F Ki µ.- C: .? i :dc i/#V0WY .? :dc ® # V0WY i ë® i¼ 2* [ : ibI V : il ?¹- # V : il# 243 *+ ® G (110)

and N 7% XZ TO TS 6 g ? ¾ ¶ :i¿ 243 * -c F; Ki µ.- C: .? i C:dc i/>VXW/Y .? :dc ® # V0WY i ë® i¼ 2* [ : ibI V : il ?¹- # V : il# 243 * i ® i[ V : ib# * ? : i V : il : ib ? : il ® Gkm (111)

E. Analytic continuation for®»§ '

Thecontour-integral representationsfor theHelmholtzoperatorsymbolsgivenin Eq.(105)andEqs.(109)-(111)for thedefocusingquadraticprofile andEq.(92)andEqs.(102)-(104)for the focusingcase,in conjunctionwith theappropriateanalyticstructureinthecomplex variable

¦associatedwith thecontourintegrals,immediatelyestablishtheextendedanalyticcontinuationresults[STVUWOQPR TS 6 ² |~] i ] [ XZ TOQPR TS 6 M ®Â§ ' M ® G - 4 (112)[ST U%WO TS 6 ² |~] i ] [ XZ TO TS 6 M ®Â§ ' M (113)

whileN 7% TVU%WOQPR TS 6 ² |~~] i ] N 7 XZ TOPR TS 6 M ®»§ ' M ® G - 4 (114)N 7% T U%WO TS 6 ² |~~] i ] N 7 XZ TO TS 6 M ® § ' Mm (115)

The equalitiesin Eqs.(112)-(115)extend the applicability of the previous analytic continuationresultssummarizedin Eq.(68)from

®åº -to® § '

, with theextensionto thepreviously omitted,integer®

valuesfollowing from the spectral(modal)summationrepresentationsandseveraladditionalcontour-integral representationsbriefly discussedin AppendixB.

Theprincipal resultsof Section3 aretheHelmholtzoperatorsymbolrepresentationsin Eq.(92)andEqs.(102)-(104)for thefocusingcase,Eq.(105)andEqs.(109)-(111)for the defocusingcase,andthe analyticcontinuationformulasin Eqs.(112)-(115)connectingthem.

4. Time-Fourier versustime-Laplacedomain

In theanalysisof two-way wave scattering,thegeneralizedBremmerseriesthatcouplestheone-way wavesplaysa fundamentalrole.Theconvergencepropertiesof thisseriesareunderstoodin thetime-Laplacedomain(DeHoop(11)) andrequireSobolev orderestimatesof thesquare-root‘Helmholtz’ operatoruniformin theLaplaceparameter. However, mostalgorithmsthatcomputetermsin thegeneralizedBremmerseriesarecarriedout in thetime-Fourierdomain.Thequadraticprofile providesa canonicalmediumin which the transformationof thesquare-root‘Helmholtz’ operatorfrom the time-Fourier to the time-Laplacedomain,andviceversa,canbecarriedout explicitly andunderstood.

In thespace-Laplacedomain,the formally exactwave equationfor (one-way) propagationin a transverselyinhomogeneoushalf-spacesupplementedwith appropriateright-traveling-wave radiationandinitial-valueconditionsis givenby (11).- le /8· 7 < (116)

(compareEq.(1)),where 7 7 "! # $ 9(' )+* "! ?a.- l#*X ,*$ 1"243 * (117)

is now thestrictly elliptic square-root‘Helmholtz’ operator(compareEq.(2)).Notethat,here,

172 L. Fishman,M.V. deHoop& M.J.N.vanStralen 5 !/z ¢%9GF K£@V0WY .? ¢0£> 5 !/z £#MmThephasespaceanalysisremainslargely thesame,andcanbesummarizedby thelatticemultivariate(path)integral,with Hamil-tonianequalto thesquare-root‘Helmholtz’ operatorsymbol(11),y 7 5 !/z 5 B ! B g 5 ? 5 B |~ FbH J . PR ´ 2 C:%c .K S^ ; 2 2 K !/ VXW/Y ? 2¡ ? i

S^ "!/ ?!/ ; 2 ,N 7OS¢ TS^ !/ 0 #(; 2 5¡£e (118)

(cf. Eq.(10)).Themappingsbetweenthesquare-root‘Helmholtz’ operatorsymbolsconstructedin thetime-Fourierdomainin Sections2 and

3 andthecorrespondingsymbolsassociatedwith thetime-Laplacedomainoperatorin Eq.(117)follow from thespecificform of thequadraticprofile representationandtheanalyticcontinuationresultsin Eqs.(113)and(115).Treatingonly theWeyl representationhere,first, for thedefocusingcase,it follows from Eqs.(2)and(113)that[ XZ TO ¢ TS 6 9 ? i

|~~] i ];¤ ¥¡¦ i ¥x¦ [ XZ TO TS 6 g ? i|~~¥¡¦ i ¥¡¦ [STVUWO TS 6 @m (119)

Substitutingrepresentation(109)into Eq.(119)yields[ XZ TO ¢ TS 6 gÁV0WY \ ? 2± c i _j¾ ¶:,¿ 243 * -c F Kui µ.- : .? i C:%c i#V0WY .? :%c i ® # VXW/YL' ?ë® i¼ ib# 1 V ibS ® V ib# * ib9 ® G - : 0 (120)

while thesubsequentchoiceof contour j B B in Fig. 6 yieldstheequivalentform (seeFishman(23, (32)-(34)))[ XZ TO ¢ TS 6 g¾ ¶c ¿ 243 * F K£e£ ; 243 * VXW/YL' ?ë® £i N £># 1 V £> ® V £>##*9 £><m (121)

(compareEq.(125)).For thefocusingquadraticprofile, it likewisefollows that[STVUWOS¢ TS 6 9 |~~] i ] [ XZ TOS¢ TS 6 g ? i|~~] i ];¤ ¥¡¦ i ¥¡¦ [STVUWO TS 6 & ? i

|~~] ; ];¤ ¥¡¦ i ¥¡¦ [ XZ TO TS 6 Xm (122)

Substitutingtherepresentations(120)and(121)into Eq.(122),thenresultsin theequivalentexpressions[STVU%WOS¢ TS 6 g¾ ¶:,¿ 243 * -c F Ki µ/.- C: .? i C:%c i>V0WY .? :%c ® # VXW/YL' i ë® i ? [ ib# 1 V ib ? i®

i[ V il##*E il ® Gk (123)

and [STVU%WOS¢ TS 6 gÁV0WY \ 2± c i _Z¾ ¶c¿ 243 * F K£e£0; 243 *bV0WYL' i ë® £ ? [ £># 1 V £> ? i®

i[ V £># * £#jm (124)

Theoperatorsymbolrepresentationsin Eqs.(120)and(123)will prove usefulin theconstructionandanalysisof thesquare-root‘Helholtz’ operatorin theright-halfof thecomplex Laplaceplane,whichwill bepresentedelsewhere.

5. Asymptotic operator symbolexpansions

Thecontour-integralrepresentationsfor theHelmholtzoperatorsymbolsdevelopedin Section3 enablethehigh-andlow-frequency,asymptoticoperatorsymbol expansionsto be derived in a straightforward fashion.Only the expressionsfor

[ TVU%WO TS 6 will bepresented;the other casescan be derived in a similar manner. (The superscriptnotation introducedin Section3 will now besuppressedsinceall subsequentresultswill applyto thefocusingcase.)


A. Low-fr equencyasymptoticoperator symbolexpansion

In the low-frequency limit, ¶ and® , enabling

®to berestrictedto

®º»-, correspondingto ¸ ¨ , wherethereare

no propagatingmodes.In theabsenceof propagatingmodes,theexpressionfor thesquare-rootHelmholtzoperatorsymbolin theWeyl calculusreducesto (cf. Eq.(61)with Eq.(60)),[ O TS 6 9 ? i

¾ ¶c ¿ 243 * F K££X; 243 *IV0WYL' ® £ ? £# 1 V £# ® ? V £###* ? £#j ®»º - m (125)

Expandingtheexponentialin theintegrandandorderingtheresultingtermsin powersof ¶ yield asumof integrals,eachof whichcan be expressedin termsof the generalizedzetafunction

µ . On the one hand,

µ is definedas a specialcaseof the Lerch

transcendentalfunctionintroducedin Eq.(82),viz.,µ 9 «¥ .? « ¬ X;· G ?L- ? : 0 (126)

andits appropriateanalyticcontinuations(29, 37). On theotherhand,µ

is representedthroughtheintegralF K£e£ 7 ; 2 V £# £>9 : *X; 7 k ¢% µ ¢ ?¹- - C: ] ª V ¢ ¼ç k (127)

following from theresultin Magnuset al. (29, p.34).Employing this integral representation,andintegratingby partsthevarioustermsin theabove mentionedexpansionthenyield §[ O TS 6 ¡¨ w i : 243 * A : µ .?L- C: - C: ¶ 243 *? : ; 2 µ .- C: - C: .)+* : * µ .?a C: - C: ´¦ * 6 *g S * ¶ ; 243 *? : ; ± µ © C: - C: .) ± ?<µ .?L- : - : .) * ´¦ * 6 * S * ? : ; 2 µ .?L- : - C: i : µ .?4 C: - : # ´¦ * 6 *& S *#* ¶ ;b° 3 *g Cém (128)

Thelow-frequency asymptoticexpansiongiven in Fishman(23, (29)) for thedefocusingquadraticprofile in conjunctionwith theanalyticcontinuationpropertygivenin Eq.(113)leadto thesameresult.As expected,

[ O TS 6 is purelyimaginaryfor® º-

, intheabsenceof propagatingmodes.

B. High-fr equencyasymptoticoperator symbolexpansion

The high-frequency, ¶ , asymptoticevaluationof[ O TS 6 startswith a delicatechoiceP of the contour in the integral

representation(102).Essentially, thecontour follows the imaginaryi-axis,circling aroundthe isolatedsingularities– leading

to anoscillatoryintegral representationfor theWeyl operatorsymbol.Changingthe variablesof integration,

i¹i£

alongthe ‘linear segments’of the contour, andi¹

i' c^C: = V0WY i ª 1 andi

i' c^C: = VXW/Y i ª 1 with ª § ' ? c > 1 alongthe‘semicircularsegments’,thenresultsin theexplicit representation[ O TS 6 g i # F 3 *X;« K£|¬ £>, F ° 3 *X;« 3 *#;« K£|¬ £>, F *~° 3 *#;« K£|¬ £>&àVXW/Y \ 2± c i _V0WY \ 2* c i ® _¼¾ ¶: ¿ 243 * -c = F K ª V0WY .? i ª ' V0WY c i ® µ.- : 3 = C:dc eV0WY .? i ª X#V0WY :dc i ® #?Qµ.- : - 3 = C:%c V0WY .? i ª X#V0WY :dc i ® # 1 V0WYL' i ë® = VXW/Y .? i ª i l = VXW/Y .? i ª ## 1 = VXW/Y .? i ª #§ Thevarioustermscontainintegralsof thetype, 7 8 óó 7% ð®~¯ Î ó Ï Ö Ø ; 7 Î° Ð Ñ Ï ; 2 s Î° Ð Ø Ï©± Î° Ð Ñ Ü ÑXÓØ Ï ,7 8 óó 7 2 ð®~¯ ° Î ó Ï Ö ØC; 7 ;b* s Î° Ð Ø Ï²± Î° Ð Ø Ü ÑXÓØ Ï × ØC; 7 s Î³° Ð Ø Ï²± Î³° Ü ÑXÓØ Ï , and7 8 óó 7 ð®~¯ ° Î ó Ï´µ¶ ¯ Î ó Ï Ö ØC; 7 ÎÒÑXÓ· Ï s Î° Ð Ñ Ï©± Î° Ü ÑXÓØ Ï × Ø%*X; 7 ÎÒÑXÓ· Ï s Î° Ð Ñ Ï©± Î° × Ø Ü ÑXÓØ Ï .

174 L. Fishman,M.V. deHoop& M.J.N.vanStralen ® ? = VXW/Y .? i ª ###*g il = V0WY .? i ª # ® G - : (129)

where¬ £> d ? V0WY \ 2± c i _Z¾ ¶:,¿ 243 * -c µ.- : £ :dc X#V0WY :dc i ® # V0WYL' i ë® £ ? £># 1 V £>Y ® ? V £>##* ? i £> (130)

and º = º c^C: . An asymptoticevaluationof

[ O TS 6 in the ¶ limit canthen,for themostpart,bereducedto astationaryphaseevaluation(38,39) of thefirst threeintegralsin Eqs.(129)-(130)in conjunctionwith aLaplacemethodevaluation(38, 39) ofthesemicircular( ª -)integralcontributionsin Eq.(129).

Theprincipalpartof[ O TS 6 is foundto be' ¶ ë®? 1 243 * 9) * ?à¦ * 6 * ?S * 243 * z

hence,theanalysisnaturallydividesinto two cases:(1)® ç ç (locally-propagatingregime)and(2) ç ® ç (locally-

evanescentregime).

Case1:® ç ç . Applying thestationaryphaseandtheLaplacemethodsto theoperatorsymbolintegral representationsgiven

in Eqs.(129)and(130),thedominantcontributionsarefoundto resultfrom exteriorandinteriorendpoints,interiorcritical points,andthesingularpointsof the integrand.SincetheCauchyintegral theorem(35) implies that therepresentationis independentoftheparticularchoiceof

=for º = º c^C: , this canbeexploited in thedetailedcalculation.Theprincipal ideais to choose

=so

that(i) thedominantcontributionscanbedividedinto threedisjointgroupswhichcanbecalculatedindependentlyandaddedin theendto producethefinal result,and(ii) thecontribution from thesingularpointsof theintegrandcanbeevaluatedin anexpeditiousmanner.

The first groupcomprisesthe contributionsfrom the exterior endpointsat£¼

and :dc . The secondgroupcomprisesthecontributionsfrom the interior critical points.Theseriesandintegral representationsof theLerch transcendentalfunction,given,respectively, in Eq.(82)andEq.(83),readilyestablishthat,in thecontext of thehigh-frequency asymptoticanalysis,in thefunction¬

, µ.- C: £ C:%c X#VXW/Y :dc i ® # is solelyanamplitudefunction,while¸B¹ £ § ' :%c 1 :µ.- : £ C:dc X#V0WY :dc i ® #GG (131)

andµ.- : £ :dc X#V0WY :%c i ® # is finite exceptat

£gG.

Thus,theoscillatorypartof¬

is theexponential,V0WYL'ië® £ ? £># 1

which implicitly containsthe‘large’ parameter¶ ; 2 . The(‘interior’) critical points,i i

£ say, associatedwith its phase,followfrom theequality

® ? V £## * , andaregivenby£ , c ? £ , c j£ , and :dc ? £ , where£ »º \ ' ë® ?- 1"243 *%_ (132) 2* i |l2¼ ^ i

Á' ë® ?¹- 1 243 *i? ' ë® ?¹- 1 243 * ` (133)

with theprincipalbranchesunderstood.(Observe, thatasë® , then

£ c ch: andthecritical pointstendto the isolatedsingularities,while as

ë® - , £ E andthecritical pointstendto theexteriorendpoints,

and :dc , andto c .)Thethird groupcomprisesthecontributionsfrom thefour interiorendpointsat

£g c^C:x½ = and£g c^C:x½ = in conjunction

with thetwo semicircular( ª -)integralsderiving from the=-neighborhoodsof theisolatedsingularpointsat

£g c^C: and£g c^C: .

With theexceptionof thetwo previouslymentionedlimits of coalescingcritical points,thecontributionsto thehigh-frequencyasymptoticoperatorsymbol evaluationfrom the threegroupsare disjoint if thereare no critical pointson the intervals on theimaginary

i-axis,

' c^C: ? = ch: =d1 and' c^C: ? = ch: =d1 . Thus,for fixed ¶ ® and ¶ , to ensurethat

£ º c^C: ? = , requiresthecondition,® º = ##*m (134)

On the otherhand,the radius=

of the semicirclesmust be sufficiently large to avoid a residualcontribution from the isolatedsingularitieswhen

£ cLc^C: .


The further refinementof thecontour in relationto the calculationof thecontributionsfrom the=-neighborhoodsof the

two isolatedsingularitiesrequiressomeattention.Theexponentialoccurringin the integrandof thesemicircularcontributionstoEq.(129)canbewritten in theformV0WYL'

iû® = V0WY .?

i ª , l = V0WY .? i ª ## 1 ÁV0WYi' N;¾ ª I iN¿ ª 1

whereN;¾ ª 9 ® = ª , 2* i ' l = V0WY .? i ª # ? l = V0WY i ª # 1 (135)

andN ¿ ª ì ® = l % ª I 2* ' l = V0WY .? i ª #I\l = V0WY i ª # 1 (136)

with ª § ' c 1 . Theestimationof thecontributionsfrom thetwo semicircular( ª -)integralsin Eq.(129)is governedbyN ¾ ª in

Eq.(135).ThefunctionN ¾

(cf. Eq.(135))canbewritten in theformN ¾ ª ì ® = ª I : = ª #l % : = l % ª # ? l : = ª # mIt follows that

N;¾ ª is a symmetricfunctionof ª with respectto ch: for ª § ' c 1 . Also, notethatN;¾ ¥ N;¾ c Q . For

sufficiently small=, it is straightforwardto establishthat

N;¾ ª is monotonicallydecreasingfor ª § ' c^C: andmonotonicallyincreasingfor ª §( c^C: c 1 with a global minimum at c^C: , transitioningfor increasing

=to a ‘double-well-like’ structurewith

a local or globalmaximumat c^C: andtwo, symmetricallylocatedglobalminima.Condition(134),while ensuringthatN B ¾ ¥Þ? N B ¾ c º , is notsufficient to imply that

N ¾ ª º ¹ ª §j c . RequiringthatN ¾ c^C: º usingEq.(135),resultingin the

additionalcondition,® º = ; 2 l = M (137)

ensuresthe latter condition.EnsuringthatN ¾ ª º ¹ ª § c implies that the exponentialfunction in the semicircular

contributions to Eq.(129)is dominatedby the exponentialdecayin the high-frequency limit. Condition (137), however, is notsufficient to ensurethat

N ¾ ª hasa single,global minimum at c^C: . RequiringthatN B B¾ c^C: ç ensuresthe latter condition,

resultingin theadditionalandfinal conditionon=,® º = ##* : = l = ?¹- @m (138)

EnsuringthatN;¾ ª hasa single,globalminimumat ch: not only ensuresthat theintegrandscorrespondingto thecontributions

from thetwo isolatedsingularpointsareexponentiallysmallfor ª §j c , but that,in addition,thedominantcontributionsin theLaplacemethod(38,39)calculationcomeentirelyfrom theneighborhoodsof themaximaat theendpoints ª G and c .

In summary, the contour is specificallyconstructedsuchthat for fixed ¶ ® and ¶ , a sufficiently small=

is chosensoasto satisfytheinfimum of inequalities(134),(137)and(138),whichsimply yields(thesubsequentinequalitieshold for sufficientlysmall

=)® º = ##* : = l = ?¹- º = ; 2 l = ºG = ##*m (139)

The choiceof=

in Eq.(139)ensuresthat (i) the dominantcontributions to the operatorsymbol integral representationgiven inEqs.(129)and(130),with thegeneralmethodof stationaryphase(38,39), canbedividedinto thefollowing threedisjointgroups:

(a) theexteriorendpointsat£g

and :%c ,(b) thefour interiorcritical pointsat

£ , c ? £ , c j£ , and :dc ? £ , and(c) thefour interiorendpointsat

£& c^C:½ = and ch:S½ = in conjunctionwith thetwo semicircular( ª -)integralsderiving from

the=-neighborhoodsof thesingularpointsat

£g c^C: and ch: ,

and(ii) thecontribution from thetwo isolatedsingularpointscanbereducedto aLaplacemethodendpointcalculation.

(a). Applicationof Eq.(84)andexploitationof theperiodicityof thetrigonometricfunctions, £, :%c 9 £> and V £i :%c 9 V £#@ (140)

reducetheexteriorendpointcontributionsfrom£g

and :dc in Eqs.(129)and(130)to asingleendpointcontribution from£g

(of thesemi-infiniteintegral representation,employing Eq.(81)),[YÀÁ ÂO TS 6 xw ? V0WY \ °± c i _Z¾ ¶c¿ 243 * F K£e£0; 243 *IV0WYL' i ë® £ ? £># 1 V £>

176 L. Fishman,M.V. deHoop& M.J.N.vanStralen ® ? V £>##* ? i £> (141)

(compareEq.(79)).Thestandardevaluationof theendpointcontribution in Erdelyi (38,pp.52-56)Ã appliedto Eq.(141)thenyieldsthealgebraicbranchof theasymptoticoperatorsymbolexpansion,[YÀÁ ÂO TS 6 ¨ w ' ¶ ë® ? 1"243 *g 2Ä ¶ * ¶ ® b' ¶ ë® ? 1 ; 3 *g|ëm l/m mßm (142)

The first two (non-vanishing)termsin the algebraicbranchof the asymptoticoperatorsymbolexpansionare Å .- and Å ¶ * ,respectively, in contrastto thestandardoperatorsymbolexpansionwhichcontainsan Å ¶ term(20). Thisdifferenceis areflectionof thesymmetryinherentin theWeyl construction(16, 17).

Equation(142) canalso directly be derived from the analyticcontinuationof the correspondingasymptoticresult for thedefocusingquadraticprofile presentedin Fishman(23, (27)) throughthe relationshipgiven in Eq.(113).The algebraicbranchcoincideswith theoutcomeof thepolyhomogeneouscalculusof operatorsymbols(1, 15, 16).

(b). Thestationaryphaseevaluationof thecontributionsfrom thefour interior critical points,accountingfor Eq.(131),is accom-plishedin thestandardmannerasin Erdelyi (38, pp.52-56)Ã . Thecontributionsfrom

£ and c ¹£ arecombinedaswell astheonesfrom c ? £ and :%c ? £ . While carryingout thesecombinations,thefollowing identity,? : ° 3 * µ© : : ? g µ© C: - : * ?<µ© C: * ] (143)

is invoked; it follows readily from Eq.(82)or Eq.(83).Theresultyieldstheoscillatorybranchof theasymptoticoperatorsymbolexpansion,[ UÆ³WO TS 6 ¡¨ w 2* ¶]c ;l° 3 * ¶ ; 243 * ' ë® ?¹- 1 ; 243 ±A i

VXW/Y c i ® µ© : -Þ?é £ dc X ? V0WY c i ® #@V0WYi' ? ië® £ ? ' ë® ?- 1"243 *% 1?Qµ/© C: £ %c X ? V0WY c i ® #eV0WY^' i ë® £ ? ' ë® ?¹- 1"243 *d 1 CàZ|ëm l/m mßm (144)

Theleadingtermin theoscillatorybranchof theasymptoticoperatorsymbolexpansionis Å ¶ , with thecontributionsof Å ¶ 243 * ,whichareproportionalto thesingularamplitudes

µ/.- C: £ :dc X>V0WY :%c i ® # , µ.- C: -Þ?a £ :dc X#V0WY :%c i ® # , andµ.- : .- C: ½ £ C:%c X.V0WY :dc i ® # , having vanished.

(c). The interior endpoint contributionsarecalculatedin thestandardmannerasin Erdelyi (38,pp.52-56)asbefore.Thespecificchoiceof

=in Eq.(139)reducesthe ª -integral contributionsto a standardLaplacemethodendpoint calculationasin Erdelyi (38,

pp.36-39);asexpected,thesecontributionsexactly cancelthecorrespondinginterior endpoint contributions.Theresultis a totalgroup(c) contributionof

to within exponentiallysmallterms.Thus,[YÆ³ÇÂO TS 6 ¨ w Å exponentiallysmallterms

Mm(145)

(a) + (b) + (c). Theasymptoticcontributionsfrom groups(a)–(c) representedin Eqs.(142),(144),and(145)canbeaddedtogetherto producethedesiredfinal result,[ O TS 6 ¨ w ' ¶ ë®G? 1"243 *g 2Ä ¶ * ¶ ® I' ¶ ë®? 1 ; 3 * 2* ¶@c ;b° 3 * ¶ ; 243 * ' ë® ?a- 1 ; 243 ±A i

VXW/Y c i ® µ© : -Þ?é £ dc X ? V0WY c i ® #@V0WYi' ? ië® £ ? ' ë® ?- 1"243 *% 1

Ã For anoscillatoryintegral, È Ö 7É 8 ó ð#ñò ù i Ê ; 2 v Î ó Ïëú%Ê Î ó Ï , where v B Î ó Ï Ö Î ó,× <IÏ©Ë ; 2 ÎÍÌ ×ó Ï /; 2 v 2 Î ó Ï with v Î ó Ï monotonicallyincreasing

on

óMÎ ù < Ü Ì ú , and Ê Î ó Ï Ö Î óI× <IÏ²Ï ; 2 ÎÐÌ × ó Ï©Ñ ; 2 Ê 2 Î ó Ï , with specialpointslocatedat

ó,Ö < andÌ

, thedesiredexpansionis È ¥eÒDÓR¥ Î Ê ; 2 Ï ×Ô ¥ Î Ê ; 2 Ï . Here,Ô ¥ Î Ê ; 2 Ï Ö × ô ¥ ; 2 ÎÍÕ Ã õ Ï ; 2 s Î4Î Ã Ð Õ Ï ÓÕ Ï ð#ñCò 2* i

Î ÃÞÐ Õ Ï ÓÕ Ê Ï 3 Ë ð#ñò ù i Ê ; 2 v Î <IÏëú;ÖV×VØÙÚ Û Ü ÎÍÝ Ï , whileÓ¥ Î Ê ; 2 Ï ÖÁ× ô ¥ ; 2 ÎÐÞ Ã õ Ï ; 2 s ÎÒÎ ÃÐßÏ ÓÞ Ï ð#ñò × 2* iÎ Ã@ÐpßÏ ÓÞ Ê Ñ 3 ð#ñCò ù i Ê ; 2 v ÎÍÌ Ïûú;ÖV×VØà Ú Û á ÎÐâ Ï , where ù ÝIÎ ó Ïûú>Ë Ö v Î ó Ï ×v Î <IÏ , Û Ü ÎÍÝ Ï Ö ù ÝbÎ ó Ïëú 2 ; Ï Ê Î ó Ïlù Ý B Î ó Ïûú ; 2 , while ù âeÎ ó Ïëú Ö v ÎÐÌ Ï × v Î ó Ï , Û á ÎÍâ Ï Ö ù âeÎ ó Ïûú 2 ; Ñ Ê Î ó Ïù â B Î ó Ïëú ; 2 . For the specific calcula-

tions, v Î ó Ï Ö Ê Î ÚQóE×Kã ´µ¶ Î ó ÏÒÏ , while Ê Î ó Ï is divided into two terms,oneproportionalto ð® Î ó Ï Ú×_ã Î ð® Î ó ÏÒÏ * andoneproportionalto

×i ð® Î ó Ï2´µ¶ Î ó Ï , to accountfor thedifferentvaluesof Õ and ß in eachcase.

Exactsquare-rootoperator symbols 177?Qµ/© C: £ %c X ? V0WY c i ® #eV0WY^' i ë® £ ? ' ë® ?¹- 1"243 *d 1 CàZ|ëm l/m mßm (146)

Theconvergenceof theLerch transcendentalfunctionsin Eq.(146),following from Eq.(82),removestherestrictionson®

, orig-inating from the contour-integral representationin Eq.(102).Most importantly, while the asymptoticexpressionin Eq.(146)wasultimatelyderivedfor a particularlyconvenientchoiceof

=, it follows from theCauchyintegral theorem(35), that this expression

holdsfor all, fixed,= §< c^C: , providing thedesirednonuniform asymptoticexpansion. è (normal propagation at 6 è ). While the limit

® (implying, in the construction,the limit= ) cannot

be taken in Eq.(146)due to the coalescingpairs of interior critical points (£ and c ? £ at c^C: and c G£ and :%c ? £ at c^C: ), necessitatinga uniform asymptoticexpansionanalysis(39), thehigh-frequency asymptoticexpansionof

[ O >¥canbe

determinedin thefollowing fashion.Choosing B , asillustratedin Fig.5, in Eq.(102)yieldstheoperatorsymbolrepresentation[ O TS 6 9GFä å KuiY¬ .? iib, :dc i ª V i 3 * ¬ .? i æ , ª V i ° 3 * ¬ .? i æ <® G - : (147)

following from theCauchyintegral theorem(35). In this case,for fixed ¶ ® and ¶ , a sufficiently small= ç canbechosensuch

that = ##* : = l = ?a- º = ; 2 l = º = ##* º¹® m (148)

With=

satisfyingthis inequality, it follows that (i) the interior critical pointsnow resideon the intervals c^C: ? = ch: = and© c^C: ? = ch: = and(ii) thecorrespondingª -integral integrandsfollowing from Eqs.(147)and(130)areexponentiallysmall

on ª § c , with thedominantcontributionsresultingfrom theneighborhoodsof themaximaat theendpoints ª » and c .Thefirst pointestablishestheabsenceof any contributionsfrom group(b), while thesecondpoint, in conjunctionwith theinteriorendpoint contributioncalculation,againestablishthatthegroup(c) contribution is exponentiallysmall.Thus,in thiscase,[ O TS 6 ¨ w [YÀÁ ÂO TS 6 iG|~¨ :%c i ª V i 3 * ¬ .? i æ , ª V i ° 3 * ¬ .? i æ ® G - : m (149)

For thecaseS 6 ( Â ), theresiduesin Eq.(149)canbeevaluatedin a straightforward fashion(35). Both, polesof first

order, associatedwith thefactor®D V £>

, andpolesof secondorder, associatedwith thefactor?

i V £# £> , occur. Addingthe

contributionsyields,[ O C¡¨ w ¶ ® 243 * ZV0WY \C°± c i _ ¶ 243 *c µ© : - : ? V0WY c i ® #VXW/Y c i ® C: 2Ä ¶ * ¶ ® X;b° 3 *&Z|ëm lm ml (150)

using,again,Eq.(143);here,theresiduesof thepolesof first orderhavecancelled.Again, therestrictionson®

areremovedby theconvergenceof theLerchtranscendentalfunctionin Eq.(150).

The connectionbetweenthe contour-integral representationin Eq.(102)andthe subsequentasymptoticexpansions,andthespectral(modal)summationbecomesmoreapparentuponexpressingtheoperatorsymbolasymptoticexpansionsin Eqs.(146)and(150)in alternative forms,viz., throughtheapplicationof theLerchfunctionalequationasgivenin Magnuset al. (29,p.34),µ g i ;ç :dc /; 2 k .-]?B A V0WYi' ? c i : 1 µ/.-]?è :dc i X; 2 |l2¼ X#V0WY .? :%c i #? V0WYi' c i # C: i : 1 µ/.-]?è -Þ?¹ :dc i ; 2 |l¼ X.VXW/Y :dc i #Ca (151)

wheretheprincipalbranchof the|l2¼ .

function is understood.Introducing é accordingto® :d¸ ?- é , é § ' : 1 (cf. the

estimatebelow Eq.(19)),andapplyingEq.(151)to Eq.(150)yield[ O C¡¨ w ¶ ® 243 * ? : ° 3 * ¶ 243 * .? !ê ' µ .?L- : ë C: ? iµ .?L- C: -Q?a ë C: # 1 2Ä ¶ * ¶ ® ;b° 3 * Z|ëm lm ml (152)


wherethegeneralizedzetafunctionis definedby theappropriateanalyticcontinuationsof Eq.(126)(29, 37), withë¼'ë é ÿ A é if é § ' - 1 é ? : if é §.- : 1 (153)ê ê é ² A -if é § ' - 1 ?L-if é §j.- : 1 m (154)

Ratherthanbeingasymptotic,bothof theexpressionsfor© [ O C in Eqs.(150)and(152),resultingfrom theevaluationof

theresiduesin Eq.(149),areexact.This followsonstartingwith© [ O >¥ in Eq.(152)andusingEq.(84)with

-to show

that ? ' µ .?L- : é/C: ?¹- ? iµ .?L- C: : ?é é/: # 1 µ .?L- C: é/: ? iµ .?L- C: -]?é é/C: #] for é §j.- : 1 (155)

establishingthatEq.(152)is valid for é § ' : 1 withë¼ é and ê - . Then,useof theidentity |ì& µ 9 µ/?- M (156)

which followsdirectly from Eq.(82),applicationof Eq.(83),andcomparisonwith© [ TVUWO >¥ in Eq.(61)for æ ¸ completethe proof. The Lerch functionalequation(151) makesexplicit the numberofpropagatingmodes in theoperatorsymbol,contour-integral representationin Eq.(102).

Case2: ç ® ç . For thehigh-frequency asymptoticevaluationof theoperatorsymbolintegral representationin Eqs.(129)and(130)in this case,thereare(i) exterior endpoint contributionsfrom

£@(and :%c , (ii) no interior critical pointson theimaginaryi

-axis,following from Eq.(132),and(iii) contributionsof exponentiallysmallorderfrom group(c), following from Eq.(139)andthe discussionprecedingEq.(145).Returningto Eq.(102)andthe original contour , a critical point is encounteredon the reali

-axis:it follows from theequality® ? V il# * , andis givenbyi º \ ' -Þ?aë® 1"243 *_ (157) 2* |l2¼í^ - ' -Þ?éë® 1 243 *-]? ' -Þ?éë® 1 243 * ` (158)

with theprincipalbranchestaken.Thecontribution from this critical point is a termof exponentiallysmall order, supplementingthe algebraicbranchderiving from group (a). The most direct way to derive the resultingexpansionis to exploit the analyticcontinuationresultin Eq.(113)in conjunctionwith thecorrespondingasymptoticresultfor thedefocusingquadraticprofile,derivedby astationaryphaseevaluation,in Fishman(23, (27),(28)).Thefinal expressionis[ O TS 6 ¨ w ' ¶ ë®G? 1 243 * 2Ä ¶ * ¶ ® I' ¶ ë®? 1 ; 3 * 2* ¶ i ;b° 3 * ¶ X; 243 *' -]?éë® 1 ; 243 ± V0WY^' ® i ? ' -]?éë® 1"243 * 1 Z|ëm lm mDm (159)

Theprincipalresultsof Section5 arethelow-frequency asymptoticexpansionin Eq.(128)andthehigh-frequency asymptoticexpansionsin Eqs.(146),(150), (152), and (159). Taken together, Eqs.(146)and(159) provide the nonuniform,high-frequencyasymptotic,operatorsymbolexpansionfor thefull rangeof thephasespacevariables

Sand 6 . As previously discussed,thehigh-

frequency resultsarenot valid in the® limit dueto the coalescingpairsof interior critical points.Thehigh-frequency

resultswill alsofail in thelimit of® - dueto thecoalescingof (i) theinteriorcritical pointat

£ andtheexteriorendpointat£g, (ii) theinteriorcritical pointat :%c ? £ andtheexteriorendpointat

£g :%c , and(iii) thetwo interiorcritical pointsat c ? £ and c é£ at

£ c . To addressthesetwo limiting cases,uniform methodsmustbe applied(39). The nonuniformexpansions,however, doestablishandillustratethefundamentaloscillatorycharacterof thesquare-rootHelmholtzoperatorsymbolin thehigh-frequency limit (1, 23). In theelliptic pseudodifferentialoperatorcalculus,only thenonuniformalgebraicbranchgivenin Eq.(142)is obtainedin theasymptoticanalysis(1, 14,15, 16, 17).


6. One-waypropagation

A. Propagatingand nonpropagatingoperator symbolconstituents

It is clearfrom thederivationof thespectral(modal)summationrepresentationspresentedin Section2 thattheHelmholtzoperatorsymbolsnaturallydivide into theirpropagatingandnonpropagatingmodalcontributions.In theWeyl calculus,in bothEq.(61)withæ ¸ andEq.(64),the first, finite sumof ¸ termsrepresentsthepropagatingmodalcontribution to thesquare-rootHelmholtzoperatorsymbol,while theremainingtermscomprisethenonpropagatingmodalcontribution,with acorrespondingdecompositionin bothEq.(33)with æ ¸ andEq.(48)in thestandardcalculus.Fromthepropagatingandnonpropagatingmodaldecomposition,adoptinganobvioussuperscriptnotation,it thenfollows that

[ O [ïîO [rð;îOwith[ îO TS 6 ÿ ª V [ O TS 6 (160)[ ð;îO TS 6 ÿ i

© [ O TS 6 m (161)

Correspondingly, decomposingthepreviously mentionedpropagatingandnonpropagatingmodalcontributionsinto their realandimaginaryparts,andfurther noting the even/oddsymmetrywith respectto

S(or 6 ) establishthat

N 7 O N 7% îO N 7 ð;îOfor the

standardsymbol,withN 7% îO V0WY .?i S 6 ñRò ª V V0WY i S 6 N 7 O TS 6 @ i ó ò © V0WY i S 6 N 7 O TS 6 £ (162)

and N 7% ð;îO ÁV0WY .?i S 6 ó ò ª V V0WY i S 6 N 7 O TS 6 ] i ñ(ò © V0WY i S 6 N 7 O TS 6 £ (163)

wheretheevenandoddpartsof a functionz 5 aredefinedbyñRò ' z 5 1 2* ' z 5 ,z .? 5 1 (164)

andó ò ' z 5 1 2* ' z 5 ? z .? 5 1 (165)

respectively, andaretakenwith respectto eitherS

or 6 in Eqs.(162)and(163).Equations(160)-(163)canbeappliedto any exactrepresentationof thesquare-root

Helmholtzoperatorsymbol.In particular, equatingthe Weyl operatorsymbolconstructions(61) and(102) in Sections2 and3,respectively, yields[ îO TS 6 9 :d¶ 243 * ! ; 2«¥ ' ? : ¬ -]?® 1"243 * O « (166)

or [ îO TS 6 9 ª V A ? VXW/Y \ 2± c i _Z¾ ¶:^¿ 243 * -c F8Kui µ.- : .? i C:%c i#V0WY :%c i ® # V0WY¼' ® i ? ib 1 V ibS ® ? V ib##* ? ibC ® G - : 0 (167)

and[ ð;îO TS 6 g ? i¾ ¶c ¿ 243 * F K£e£X; 243 *IV0WY ë® £#J I £> ? I @ £ï ¸ L (168)

or [ ð;îO TS 6 & i© A ? V0WY \ 2± c i _j¾ ¶:,¿ 243 * -c F Kui µ.- : .? i C:%c i#V0WY :%c i ® # V0WY¼' ® i ? ib 1 V ibS ® ? V ib##* ? ibC ® G - : 0 (169)

with analogousresultsfollowing from theapplicationof Eq.(64).Following from Eqs.(166)-(169)andtheconstructionspresentedin Section5, thehigh-andlow-frequency, asymptoticexpan-

sionsfor the propagatingandnonpropagatingconstituentsof the square-rootHelmholtzoperatorsymbolcanbe obtained.In the


high-frequency, ¶ , limit, for example,utilizing Eqs.(146)and(159)resultin[ îO TS 6 ¡¨ w ôõ ö ª V rhsof Eq.(146) ® ç ç kª V rhsof Eq.(159) ç ® ç k (170)

with a correspondingsetof expressionsfor the nonpropagatingoperatorsymbolconstituent.The decompositioninto the propa-gatingandnonpropagatingconstituentsin conjunctionwith thehigh-frequency asymptoticexpansionsrelatethenumberof propa-gatingmodes to specificstructuralfeaturesof thesquare-rootHelmholtzoperatorsymbol.TheoscillatoryasymptoticbranchinEqs.(144)and(146)is governedby theexponentialphasefunctions,½ \ ® £ ? ' ë® ?a- 1 243 * _ . Taking

® p÷ - andS * ÷ ¦ * 6 * in Eqs.(132)and(144)thenyield thedominantexponentialphase

function(to within asign),\ ® £ ? ' ë® ?¹- 1 243 * _ø 3ùúk2w \ ®Á c^C: ? : ® 243 * 243 * _ á ú ] w \ ®Á ch: ? : ® 243 * ¶ ; 243 * S _@mThus,in thelowestorder,® / ¶ c^»û á #* (171)

where û á is the periodof oscillationof[ îO

as a function ofS

. An estimateof the numberof propagatingmodes ¸ followsimmediatelyfrom theestimatefollowing Eq.(19),with moresophisticatedexpressionsfollowing in anaturalmanner.

Moreover, theadditionof onepropagatingmodethroughtheincreasein) * and/ordecreasein ¶ , mathematicallyrepresented

by taking ¸ ¸ - (® ® : ), correspondsto a changein thesignof thescaled,nonpropagating,square-rootHelmholtz

operatorsymbolconstituent(i© [ O >C Þ¶ ; 243 * ) in Eq.(152)in thehigh-frequency limit.

B. The one-waypropagator

Theinfinitesimal,one-way propagator, in conjunctionwith thespecialstructureof theWeyl andstandardsymbolsfor thesquare-rootHelmholtzoperator, revealboththemultiresolutionandgeneralizedscreennatureof thepropagationprocess.Thefundamentalsolutionof theone-waywaveequationcanbewrittenin theform y 5 !/z 5 B ! B 9 5 ? 5 B s 5 !/z 5 B ! B with s 5 !/z 5 B ! B denotingtheone-waypropagator. Following from Eq.(10)with

-, theinfinitesimal,one-way propagatorfollowsass 5 !z 5 B ! B 9 F H J :dc .K S VXW/Y i Sh"!?<! B i [ O \ S 2* "! ! B _ 5 ? 5 B £ m (172)

In thelimit 5 E 5 B , thepropagatorcanbeasymptoticallyexpandedto thelowestorderin theforms 5 !/z 5 B ! B (w = "!?<! B i 5 ? 5 B QF H J :dc .K S V0WY i Sh"!8?! B [ O \ S 2* "! ! B #_ (173)

wherethe integral termis recognizedastheSchwartzkernelA "! ! B

(cf. Eqs.(6)and(8)) andis understoodin thedistributionalsense(4, 30).

Recognizingfrom Eq.(63)that theWeyl operatorsymbol[ O TS 6 canbewritten asthesumof anabsolutelyanduniformly

convergent infinite seriesand a well-definedintegral, it follows that in the limitï S ï

, the seriescontribution tendsto

while the integral contribution reproducesthe pseudodifferentialoperatorlimit[ O TS 6 w i

ï S ï. In the limit

üED, but

ýG ,

discardingthe integral contribution and approximating[ O TS 6 by the infinite seriesaloneresult in a convergent integral in

Eq.(173),correspondingto an essentiallycorrecttreatmentof all of the modalcontributionsexcept the ‘large ¬ ’ valuesin the(generally)deepevanescentregime.Denotingthisapproximationby s 5 !z 5 B !CB xw s 5 !z 5 B !CB , theexpansionin Eq.(173)takestheforms 5 !/z 5 B ! B (w = "!?<! B , i 5 ? 5 B : «¥ A ' ? ¶ : ¬ -]?<® 1"243 *g : i ¾ c8¿ 243 * ¶ 243 *C' ? : ¬ -Þ?® 1 2 í 2 .- : z : z? : ¬ -Þ?® #C FbH J C:%c .K S¼.? « ¸ ~ « : MV0WY .? V0WY,' i S^"!L?<! B 1 (174)


whereèþ ¶ ; 2 \ 2± ¦ * "! ! B * S * _ . Thelastline of thisequationcanbewritten in theformFIH J C:%c .K S.? « ¸ ~ « : MV0WY .? VXW/Y,' i Sh"!8?<! B 1 F H J C:%c .K S V0WY .? ¶ ; 2 S * V0WY : i Sb! .? « ¸ ~ « : MV0WYp ? 2± ¶ ; 2 ¦ * "! ! B #*^V0WYi' i Sh"! ! B 1 m (175)

In Eqs.(173)-(175),a forwardconstituenttransformis identifiedasthewindowedFouriertransform-ÿ : FbH JLK ! B VXW/Y # ? [ : "! ! B *: &QV0WY # ? i S ÿ : "! ! B ÿ : & where

[ ¦ is a dilation parameter, with a correspondingconstituent‘inverse’ transform,ÿ : 7 H J C:%c .K S , revealing the

multiresolution(40) natureof thepropagationprocess.Thestandardcalculusanalogueof the(modal)expansionin Eq.(174)is givenbys 5 !/z 5 B ! B (w = "!?<! B , i 5 ? 5 B : 243 * «¥ -¬E ¾ i:^¿

« ³ « \ ´¦ / 243 * ! _A ' ? ¶ : ¬ -Þ?<® 1 243 * : i ¾ c ¿ 243 * ¶ 243 * ' ? : ¬ -Þ?à® 1 2 í 2 .- : z C: z%? : ¬ -Þ?à® #C F H J C:%c .K S ³ « \ ¦ 243 * S _ V0WY .? i Sl! B Mm (176)

Equation(176)revealsthestructureof a ‘generalizedscreen’expansionrepresentation(41):aFouriertransform(!B S ), followed

by amultiplication(Hermitepolynomials)in theS

-domain,followedby a(Gaussianweighted)inversetransform(S ! ), followed

by amultiplicationin the!-domain.

The sequenceof operationsin the infinitesimalpropagationprocess,outlinedin the multiresolutionandgeneralizedscreenanalysesabove, is inherentin the path-integral structureof the fundamentalsolution (1, 2, 11, 18, 19, 20, 21, 36) in Eq.(10).Moreover, thepath-integral representationreducesto thespectral(modal)representationfor thefundamentalsolution,which, forthe(range-independent)focusingquadraticprofile,takestheformy 5 !/z 5 B ! B g 5 ? 5 B «¥ VXW/Y i ' ? ¶ : ¬ -]?® 1Ò243 * 5 ? 5 B £ V0WY A ? : ¾ c¿ 243 * ¶ 243 *' ? : ¬ -]?® 1 2 í 2 .- : z C: z? : ¬ -Þ?® # 5 ? 5 B C ¶ 243 * c 243 * -¬E : « ¯ « "! ! B Mm (177)

(ForpE

, but'G

, Eq.(177)correspondsto theapproximationassociatedwith s outlinedabove,while forà

, it representstheexactfundamentalsolution.)In Eq.(177),a transformpair canbeidentifiedthroughtheidentity «C ¶ 243 * c 243 * -¬E : « ¯ « "! ! B g «¥ ´¦ / 243 *c 243 * -¬E : « ³ « \ ´¦ / 243 * ! _E³ « \ ´¦ 243 * ! B _8 = "!8?<! B MmThus,the

³ «, subjectedto the appropriatenormalization,form an orthonormalbasis.With Eq.(177),in the actionof the propa-

gator, theintegrationover! B

constitutesa ‘forward’ transform,whereasthesummationover ¬ constitutestheassociated‘inverse’transform;thepropagatoris ‘diagonal’ in thetransformdomain.Unlike theglobaloperatordiagonalizationin thespectral(modal)propagatorrepresentation,theinfinitesimalpropagatorrepresentationscanbeviewedasa ‘diagonalization’in thephasespacestripabouta localizedcoordinatepoint (24).

7. Exact symbolsfor operator rational approximationsto The square-rootHelmholtzoperatorcanbe approximatedby operatorrationalapproximationsin general,andadditive, operatorrationalapproximationsin particular(1,3,4,5, 6,10, 23). Theseoperatorapproximationsimplicitly correspondto uniformoperator


symbolconstructionsoverappropriateregionsof phasespace,andimmediatelyresultin approximate,partialdifferential,one-waywave equations(1, 3, 4, 5, 6, 10, 23). With a continued-fractionapproach,thesquare-rootHelmholtzoperator

is approximated

to æ th-orderby¨' 1"243 * / ã 2 ' ¤ ã 1 ; 2 ¤ ã (178)

where * ? (' ) * "! ?- k .- l * *$ 1 (179)

theoperatorsumin Eq.(178)is supplementedwith a right-traveling- (outgoing-)wave radiationcondition,andtheapproximationcoefficients ¤ ã ¤ ã aredeterminedin thehomogeneousmediumlimit by a varietyof approximation-theoreticcriteria(1, 3,4, 5, 6, 10, 23, 42). For everycasewhereanexactsquare-rootHelmholtzoperatorsymbolcanbeconstructed,theoperatorsymbolscorrespondingexactlyto theoperatorsumin Eq.(178)canbewrittenin closedform (1,23). Let theWeyl symbol

[ É PR TS 6 correspondwith theoperator' 1"; 2

, thenEq.(178)canbesimplifiedas / J + ã 2 ¤ ã ' ? ' ¤ ã 1 ; 2#1 sothattheexact,additive, rationalapproximationoperatorsymbolin theWeyl calculusis givenby (1, 23)[ J O TS 6 9 - ã 2 ¤ ã ' -Þ? [ n PR TS 6 1 (180)

where ¤ ã ¤ ã ¤ ã m (181)

Theoperatorsymbol[ É PR TS 6 is thefundamentalfunctionin theconstruction.It follows from (i) therelevantresults

in Fishman(23) for the defocusingquadraticprofile in conjunctionwith the analyticcontinuationresult in Eq.(112)and(ii) thepreviousconstructionof

[ OQPR TS 6 for thefocusingquadraticprofile in Eq.(56)that[ É PR TS 6 & :¶ «¥ ' ? : ¬ -]? ® 1 ; 2 O « (182)

for thespectral(modal)summationrepresentation,andfrom (iii) theobservationthattheappropriateLerchtranscendentalfunctionµ involvedin theconstructionof[ É PR TS 6 appearsin theanalogueof Eq.(92)with

G, that[ É ! PR TS 6 & ? -¶ --Q? V0WY :dc i ® (183) F KuiV0WYKJ ® i ? ib L V ibM ® G - :

for thecontour-integral representation.In Eqs.(182)and(183),® ® ¶ ; 2 ' , ?a- 1 m (184)

AppendixB containsanadditionalrepresentationfor[ É ! PR TS 6 . CombiningEqs.(182)and(183)with Eq.(180)resultsin

the exact, closed-formrepresentationsof the Weyl symbol for the additive, rationaloperatorapproximationsof the square-rootHelmholtzoperator.

Finally, expressionsanalogousto Eqs.(182),(183),and(B3), in conjunctionwithEqs.(180)-(181),canbewritten for theoperatorsymbolsin thestandardcalculusin astraightforwardmanner.

8. Numerical results

The square-rootHelmholtzoperatorsymbol for the focusingquadraticprofile canbe numericallycomputed,in both the Weylandstandardcalculi, from the spectral(modal) summationand contour-integral formulas.For the spectral(modal) summation


Figure7. v 7 O Î ÷ Ü ÛCÝ " Ï vs.÷

for thefocusingquadraticprofile.Theexactstandardoperatorsymbolis computedfromEq.(104).



representations,for thestandardoperatorsymbol,for example,Eq.(48)is computed.Thefinite andinfinite sumsarerewritten andevaluatedin themanneroutlinedin detailbyVanStralen(4), while theremainingfinite integralis computedbyanadaptiverecursiveNewton-Cotes8 panelrule(4,43), with

chosenrelativeto themagnitudeof theintegrandin amannerwhichbalancesthelocation

of thesecond,infinite serieswithin its circle of convergenceagainstthemagnitudeof theintegrandandtherangeof integrationinthatfinal term.In this scheme,it immediatelyfollows from theestimatesin Eq.(42)andEq.(44)thatthelimit

correspondsto theinfinite sumapproachingits radiusof convergence,requiringanever increasingnumberof termsfor anaccuratenumericalcomputation,anda relatively simplenumericalintegration,while thelimit

correspondsto thesumapproachingthecenterof its circle of convergence,requiringan ever decreasingnumberof termsfor an accuratenumericalcomputation,anda moreinvolvednumericalintegration(44).Theexpressionin Eq.(48)is independentof theparticularchoiceof

, which wasverifiedin

thenumericalcomputations.Thecorrespondingcomputationin theWeyl calculusof Eq.(64)is treatedin asimilar fashion.For the




contour-integral representations,theWeyl andstandardoperatorsymbolsarecomputedfrom Eq.(102)andEq.(104),respectively.In both cases,the contour in the complex

i-planeis chosenas j B B in Fig. 6 with

ªtaken to be finite. Applying Eq.(84)and

exploiting theperiodicityof thehyperbolicfunctionsin Eq.(107)in thesamemannerusedin proceedingfrom Eq.(106)to Eq.(108)reducethecontourintegral to asingleintegralalong

k B B2 andtwo integralsalongk B B* . TheLerchtranscendentalfunctionis computed

from Eq.(83)by theRomberg numericalintegrationmethod(45) appliedto thethreeresultingintegrals.ª

is chosento makeªD®

an Å .- quantity, therebybalancingthenumericaleffectsof thesingularitiesati

i ch: andi ch: for thelimit

ª with theattemptto constructan Å .- quantityfrom theintegrationof extremelylargemagnitudeintegrandsin thelimit

ª . In boththeWeyl andstandardcases,thecomputationsare,in principle,independentof thechoiceof

ª, which wasverifiednumericallyfor a

reasonablerangeof theparameter. In thesubsequentnumericalcomputations,thespectral(modal)summationandcontour-integralmethodsresultedin identicalcurves.Further, for thecase

® º¨-, thesetwo computationalmethodswerefoundto bein complete

agreementwith theresultsobtainedfrom numericallyintegratingEq.(125)andits standardcalculusanalog,Eq.(33),with æ .Theexactoperatorsymbolcurvespresentedin thissectionwerecomputedfrom thecontour-integral representations.

It follows from Eqs.(48)and(104) that thestandard,square-rootHelmholtzoperatorsymbolN 7 O TS 6 is (i) invariantunder

the interchangeS# ¦ 6 , (ii) a symmetricfunction of

S( 6 ) for 6 (

S)è

, (iii) an asymmetricfunction ofS

( 6 ) for 6 (S

)Gè

,and (iv) a (an) symmetric(antisymmetric)function of

Sand 6 for the imaginary(real) part of the symbol for

® º -in the

absenceof propagatingmodes.The third and fourth pointsare illustratedin Figs. 7-10 by plottingN 7 O TS >m 4

for the specificquadraticcase

) * 6 -¼? 6 * and 4 m 4,- m 4

, m 4

, andm 4

, respectively. Figure10 for m 4 illustratesthe fourthpoint. The sequenceof figuresalsoillustratesthe transitionfrom the high- to the low-frequency regime for a choiceof q withinthe well, with the square-rootfunction plus superimposedoscillatorybehavior, characteristicof the locally-homogeneous,high-frequency limit (1, 4, 23), graduallytransformingto theabsorption-dominatedcurves,correspondingto theabsenceof propagatingmodes,in the low-frequency limit. Figure11 displays

N 7 O TS - m 4 for the sameprofile and a - m 4 , illustrating the dominant

absorptivebehavior for achoiceof q outsidethewell. Figure12 illustratesN 7 O 6 for thesameprofileand ì - m 4 , andwill be

appliedin demonstratingthewaveguidingpropertiesof thefocusingquadraticprofile in thefinal Sec.??. All of theFigs.7-12areconsistentwith theappropriateanalyticcontinuationof thecorrespondingresultsfor thedefocusingquadraticprofilepresentedbyFishman(23).

It follows from Eqs.(64)and(102) that the Weyl, square-rootHelmholtzoperatorsymbol[ O TS 6 is (i) solely a function

of thevariable , following from thesymplecticstructurein theWeyl compositionequation(11) andthequadraticdependenceof) * 6 (16, 23), (ii) invariantunderthe interchangeS$# ¦ 6 , (iii) a symmetricfunctionof

Sand 6 , and(iv) purely imaginaryfor®èº -

in theabsenceof propagatingmodes.Figures13-17illustrate[ O TS >C

for thespecificquadraticcase) * 6 -8? 6 *

and Á 4 m 4,- m 4

, m 4

,m % 4

, andm -

, respectively. As in the casefor the standardoperatorsymbol,the sequenceof figuresagainillustratesthecharacteristicbehavior andtransitionfrom thehigh- to the low-frequency regimefor a choiceof q within the


Figure 10. v 7 O Î ÷ Ü4ÛCÝ " Ï vs.÷

for the focusingquadraticprofile. The exact standardoperatorsymbol is computedfrom Eq.(104).

Figure 11. v 7 O Î ÷ Ü Ñ Ý " Ï vs.÷

for the focusingquadraticprofile. The exact standardoperatorsymbol is computedfrom Eq.(104).

well. Analogousbehavior to thatdisplayedfor thestandardoperatorsymbolin Fig. 11, for a choiceof 6 outsidethewell, followsimmediatelyfrom Fig. 14 andthevariabledependenceon . Onceagain,all of theFigs.13-17areconsistentwith theappropriateanalyticcontinuationof thecorrespondingresultsfor thedefocusingquadraticprofilepresentedby Fishman(23).ThedifferencesinthesymmetrypropertiesbetweenthestandardandtheWeyl, Helmholtzoperatorsymbols,exhibitedin theformulasandillustratedin theprecedingfigures,area reflectionof thedifferentoperator-orderingschemesunderlyingthetwo pseudodifferentialoperatorcalculi (17).

Theeffectivenessof boththelow- andhigh-frequency asymptotic,Helmholtzoperatorsymbolexpansionsderivedin Sec.?? isreadilydemonstrated.Figures16 and17 comparetheexactresultin Eq.(102)andthelow-frequency asymptoticresultin Eq.(128)for ¶ - m 4 : 1 and

- , respectively, suggestingtheincreasingaccuracy as ¶ andthemannerof breakdown of Eq.(128).The

sameresultsareobtainedusingEq.(125)for theexactoperatorsymbolcalculationfor®èº»-

. In Figs.13-15,theexact resultin


Figure 12. v 7 O Î ÛCÜ.ø Ï vs.

øfor thefocusingquadraticprofile.Theexactstandardoperatorsymbolis computedfrom

Eq.(104).

0.0&

0.5 1.0 1.5 2.0-0.5

0.0&0.5&1.0

1.5

2.0

p'

Exact Re Exact Im HF Re HF Im

ΩB(

ΩB(ΩB

ΩB(K

)q q2 21(*

)+

= − k,

= 50.5-

(p ,

0)Ω

B

Figure13. .0/21 346587 vs. 3 for thefocusingquadraticprofile.TheexactWeyl operatorsymbol(Eq.(102))iscomparedwith theWeyl, high-frequency (HF) approximate,operatorsymbol(Eqs.(146)and(159)).

Eq.(102)andthehigh-frequency asymptoticresultsin Eq.(146)andEq.(159)arecomparedfor 9;:=<?> <@BADC , <> <EAEFEG , and <?> GCHFDI ,respectively. Thesequenceof figuresdemonstratestheaccuracy of thenonuniform,asymptoticexpansionsoutsidethe JKLMN@and JOK8LMQP regimesas 9;MQ< . In particular, thenonuniformityassociatedwith the JOKLMQP regimeis seento beconfinedto a very narrow region aboutR:S< . Thelowest-orderestimateof thenumberof propagatingmodesT , following from Eq.(171),yields GDF , U , and @ , respectively, for thepreviouslymentioned9 sequence.Comparisonwith theexactnumberof modes,GDF , F , andG , respectively, againillustratestheincreasingaccuracy of theasymptoticanalysisas 9MV< . In Fig. 18,theexactresultin Eq.(102)for W /YX <?Z[<H\ is comparedwith thecorresponding,high-frequency asymptoticexpressionin Eq.(150)for thespecificquadraticcase]_^ Xa` \;:b@2c ` ^ and degfih <?ZjG8k ( 9 f X P Z[<?> FBk ), illustrating the increasingaccuracy of Eq.(150)as 9lMm< . Figure18 furtherillustratestheexact,ratherthantheasymptotic,natureof theexpressionfor n6oqpW /rX <Zs<H\[t in Eq.(150).


0.0u

0.5u

1.0 1.5 2.0-0.5

0.0u0.5u1.0

1.5

2.0v

p'

Exact Re wExact Im wHF Re HF Im x ΩB

ΩB(

ΩB(

ΩB

(p ,

0)Ω

B

K q q2y

2y

1(*

)+

= − k =10.5

Figure14. . / 1 346587 vs. 3 for thefocusingquadraticprofile.TheexactWeyl operatorsymbol(Eq.(102))iscomparedwith theWeyl, high-frequency (HF) approximate,operatorsymbol(Eqs.(146)and(159)).

0.0z

0.5z

1.0 1.5 2.0-0.5

0.0

0.5

1.0

1.5

2.0

p

Exact Re Exact Im HF Re HF Im

ΩB|

ΩBΩBΩB|

K q q2

2

1(~

)

= − k = 3.5

(p ,

0)Ω

B

Figure15. . / 1 346587 vs. 3 for thefocusingquadraticprofile.TheexactWeyl operatorsymbol(Eq.(102))iscomparedwith theWeyl, high-frequency (HF) approximate,operatorsymbol(Eqs.(146)and(159)).

In thehigh-frequency, 9MV< , limit, theWeyl compositionequation(11)canbeapproximatedbyW / X RZ ` \: ] ^ Xa` \cR ^O h W /rX RZ ` \6k ^ Z (185)

whichcanserve asthebasisfor anapproximate,high-frequency reconstruction(23). This is illustratedin Fig. 19 where W ^/ X <Z ` \(computedfrom Eq.(102))is comparedwith

]_^ Xa` \0:@c ` ^ for de :F<> F , @<> F , and ?> F ( 9:g<> <?@BAEC , <> <EAEFEG , and <?> GCHFDI ). Theaccuracy of thereconstructionincreasesas 9Mb< , with thedeviationfrom zeroimaginarypartserving,in somesense,asameasureof theaccuracy of theprofilereconstructionfor realprofiles(23).


0.0z

0.5z

1.0 1.5 2.0-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

p

Exact Re Exact Im LF ReLF Im

ΩB|

ΩB|

ΩBΩB|

(p ,

0)Ω

B

K q q2

2

1(~

)

= − k = 0.95z

Figure16. . / 1 346587 vs. 3 for thefocusingquadraticprofile.TheexactWeyl operatorsymbol(Eq.(102))iscomparedwith theWeyl, low-frequency (LF) approximate,operatorsymbol(Eq.(128)).

0.0z

0.5 1.0 1.5 2.0-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

p

Exact ReExact Im LF Re LF Im

ΩBΩB|

ΩB|

ΩB

K q q2 21(~

)

= − k = 0.1z

(p ,

0)Ω

B

Figure17. . / 1 346587 vs. 3 for thefocusingquadraticprofile.TheexactWeyl operatorsymbol(Eq.(102))iscomparedwith theWeyl, low-frequency (LF) approximate,operatorsymbol(Eq.(128)).

9. Discussion

TheHelmholtzoperatorsymbolsfor thefocusingquadraticprofilehavebeenexactlyconstructedby two complementarymethods:(i) aspectral(modal)summationapproachin Section2 deriving from standardrepresentationsandconstructionsin spectralanalysisand(ii) a contour-integral approachin Section3 basedon extractingtheoperatorsymbolsfrom theappropriateGreen’s function,or correspondingparabolic(Schrodinger)propagator, data.As thenameimplies,theformermethodis naturalfor partitioningtheoperatorsymbol into its propagatingandnonpropagatingcomponents,aswell as for examining the individual modalcontribu-tions, while the latter methodis naturalfor examining the total operatorsymbol, in particular, for deriving both the high- and


0.00 0.25z

0.50 0.75 1.00 1.25 1.50 1.75 2.00-1.4

-1.0

-0.6

-0.2

0.2

0.6

1.0

1.4

Exact Re

Exact Im HF Re

HF Im

ΩBΩB|

ΩBΩB|Ω

B(0

, 0

)

k

K q q2

2

1(~

)

= −

Figure18. . / 15E4587 vs. for thefocusingquadraticprofile.TheexactWeyl operatorsymbol(Eq.(102))iscomparedwith theWeyl, high-frequency (HF) approximate,operatorsymbol(Eq.(150)).

0.0z

0.5 1.0 1.5-1.2

-0.6

0.0

0.6

1.2

1.8

q

Re

Im

1 - q

ΩB|2ΩB|2

2

(0 ,

q)2

ΩB

K

q q2 21(~

)

= −

k

= 50

5.

k

= 50

5. k

= 105.

k

= 105.

k

= 3

5.

k = 3

5.

Figure19. . ^/ 15E47 vs. for thefocusingquadraticprofile.Theexactfocusingquadraticprofile, ^ 1B7 ^ ,is comparedwith thesquareof theexactWeyl operatorsymbolcomputedfrom Eq.(102).

low-frequency asymptoticexpansions.Thetwo approachescanbecombined,asin Section6, to derivebothintegralrepresentationsandasymptoticexpansionsfor theindividualpropagatingandnonpropagatingmodalsums,asindicatedin Eqs.(166)-(170).In par-ticular, for thepropagatingcontributionto thesquare-rootHelmholtzoperatorsymbol,in thehigh-frequency limit, asumof afinite,but ever-increasing,numberof terms(cf. Eq.(166))is asymptoticallyanalyzedin a straightforward mannerthroughthe contour-integral equivalenceasindicatedin Eq.(170).Furthermore,even thoughtheperiodicityof theassociatedparabolic(Schrodinger)propagatoris explicitly exploitedin thecontour-integral constructionandthediscretenatureof thespectrumis, likewise,exploitedin thespectral(modal)summationconstructionfor thefocusingquadraticcase,thetwo complementaryoperatorsymbolconstruc-tion proceduresandtheir combinedusagearequitegeneral,andapplicableto otherprofiles.Indeed,for the Helmholtzoperator


symbolsin thedefocusingquadraticcase,which wereconstructedby thesameprocedures(4, 23), theassociatedspectrumhasnodiscretecontributionsandtheassociatedparabolic(Schrodinger)propagatoris notperiodic.

Thefractional,Helmholtzoperatorsymbolsconstructedin Sections2 and3 representtheappropriateHelmholtzoperatorrootsassociatedwith thephysical,right-traveling wave field. In thespectral(modal)summationconstruction,this conditionis enforcedthroughthe appropriateinfinitesimalshifting of the resolvent singularitiesin the integral representation(1, 4, 23), while, in thecontour-integral construction,the correspondenceto the physicalroots follows immediatelyfrom the extractionof the operatorsymbolsfrom the physical,outgoingwave Green’s function or the correspondingparabolic(Schrodinger)propagator(23). Assuch,the Helmholtzoperatorsymbolsmust satisfy the appropriatecompositionequationsandbe consistentwith the physical,right-traveling- (outgoing-)wave radiationcondition(1, 4, 23), aswasdemonstratedfor the defocusingquadraticprofile casebyFishman(23).This is briefly outlinedfor thefocusingquadraticprofilecasein AppendixC. Satisfyingtheappropraitecompositionequationsaloneis notsufficientto ensuretheconstructionof thephysical,Helmholtzoperatorsymbols;thecorrespondencewith theradiationconditionis essential.For example,if the MQc i limit is taken in theanalyticcontinuationformulas(112)-(115),theresultingsymbolswill still satisfytheappropriatecompositionequations(at leastin a formal,asymptoticsense(15, 17)), however,they will no longerbeconsistentwith thephysicalradiationcondition,and,thus,will notcorrespondto thephysicaloperatorroots.Theconstructionof thephysical,Helmholtzoperatorsymbolshasbeenbriefly discussedby Fishman(23).

Theexact,square-rootHelmholtzoperatorsymbolconstructionsgivenin Eqs.(102)and(104)in conjunctionwith thenumeri-calresultspresentedin Section8 illustratethewaveguidingpropertiesof thefocusingquadraticprofile.At thelevel of themarchingrangestep,which follows immediatelyfrom Eq.(10),theinfinitesimal,down-rangewavefield is givenby (19, 20, 21, 23) Xa¡r¢£Y¡ Z¤¥\ §¦¨ © X de KEG8ª\«R¬¯®° X i de R¤H\¬¯®° h i de±²/ X RZ³¤¥\ £Y¡ k´ Xa¡ ZaR\µ> (186)

Takingtheinitial wave field at ¡ to berepresentedby a very broadGaussianfunction,which is essentiallyconstantover thenon-absorptive rangeof the profile (

] ^ X ¤H\·¶¸< ), resultsin a very narrow Gaussianfunction,sharplypeaked about R:¹< , for thecorresponding Xa¡ ZR!\ , leadingto theapproximation Xa¡r¢£Y¡ Z¤¥\ ¬¯®° h i de±º²/ X <Zs¤H\ £Y¡ k> (187)

For the casescorrespondingto propagatingmodes,it follows from Fig. 12 (andthe R»¼ ` invarianceof± ² / X RZ ` \ in the qua-

dratic case)that the energy within the effective waveguide will be redistributed,with the wave field in the absorptive regionsbeingsuppressed.In particular, the oscillatorycharacterof the operatorsymbolensuresthe strict conservation of the integratedenergy flux (1, 19), while thephasespaceregionswith no½p ± ² / X RZ ` \[tY¾g< ultimatelyallow for thedown-rangefocusingcorrectlycorrespondingto themodalenergy distribution.

While the focusingquadraticprofile is, in somerespects,nonphysical,thecorrespondingHelmholtzoperatorsymbols,nev-ertheless,establishcanonicalsymbol featuresfor more generalprofiles containinglocally-quadraticwells. This shouldnot besurprising.In thecontext of a modalanalysis,the low-lying eigenfunctionsandcorrespondingeigenvaluesin suchwells will ex-hibit a quadraticcharacter. This is thesamephenomenonexpressedat thelevel of theoperatorsymbol.This will be illustratedindetailelsewhere.

The Helmholtzoperatorsymbolresultspresentedin Sections2-7 canbe immediatelyextendedto thegeneralized,focusingquadraticprofiledefinedby] ^ X ¤¥\: ] ^¿ ¢ GEÀ¤c_ ^ ¤ ^ Zwith

] ¿ Z f n Á and À f n Á . Sincethe effect of the linear term is to shift the equilibrium positionandthe overall phaseinthecorrespondingharmonicoscillatorproblem,it immediatelyfollows from Fishman(23) andtheanalyticcontinuationresultsinSection3 thatall of theHelmholtzoperatorsymbolrepresentationsderivedfor thefocusingquadraticprofile ( À:§< ) hold for thegeneralizedprofilewith thefollowing identifications,L cM 9?ÂÄÃÆÅ ^ Xa` cÀK ^ \ ^ ¢ R ^sÇ Z (188)J cM 9 ÂÄÃ Å ] ^¿ ¢ À ^ KB ^ Ç Z (189)È cM 9 ÂÄÃ Å Xa` cÉÀKB ^ \R Ç Z (190)

while` cM ` cÊÀKB ^ > (191)

Thelineartermis seento increasetheeffective J parameterin Eq.(189),tending,in general,to increasethenumberof propagatingmodalcomponents,consistentwith theincreasedability of thepositive (non-absorptive) partof thegeneralizedfocusingquadratic


profile to support‘bound states’.Exact symbol constructionsfor the Helmholtz operatorscan also be extendedto the higher-dimensional,separableand coupled,focusingquadraticcasesfollowing from the resultsin Fishman(23) and the appropriateincorporationof theperiodicityin theassociatedparabolic(Schrodinger)propagator.

The general,spectralandcontour-integral, Helmholtzoperatorsymbolconstructionprocedurespresentedhereandin Fish-man(23) canbe applied,aspreviously suggested,to othercasesof interest.The hyperbolicfunction profile,

]_^ X ¤¥\q: ]_^¿ ¢Ë;ÌsÍDÎÏ XÐ ¤H\ ¢ÒÑÓ ¬Ô Ï^ XÐ ¤¥\ (23, 46), is appropriatefor physicalmodeling,encompassesa rangeof perturbationlimits, illustratesthe operatorsymboltransitionfrom high to low frequency, andaccommodatesthe effectsassociatedwith both symmetricalandasymmetricalwells in addition to large gradients,for example.Particularly interestinglimiting casesinclude the delta profile,] ^ X ¤¥\: ] ^¿ ¢ GÕÖ X ¤¥\ (47,48), thediscontinuityprofile,

] ^ X ¤¥\: ] ^Ã ¢£ ] ^× X ¤¥\ , where× XØ \ is, again,theHeavisidefunc-

tion and £ ]_^ : ]_^^ c ]_^Ã (27,49), andthereflectionlessprofileswith T prescribedboundstates(49, 50). In addition,rectangularwells (4, 27) alongwith a combinationof the deltaandquadraticprofiles(48) canbe considered.Thesedetailedconstructionswill bepresentedelsewhere,andareof particularinterestowing to their ability to illuminatethedifferencesin thehigh-frequencyasymptotic,operatorsymbol structurebetweensmoothand nonsmoothprofiles.Moreover, the very recent,extensive listing ofexactsolutionsfor theSchrodinger(time-dependent)andcorrespondingHelmholtz(time-independent)quantummechanicalequa-tions(51) in conjunctionwith theoperatorsymbolconstructionprocedurespresentedhereandin Fishman(23) allow, in principle,for theconsiderationof additionalcases.

TheHelmholtzoperatorsymbols(in thefrequency domain)lie outsideof thewell-establishedtheory and correspondingcalculusof elliptic pseudodifferential operators(1, 15). As a result, that well-developedcalculuscannotprovide theuniformcharacterizationof theseoperatorsymbolsoverphasespacewhich is crucialfor many applica-tions(1). In thehigh-frequency asymptoticapproximations,in additionto thealgebraictermsassociatedwith theelliptic calculus,thecontributionsof exponentialorder, correspondingto the infinitely smoothpartof thekernel,mustbe properlyincluded.Thishasbeendiscussedin detailby Fishmanet al. (1), whereuniform, high- andlow-frequency approximationswerederived for thesquare-rootHelmholtzoperatorsymbol.(Thecorrespondinganalysis,at the level of theoperatorkernel,is presentedin De HoopandGautesen(9).) Exactoperatorsymbolconstructionsandtheircorrespondingasymptoticexpansionsfurtherilluminatethisnewasymptoticstructure.Specifically, for thefocusingquadraticprofile, theexactoperatorsymbolplotsfor thehigh-frequency regimegiven in Figs.7-9 and11-15andthehigh-frequency asymptotic,operatorsymbolexpansionsgiven in Eq.(146)andEq.(159)andillustratedin Figs.13-15togetherdemonstratetheoscillatorycharacterassociatedwith theHelmholtzoperatorsymbols(1,23). Theresultspresentedhereandin Fishman(23) furtherillustratethat,for thequadraticcase,thelocationof theoscillatorybranchof theWeyl operatorsymbolis confinedto (i) the locally-propagatingregimein thefocusingcaseand,correspondingly, (ii) thelocally-evanescentregime in the defocusingcase.Furthermore,the discussionsurroundingEq.(171)establishesthe detailedconnectionbetweentheseoscillationsandtheunderlyingprofile.More generally, theanalysisunderlyingtheresultspresentedin Fishmanetal. (1) establishesthat for boundedprofiles,

]_^ÙÛÚ Ü ¾ ]_^ X ¤¥\q¾ ]_^ÙÛÝÞ , the oscillatorybranchof the Weyl, Helmholtzoperatorsymbolis of finite extentwith respectto R , essentiallylying within theinterval R fh ] ÙÛÚ Ü Z ] ÙÛÝÞ k for R fh <ZjP\ . Theoscillatorybranchlocationcorrespondencesbetweenthelocally-propagatingregimeandfocusingporfilesandthe locally-evanescentregimeanddefocusingprofiles,illustratedabove for thequadraticcase,arefoundto hold in themoregeneralcases.

The exact constructionof the Helmholtzoperatorsymbolsfor the focusingquadraticprofile impactsseveral areasof directand inversewave propagationmodelingin extendedinhomogeneousenvironments.Substitutingthe exact resultsin Eqs.(102)and (104) into the one-way wave equations(8) and (7), respectively, and fixing the rangepoint at ¡ : ¡ß provide an exactrealizationof thecomputational(nonreflecting)boundaryconditionfor thetransverselyinhomogeneousHelmholtzequationfor thefocusingquadraticcase(1). CombiningEq.(104)andEq.(10)resultsin aformallyexact,explicit, path-integralrepresentationfor thefundamentalsolutionof theone-way Helmholtzequation(7) for thefocusingquadraticprofile (1, 2, 18, 19,20, 21). Furthermore,Eq.(102)providesan explicit exampleof the function which effects the formally exact ‘ à -integration’ in the Feynman/Fradkinpath-integral representation(52, 53). Moreover, the formal, phasespacepath-integral representation,in termsof the square-rootHelmholtzoperatorsymbol,canbecomparedto themathematicallyrigorousconstructionsrecentlypresentedby LaChapelle(54),basedon anextensionof theCartier/DeWitt-Morettefunctionalintegrationscheme(55), andDynin (56),motivatedby abackwardEuler approximationof the corresponding,first-order, pseudodifferential (phasespace)evolution equationand productintegralconstructions.The Dynin analysis,in particular, may ultimately provide the meansto establisha rigorousmathematicalbasisfor the formal, phasespacepath-integral constructions.The exact symbol representationsfor the well-known, operatorrationalapproximationsof the square-rootHelmholtzoperator, which provide the basisfor the practicalcomputationalrealizationof the‘parabolicequation’method(1, 3, 4, 5, 6, 10, 23), areconstructedin Section7 for the focusingquadraticprofile. Previously, thecorrespondingoperatorsymbolsfor thedefocusingquadraticprofile werenumericallycomparedwith theexactconstructionsforseveral,recent,rationalapproximationschemes(1, 10, 23). TheHelmholtzoperatorsymbolscanbeconnectedto Galerkin(basis


setexpansion)methods(57, 58, 59) for theHelmholtzequationthroughEq.(7),Eq.(47),andexpansionin theoscillatorbasissetá!â.

The path-integral representationfor the fundamentalsolution(one-way propagator)inherentlycontainsboth the asymptoticray andmodalrepresentationsof thewave field, and,in this sense,is a particularlyusefulrepresentation.Concerningthenumer-ical evaluationof the one-way propagator, although,for the quadraticprofile, the propagatorpath-integral representationcanbeexactly evaluatedthrougha spectralsummation(the square-rootHelmholtzoperatoris thencompact),for moregeneralprofiles,the infinitesimalpropagatorcontrolsany wave field continuationalgorithm.The infinitesimalpropagatoris expressedin termsofthe square-rootHelmholtzoperatorsymbol.From the quadraticprofile case,it is seenthat (i) usingthe Weyl calculus(andas-sociatedWeyl transform),the infinitesimalpropagatorfollows a multiresolutionanalysis,and(ii) usingthe standardcalculus,ascreenrepresentationfor thepropagatoris obtained.Thus,themicrolocalanalysisassociatedwith theoperatorsymbolsis tied to amultiresolutionanalysisfor wavepropagation.

Returningto thefully-coupled,two-way, elliptic wave propagationproblem,thegeneralizedBremmerseries(9, 10, 11) pro-videsa meansto incorporatethe one-way constructionsdirectly into the two-way scatteringprocess.The generalizedBremmerseriescouplestheone-waywaveconstituents,andgeneratesall multiply-scatteredwaves.Theconvergencepropertiesof thisseriesareunderstoodin thetime-Laplacedomain,andrequireSobolev orderestimatesof thesquare-root‘Helmholtz’ operatoruniformin the Laplaceparameter. However, mostalgorithmsthat (approximately)computetermsin the generalizedBremmerseriesareaccomplishedin thetime-Fourierdomain(10). Theclosed-formexpressions,derivedfor thesquare-rootHelmholtzoperatorin thequadraticprofile,shouldprove to beusefulin obtainingtheseSobolev orderestimatesin thetime-Fourierdomain.It is anticipatedthatsuchanestimatewould hold for moregeneralprofiles.Furthermore,thegeneralizedBremmerseriesrepresentationfor wavescanbeinterpretedasamethodof ‘tracingwaves’,thewave-theoreticalanalogueof thegeometrical(asymptotic)methodof ‘tracingrays’ (Weinberg andBurridge(60)). Suchmethodsplay a key role in the ‘bootstrapping’approachto inversescattering(see,forexample,Claerbout(61)).

Equation(171)canbeviewedfrom botha ‘direct’ and‘inverse’perspective. Fromthe‘direct’ perspective, viewedasa char-acterizationof theoperatorsymbol,given 9 , thedimensionlessparametercharacterizingthemediumvariability on thewavelengthscale,andmeasuringã , thenumberof propagatingmodesT followsimmediatelyfrom Eq.(171).Theaccuracy of this lowest-orderestimatewasillustratedin Section8 in connectionwith Figs.13-15.

Fromthe ‘inverse’perspective, viewedasa reconstructionfrom datawith known de , combiningEq.(150)with Eq.(171)andtreating ã and W /YX <?Z[<H\ as the dataresult in estimatesfor both the focusingquadraticprofile parametersand the numberofpropagatingmodesT . While the relationshipbetweenthe profile andthe asymptotic,Helmholtzoperatorsymbolstructurehasbeenillustratedherefor thespecificcaseof thefocusingquadraticprofile, it canbeextendedto generalprofilesthroughboththeuniform, high- andlow-frequency asymptoticexpansionspresentedin Fishmanet al. (1). Of course,

] ^ Xa` \ can,in principle,bereconstructedfrom thecompositionequationEq.(11),whichwasillustratedin Section8 andby Fishman(23) for thefocusinganddefocusingquadraticcases,respectively, in thehigh-frequency limit. Theseideashaveanaturalextensionto thescatteringandDtNoperatorsymbolsin theinverseanalysisof thegeneral,range-dependentHelmholtzequation(1, 2, 12, 13).

In summary, theinversesquare-rootandsquare-rootHelmholtzoperators(and,subsequently, thescatteringandDirichlet-to-Neumannoperators)play a fundamentalrole in many directandinversescattering/propagationproblems.Theseoperatorsdo notbelongto theclassof strictly elliptic pseudodifferentialoperatorsandtheircalculus.Eventhoughtheoperatorsymbolconstructionsandsubsequentcharacterizationspresentedin this paperare restrictedto the specific,quadraticprofile, many of the resultsarebelievedto be(at least,qualitatively) canonical,andshouldapplyto awideclassof, evensingular, mediumprofiles.In this regard,this paperprovidesa hint on theextensionof this polyhomogeneouscalculusfor symbolsof elliptic pseudodifferentialoperatorsto a calculusfor non-strictlyelliptic operators,suchasthe oneslisted above. This shouldimpactthe continuingdevelopmentofuniformasymptoticsymbolexpansions(1, 9) for thesepropagationandscatteringoperators.

Acknowledgments

Theresearchreportedin thispaperhasbeenfinanciallysupportedby theNaval ResearchLaboratory, thesponsorsof theConsortiumProjectat theCenterfor WavePhenomena,aSpecialResearchFundof theExecutiveBoardof theDelft Universityof Technology,Delft, theNetherlands,andresearchgrantsfrom theStichtingFundfor Science,TechnologyandResearch(a companionorgani-zationto theSchlumbergerFoundationin theUSA). Theauthorsgratefullyacknowledgeall of thesesourcesof support.RichardKeiffer at theNaval ResearchLaboratoryis alsospecificallyacknowledgedfor computingFigs.7-19andJeromeLe RousseauattheColoradoSchoolof Minesfor makingFigs.1-6.


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Appendix A: Operation interchangeand theRiesz/Young theorem

Thisappendixprovestheidentity¦Sç¿ «è¥è ÂÃ6é ^ ¬j®° h X Jêc@8\6èkë çìâHíî @ïÛð$ñ iGÄò âOó â X RZ ` \¬¯®° X cÆG ï è³\õô:gª Ã6é ^ çìâHíî @ïð @X G ï ¢ @cÊJ;\ Ã6é ^ ñ iGò âOó â X RZ ` \Z (A1)

which was applied in establishingEq.(28). The result inEq.(A1)follows on establishingthevalidity of theinterchangeof integrationandsummation,whichis adirectconsequenceof

establishingthatthepower seriesö X÷ \Û: çìâHí ¿ @ïð ñ iG ò âOó â X RZ ` \ ÷ â Z (A2)

with ÷ f ø is uniformly convergentfor ÷ f X <?ZB@¯k .Thepower seriesin Eq.(A2) hasa radiusof convergenceùÆú

equal to one,which follows directly from the derivationof the Mehler formula (33) or from the definition (31) of

ùÆúandthepreviously referencedasymptoticestimatesof theHer-mite polynomialsandthe gammafunction.Hence,the powerseries

ö X÷ \ is uniformly convergent (31) for ÷ f X <ZB@8\ . Ex-tensionof uniform convergenceto ÷ :û@ ( è:ü< ), and theconsequentcompletionof theproof, follow onestablishingtheconvergence(31)of

ö X @B\ .The convergenceof

ö X @B\ is an immediateconsequenceof the following theoremassociatedwith M. RieszandW.H.Young(32).

Theorem. Ifý X÷ \: çì âHí ¿þ â ÷ â Z ÷ f ø Zis a power serieswith a finite radiusof convergence(takenforconvenienceto beunity) andÿ oâ ç þ â :g<;Zthen the seriesis convergent at every point of the unit circlewherethefunction

ý X÷ \ is regular.

For the seriesunderconsideration,the radiusof conver-genceis one,and,further, thepreviouslyreferencedasymptoticestimatesof theHermitepolynomialsandthegammafunctionestablishthatþ â : @ïÛðlñ iGÄò â ó â X RZ ` \: X ï ÂÃ6é ^ \ as ï MVP . (A3)

Applying theRiesz/Youngtheorem,theconvergenceofö X @8\ is

determinedby thebehavior of thefunctioný X ¬¯®° X cÆGBè[\³\:G ÂÄÃ6é ^ ¬¯®° X è[\ X è[\evaluatedat è0:g< , where XØ \ is definedin Eq.(29)(andis thegeneratorin accordancewith the Mehler formula (33)). Sinceè½: < is a regular point of the function XØ \ , it follows thatö X @B\ is convergent.

Appendix B: Alter nativeHelmholtz operator symbolintegral representations

Following from Section3, a numberof alternative, equivalentintegral representationscanbe derived for the Helmholtzop-eratorsymbolsfor the focusingquadraticprofile case.Onlytwo exampleswill be notedhere.The first one,a counterpart


of Eq.(92),W X RZ ` \: @X G9D\ Ã6é ^ ª ¬¯®° X cOLÛ\ ¦ «EÂÃ¬¯®° X GÄLÛ\ ÷ºX @BKEG?Z X @OclJ;\³KEG?Z X @c8\³K8\cÆG h X @c\³K¯k ÷ºX @BKEG?ZsT ¢X @c$J \³KDG?Z X @Oc8\³K8\ÒZJ:ê@DZ[ZjF¥Z ØØØ Z (B1)

where! is thecontourin thecomplex -planeassociatedwiththeinverseLaplacetransform(35).This representationderivesfrom thespectral(modal)summationrepresentationin Eq.(56)andthe inverseLaplacetransformof the Laguerrepolynomi-als (34) in combinationwith theappropriaterepresentationoftheLerchtranscendentalfunction(37).

Thesecondexampleis therepresentation,W X RZ ` \:êc¬¯®°#"$%?ª i & G9 Ã6é ^ ª¦§ç¿ «' @X @BKEGD\c X)(J;KDGE\* çì âEí ¿ X cOL\ âX³X EKEGE\Ûc X (JYKDGE\³\ â^ Ã X ï ¢ @EZ X @BKEGE\Ûc X (J KDGE\+ ï ¢gX HKDGE\Ûc X (JYKEGD\+Bc@8\ZJ:ê@DZ[ZjF¥Z ØØØ Z (B2)

where (J : J ¢ i ' ^ and ^ Ã XØ Z Ø + Ø + Ø \ is theGausshypergeo-metricfunctiondiscussedin Erdelyi etal. (37, chap.ii)(notetheoccurrenceof Pochhammer’s symbol).Equation(B2) followsfrom Eqs.(B17)-(B19)in Fishman(23)andutilizing propertiesof the gammafunction.RepresentationssuchasEq.(B2)pro-vide anothermeansto establishthe extendedanalyticcontin-uation resultsgiven in Eqs.(112)-(115).In addition,Eq.(B2),while expressedin a form which masksthe underlyingperi-odicity (soconvenientlyexpressedby theLerchtranscendentalfunctionandtheintegrationendpointsin Eq.(92))in Gausshy-pergeometricfunctionintegrals,is, nevertheless,anaturalformfor the exact operatorsymbolconstructionsin the hyperbolicprofilecaseandthesubsequentillustrationof thecanonicalna-tureof thefocusingquadraticprofile.

In a similar manner, alternative integral representations,analogousto theonesconstructedin Eqs.(B1)and(B2), canbederivedfor theexact,Helmholtzoperatorsymbolscorrespond-ing to theadditive, rationaloperatorapproximationspresentedin Section7.Forexample,in conjunctionwith Eqs.(180)-(184),W-, .0/ 214365 X RZ ` \:c @987 @X @BKEGE\Ûc X:9J;KEGD\* çì âHí ¿ X cOLÛ\ âX³X HKDGE\c X 9JYKEGD\³\ â^ Ã X ï ¢ @EZ X @BKEGE\Ûc X 9J;KEGD\+ ï ¢gX HKEGD\c X 9JYKEGE\+Bc@B\Z

9J;:ê@EZsZ[F?Z ØØØ > (B3)

Appendix C: Verification of Helmholtz operatorsymbolcompositionequations

Both theinversesquare-rootandsquare-rootHelmholtzopera-tor symbols,for thefocusingquadraticprofilein onetransversedimension,must satisfy the usualcompositionequationsandbe consistentwith the appropriate,right-traveling- (outgoing-)waveradiationcondition(1,4,23). Thecompositionequations(cf. Eq.(23))to beverifiedaredeG8ª ¦ ¨ © «=<«6 ±² > X R cEZ ` \ ±º² ? X RÄZ ` c@<!\ ¬¯®° X c i de A<!\: ±² > X RZ ` \ Z (C1)

anddeG8ª ¦ ¨ © «=<«6 ±² > X R cEZ ` \ ±º² X RZ ` cB<!\¬j®° X c i de A<!\: ±² / X RZ ` \:@Z (C2)

anddeG8ª ¦ ¨ © «=<«6 ±² X R cCEZ ` \ ±² X RÄZ ` c@<!\ ¬¯®° X c i de A<!\: ±² X RZ ` \0: ] ^¿ c ^ ` ^ c·R ^ > (C3)

Thecalculationsarebriefly outlinedin thestandard(left) cal-culus wherethey are technically lesscumbersome,however,the analogousresultshold in the Weyl calculus.The verifica-tion outlinesareillustratedfor both thespectral(modal)sum-mationand contour-integral operatorsymbol representations,given, respectively, by Eq.(21) and Eq.(103) for the inversesquare-rootHelmholtzoperatorsymboland,for the purposesof thisappendix,extendedto thesquare-rootHelmholtzopera-tor symbolthroughthecompositionrelationshipin Eq.(24).

For the verification of Eq.(C1) for the spectral(modal)summationoperatorsymbol representation(21), substitutingEq.(21) into Eq.(C1), interchangingthe order of the integra-tionsandsummations,applyingtheHermite,Fouriertransformresultgivenin Eq.(20),andexploiting theHermiteorthogonal-ity relationship(29),¦SçÂ ç « ÷ ¬¯®° X c ÷ ^ \ × â X÷ \ ×ED X÷ \: ª Ã6é ^ G â ïÛð Ö â D Z (C4)

leadto thefinal result(compareEq.(21))±² > X RÄZ ` \:êc G Ã6é ^9 çìâHí ¿ @ïÛð @X G ï ¢ @c$JY\ ñ iGò â ó â X RÄZ ` \> (C5)

Theexpression(C5) is in agreementwith theoperatorsymbolcalculationsdonein conjunctionwith the rationalapproxima-tion operatorsymbolconstructionsin Section.

To establishEq.(C1) for the contour-integral operator


symbol representationEq.(103),it is convenient to first con-sider the case J ¾ @ . Choosingthe contour û:GFIH Hl:J H HÃ ¢ J H H^ ¢ J H H$ , asillustratedin Fig. 6 for thedefocusingqua-draticprofileanalysis,thenallows for Eq.(103)to beexpressedas(compareEq.(65))± ² X RÄZ ` \:êc i ñ @ª9!ò Ã6é ^ ¦§ç¿ «?è¥è¯ÂÃ6é ^* ¬¯®°ÆÅ J2èÛc Ã^ L ÌsÍÎºÏ X G8è[\¢ i

È XaÓ ¬Ô Ï X G8è³\Ûc@B\ ÇXaÓ ¬¯Ô Ï X G8è[\³\jÃ6é ^ ZiJ ¾@> (C6)

SubstitutingEq.(C6)into Eq.(C1),interchangingthe orderofthe integrations,evaluatingtwo successive Gaussianintegrals(in < and ), andapplyingstandardhyperbolicfunctionidenti-tiesyield theresult±º² X RÄZ ` \:êc ñ @ªÄ9ò ¦ ç¿ «è ¦ ç¿ «?è H X è¥è H \ ÂÄÃ6é ^* ¬j®° Å J X è ¢ è H \c Ã^ L ÌsÍÎºÏ X G X è ¢ è H \³\¢ i

È XaÓ ¬Ô Ï X G X è ¢ è H \³\c @B\ Ç" Ó ¬Ô Ï X G X è ¢ è H \³\K& Ã6é ^ ZJi¾@> (C7)

Exploiting the èè H -symmetry of the integrand in Eq.(C7)throughthe introductionof the new variables< : è ¢ andL : ècM reducesthedoubleintegral to asingleintegralof theform±º² X RÄZ ` \:êc @9 ¦Sç¿ «=<¬¯®°Å JN<½c Ã^ L ÌsÍDÎÏ X G8<!\ ¢ i

È XaÓ ¬Ô Ï X G4<\c@8\ ÇXaÓ ¬¯Ô Ï X G4<!\³\ Ã6é ^ Z=J ¾§@Z (C8)

or±º² X RÄZ ` \:êc @9 @@cÊ¬j®° X Gª i J;\ ¦ «à¬¯®°Å J2à c Ã^ L ÌsÍÎºÏ X Gà\ ¢ iÈ XaÓ ¬¯Ô Ï X Gà\c@8\ ÇXaÓ ¬¯Ô Ï X Gà\³\ Ã6é ^ Z=J:g<ZB@DZsG?Z ØØ¯Ø > (C9)

Thefinal equalityin Eq.(C9),establishingthe full rangeof Jvalues,follows from analyticcontinuationarguments.Theex-pressionin Eq.(C9)is in agreementwith the operatorsymbolcalculationsdonein conjunctionwith the rationalapproxima-tionoperatorsymbolconstructionsin Section. Theequivalenceof the expressionsin Eq.(C5) and Eq.(C9) follows from theMehlerformula(33),uniformconvergencearguments(31)andtheRiesz/Youngtheorem(32), andelementaryintegration.

Examiningthe compositionequations(C2) and (C3), itis seenthat the Fourier integrals involving the square-rootHelmholtzoperatorsymbol

± ² X RZ ` \ do not exist in theusualsense,but, rather, mustbeunderstoodin thecontext of gener-alizedfunctions(23,30). This essentiallymeansthatthecom-positionintegral is givenmeaningthroughtheanalyticcontin-

uationin theparameterO of theLerchtranscendentalfunctionappearingin Eqs.(103)and(24), for example(23, 30). Opera-tionally, theimplicationis thatthebasicprocedureusedin go-ing from Eq.(C6)to Eq.(C9)in theverificationof compositionequation(C1)canagainformally beappliedin evaluatingcom-positionequations(C2) and(C3) to derive the correctresults.Thiswasthemethodemployedin establishingthecorrespond-ing compositionequationresultsfor the defocusingquadraticprofile in Fishman(23). SubstitutingEqs.(103)and (24) intoEq.(C2)andfollowing the previously outlinedprocedureleadto theresult±² / X RZ ` \:êc ¦ ç¿ «=<««=<QP ¬¯®°Å JN< c Ã^ L ÌsÍÎºÏ X G8<!\ ¢ i

È XaÓ ¬Ô Ï X G8<!\c@8\ ÇXaÓ ¬¯Ô Ï X G4<!\³\ Ã6é ^0R :ê@Z=Ji¾@> (C10)

Theanalogoustreatmentof Eq.(C3)yields±² X RÄZ ` \:êcO9 ¦ ç¿ «S<««=< P ¬¯®°Å JT< c Ã^ L ÌsÍDÎÏ X G8<!\ ¢ iÈ XaÓ ¬Ô Ï X G4<\Ûc@8\ ÇXaÓ ¬¯Ô Ï X G4<!\³\jÃ6é ^* Å Jêc$L XaÓ ¬Ô Ï X G4<\³\ ^ cÉG i È Ó ¬Ô Ï X G4<\ ÌsÍDÎÏ X G8<!\c ÌsÍÎºÏ X G4<!\ Ç=R: ] ^¿ c_ ^ ` ^ c·R ^ Z=Ji¾@> (C11)

Analytic continuation arguments extend the results inEqs.(C10)and(C11)to thefull rangeof J values.

In additionto formally satisfyingthe compositionequa-tions (C1)-(C3),the Helmholtzoperatorsymbols

± ² ? X RZ ` \and

± ² X RZ ` \ are consistent with the appropriate, right-traveling- (outgoing-)wave radiation condition by construc-tion (1, 4, 23), with thecorrectevanescentbehavior beingap-parentin Figs.7-12for

± ² X RÄZ ` \ .

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Exact constructions of square-root Helmholtz operator