Spectral analysis of the Pekeris operator in the theory of acoustic wave propagation in shallow water

Spectral Analysis of the Pekeris Operator in the Theory of Acoustic Wave Propagation

in Shallow Water

CALVIN WILCOX

Communicated by S. AGMON

Abstract

The PEKERIS differential operator is defined by

au=-c2(x.)p(x.)V" (p~x.) Vu ),

where c(x.), p(x.) satisfy

and

c(x.) = r~ c,, (C2,

V=(8/3xl,8/Sx 2 .... ,8/8x,), and the functions

0 < x . < h ,

x,>h,

x =(x l , x2, ... , x,)eR",

p(x,)=~Pl, O<=x. <h, (P2, xn~h,

where q , c2, Pl, P2, and h are positive constants. The operator arises in the study of acoustic wave propagat ion in a layer of water having sound speed q and density p~ which overlays a bo t tom having sound speed c 2 and density P2.

In this paper it is shown that the operator A, acting on a class of functions u (x) which are defined for x, > 0 and vanish for x, = 0, defines a selfadjoint operator on the Hilbert space

Jr ~ = L 2 (R+, c - z (x.) p-'(x.) dx),

where R+ ={x~R" : x , > 0 } and dx=dx x dx2...dx . denotes Lebesgue measure in R+. The spectral family of A is constructed and the spectrum is shown to be continuous. Moreover an eigenfunction expansion for A is given in terms of a family of improper eigenfunctions. When c 1 > c 2 each eigenfunction can be interpreted as a plane wave plus a reflected wave. When c 1 < c 2, additional eigenfunctions arise which can be interpreted as plane waves that are trapped in the layer 0 < x, < h by total reflection at the interface x. = h.

260 C. WILCOX

I. Introduction

This paper presents a spectral analysis and eigenfunction expansion for a selfadjoint operator generated by the PEKERIS differential operator

(1.1) A u = - c 2 ( x , ) p ( x . ) V. p ~ , ) Vu

acting on functions u(x 1, x 2 , . . . , x,) defined in the half-space

(1.2) R+ = {x=(xl, x 2 . . . . , x,): x ,>0}

and subject to the Dirichlet boundary condition

(1.3) u(x 1 . . . . . x , _ l , 0 ) = 0 for all (x 1 . . . . , x , _ I , 0 ) E O R + .

In (1.1), V=(O/~xl , O/Ox z . . . . , O/~x,) is the gradient operator and the coefficients c (x,), p (x,) are discontinuous functions defined by

0_<x,<h, (1.4) c(x") = {;12', x , > h ,

(1.5) p ( x n ) = ~ p i , O ~ x , < h , (P2, X n ~ h,

where c~, c2, Pl, P2, and h are positive constants. The operator A was introduced by C.L. PEKERIS [4] in his study of acoustic

wave propagation in shallow water. In his work the dimension n = 3 and the waves are described by a velocity potential function u(t ,x) which satisfies the wave equation

02u c2(x,,)P(xn)V.( 1 ) (1.6) ~t 2 P~fn) Vu = 0 .

This equation governs the propagation of sound in a uniform layer of water having depth h, sound speed cl, and density Pl, which overlays a bottom having sound speed c: and density P2. The boundary condition (1.3) represents the free surface of the water. Equation (1.6) with n=2 also arises in the study of electromagnetic wave propagation along a dielectric-clad conducting plane. In this case u (t, x) represents the electric field, Maxwell's equations reduce to (1.6), the boundary condition (1.3) represents the surface of the conducting plane, c 1 and Pl represent the dielectric constant and magnetic permeability in a dielectric layer of thickness h, and c 2 and P2 represent the corresponding quantities in the space adjacent to it. Details of the reduction of Maxwell's equations to (1.6) are given in [9].

The operator A is formally selfadjoint with respect to the inner product

(1.7) (u, u) = ~ u(x)/) (x) c - 2 (Xn) p- 1 (Xn) dx R"+

on the Hilbert space

(1.S) ~ = L 2 (R+, c -2 (x,) p- l(Xn) dx) ,

Spectral Analysis of Wave Propagation 261

where dx = dx 1 dxz. . , dx, denotes Lebesgue measure on R+ and ~ is the complex conjugate of u. In Section2 a domain D ( A ) c J f is defined which incorporates the boundary condition (1.3) and makes A a genuine selfadjoint operator in )f'. The remainder of the paper presents a spectral analysis and eigenfunction expansion for this operator.

It is shown in Section 5 that A has a purely continuous spectrum; that is, A has no eigenfunctions in W. Thus the eigenfunction expansion must be based on a family of improper eigenfunctions. These functions are constructed in Section 3. In the special case where q = c 2 and p~ = P2 the operator A reduces to the negative Laplacian in R~ with Dirichlet boundary condition. In this case a complete set of improper eigenfunctions is defined by

( 1 . 9 ) t~(x,p)=ei(xtp,+...+ . . . . p,,_l+x.p.)_ei(Xlm§ . . . . p._,-x.p,),

where P=(PI,P2,. . . ,P,)eR", and the eigenfunction expansion reduces to the Fourier integral representation for functions f e L: (R") with the symmetry property

(1.10) f ( x 1 . . . . . x ,_ l , - x , ) = - f ( x a , ..., x,_ 1, x,).

Physically, (1.9) describes a plane wave propagating toward the boundary of R+ together with the specularly reflected wave. If c 1 > c 2 in (1.4) the improper eigenfunctions of A are much like (1.9), consisting of an incident plane wave plus a reflected wave. However, if q < c 2 a completely new phenomenon appears. In addition to improper eigenfunctions like (1.9) a new class of improper eigenfunctions arises. These functions are localized near the region O < x , < h and correspond physically to waves that are trapped by the phenomenon of total reflection at an interface between two media. It is because of this phenomenon, and its implications for the wave propagation problems mentioned above, that the spectral analysis of A is particularly interesting.

The remainder of the paper is organized as follows. Section 2 contains the definition of D(A) and the proof that A is selfadjoint. Section 3 is devoted to a

construction of the improper eigenfunctions of A. Section 4 contains an analysis of A based on Fourier analysis in the variables Xl, x2, . . . , Xn_ 1" This transforms A into the ordinary differential operator

(1.11) A p = - c 2 ( x , ) p ( x , ) (p(x,) dx, ] - p - ~ , ) J

where p = (Pl, P2 . . . . , p,_ ~) e R"- 1. The section contains a spectral analysis and eigenfunction expansion for Ap acting in the space

(R +) = L 2 (R +, c- 2 (x,) p - 1 (x,) dx,),

where R+ = {x,: x, > 0}. In Section 5 the results of Section 4 are used to construct the resolvent and spectral family of A. The construction of the spectral family is based on the improper eigenfunctions of Section 3. In Section 6 the spectral family is used to derive the eigenfunction expansion for A. Section 7 contains concluding remarks about various extensions of the theory and its applications to the study of wave propagation phenomena.

262 C. WILCOX

2. The Seifadjoint Operator A

Several function spaces are needed for the discussion of A. They include the Schwartz space ~(R~) of infinitely differentiable functions on R~ with compact support and the dual space ~'(R~_) of all distributions on R~_ [6], The remaining spaces can be regarded as linear subsets of ~'(R~). They include the Lebesgue space L2(R~) and the Sobolev spaces

( 2 . 1 ) L~(R"+)=L2(R~)n{u: D~'u~L2(R"+) for ]~]<m}.

In (2.1), m is a positive integer and the multi-index notation is used for derivatives [6]. E'~ (R~_) is a Hilbert space with inner product

(2.2) (u, v)~,= ~ ~ D �9 u(x) O ~ v(x) ax,

and ~(R+) defines a linear subset of L~(R+) for each m. In particular,

(2.3) L�89 ~ closure of ~ (R L) in L�89 (R'+)

is a closed linear subspace of L~ (R~). It is known that all the functions in L~ ~ (R~) satisfy the Dirichlet boundary condition; see [3] and Corollary 2.7 below.

The following notation will be used in the remainder of the paper:

(2.4) ~ x=(xl' x2, " ' ' ' x"-l)eR"-l' y~R+, ((x,y)eR" l x R + = R ~ .

Moreover, dx dy=dXl.., dx,_ 1 dy will denote the element of Lebesgue measure on R~.

With this notation, A may be defined as the operator on the Hilbert space

Jt~= L 2 (R~ , c - : (y) p- ~ (y) dx dy) (2.5)

with domain

( 2 . 6 ) D(A)=-L�89176 V. (-lp(y) Vu]~Lz(R~+)

and action defined by

( 2 . 7 ) Au=-cZ(y)p(y)V �9 ~y~VU forall ueD(A).

The "divergence" operator in (2.6) is to be understood in the sense of distri- bution theory. Thus, if V=(V~, V 2 . . . . . 11,) with Vs~Lz(R~) , then V. V=veL2(R~+) if and only if

(2.8) ~I? .g t#dxdy=- ~vOdxdy forall Oe~(R~+). R~ R~

Note that the Hilbert spaces Jig and L:(R~) are equivalent because ca(y)p(y) has positive lower and upper bounds. It follows easily that (2.6), (2.7) define a linear operator on ~ . The principal result of this section is

Theorem 2.1. The operator A is seffadjoint on ~.


The proof o f Theorem 2.1 will be developed in a sequence of lemmas.

Lemma 2.Z For all ueD(A) and veL~~ we have

(2.9) (Au, v ) = - ~V. (~7S,,~ Vfi) v d x d y = ~Vfi. Vvp- ' ( y )dxdy ,

where VFt. Vv = Ofi/Sx I dv/t3x 1 +... + t3~/~x,_ 1 ~v/Ox,_ ~ + ~ / ~ y t3v/~y.

Proof. The first equation of (2.9) follows from (2.7) and the definition of the inner product in ~ . To prove the second, note that it is valid by (2.6) and (2.8) if ve ~ (R~). To prove it in general let {vj} be a sequence such that vje ~(R"_) and

L 2 (R+). Such sequences exist by (2.3). Then writing (2.9) for v; and making v j -* v h'l 1 n j ~ ~ gives (2.9) in the general case.

Lemma 2.3. The operator A is symmetric; that is,

(2.10) A c A*.

Proof. The notation in (2.10) means that A* is an extension of A. To prove (2.10), note that the set

(2.11) ~0 (R~_) =~(R~_) ~ {u: u(x, y)=0 in a neighborhood o f y = h }

is a subset of D(A): ~o(R~)cD(A). Moreover, it is easy to verify that ~0(R~_) is dense in ~ . It follows that D(A) is dense in ~ and hence that the adjoint operator A* is uniquely defined. To verify (2.10), note that if u and v are both in D(A) then (2.9) holds for u and v and also for v and u. Thus

(Au, v)= ~ V ~ . F v p - l ( y ) d x d y = ( u , Av) forall u, veD(A), R~

which is equivalent to (2.10).

Lemma 2.4. The symmetric operator A is non-negative:

(2.12) A > 0 .

Proof. Putting v = ueD(A) in (2.11) we obtain

(2.13) (Au, u)= ~[Vul2 p - l ( y )dxdy>O for all ueD(A) g~

which implies (2.12).

Lemma 2.5. The range of 1 + A is ~ :

(2.14) R(1 + A)= ~ .

Proof. Note first that i f feR(1 +A) then there exists an element ueD(A) such that u + A u = f It follows by Lemma 2.2 that

(2.15) (Au, v)= ~ V~t. V v p - ~ ( y ) d x d y = ( f - u , v ) for all veL~'~ R$

264 C. WILCOX

and hence

(f, v) = j f v c- 2 (y) p- 1 (y) dx dy

(2.16) 1r = j{Vfi. Vvp l(y)+five 2(y)p l(y)}dxdy

R')

for all v a L~ o (R+). Now the right-hand side of (2.16),

(2.17) {u,v}= S{vft. Vvp-l(y)+Ftvc-Z(y)p l(y)}dxdy, R~

defines an inner product on L~ ~ (R+) which is equivalent to the inner product (u, v) 1 defined by (2.2). Moreover, for all fa~t~ and v~LL~

(2.18)

because

(2.19)

I(f, v)l~ IIfH Iivll ~ lifll {v,/)}1/2

Ilvlt 2= j l ax , y)l 2 e-2(y) p-l(y)dxdy<= {v, v} R~_

for all veL~~ It follows by the RIESZ representation theorem that for each f e ~ there is a ueL~~ such that (2.16) holds for all veL~~ It will be shown that u eD(A) and u + A u =f. Now u eL~ ~ (R"+) by construction. Moreover, (2.16) with ve@(R"+ )= L�89176 together with the definition (2.8) implies that

I(y) EL2(R+). \

This shows that uED(A) and u+Au=f , which completes the proof.

Proof of Theorem 2.1. The fact that A is selfadjoint is a direct consequence of Lemmas 2.3-2.5 and standard results in the theory of linear operators in Hilbert space (which can be found, for example, in [2]). Lemma 2.4 implies that the deficiency indices of A are equal [2, p. 268], and then Lemma 2.5 implies that they are both zero. It follows that A is selfadjoint.

The remainder of this section presents some additional regularity properties of functions in D(A) that are needed for the construction of the eigenfunctions of A in Section 3. To formulate them it will be convenient to introduce the notation

~Ii={y:O<y<h}, I2= {y: y>h}, (2.21) [E21=R l x I x and ~ 2 = R n - l x l 2 .

Since f21 ~ f22 differs from R+ by a Lebesgue null set,

(2.22) L 2 (R~.): L 2 (Q1) t~)L2 (~t-~2) ;

that is, every u e L 2 (R~) has a unique decomposition

(2.23) u(x, y)=ul(x, y)+ u2(x, y), uj~L2(O;).

Lemma 2.6. I f u~D(A) then

(2.24) useL 2 (f2), j= 1, 2.


Proof. It is easy to verify that if ueD(A) then AufiL2(Qj), j = 1, 2, where A denotes the n-dimensional Laplacian. Hence, (2.24) follows from elementary regularity theory for elliptic operators, or directly by Fourier analysis in Oj.

Corollary 2.7. I f uED(A) then

(2.25) ueC'(fj, L2(R"-a)) for j = l , 2,

(2.26) u( ' , 0) = 0 in L 2 (R"- 1),

(2.27) u ( ' , h + ) = u ( ' , h - ) in L2(R"-I),

and

(2.28) 1 ~u( . ,h+)_ 1 Ou(. ,h-) in L2(R,_~)" P2 OY Pl ~Y

In (2.25), ij denotes the closure of I~ and C 1 (i~, L 2 (R n- 1)) denotes the class of functions from Ij to Lz(R "-1) which are continuously differentiable on Jj. Thus conditions (2.26)-(2.28) are meaningful for these functions.

Proof of Corollary 2.7. The inclusion C 1 (ij, L 2 (R n- 1)) c L22 (f2) is a simple case of the Sobolev embedding theorem; see, e.g., [-3]. Thus (2.25) follows from Lemma 2.6.

To verify (2.26) recall that D(A)cL~~ Also, (2.26) is a well-known property of L~~ see [3]. Similarly, (2.27) follows from the embeddings D(A)~L�89 C(R+, L2 (R"- 1)).

The proof of (2.28) is longer. Recall that if u ~ D (A) then ~u/~x; (j = 1 .... . n - 1) and c~u/~y are in L2(R~). Hence if Vu=(~u/~x 1 .... . Ou/x,_ 1, ~u/~y) then

(2.29) V. ( p ~ VU)=vEL2(R+);

that is, if ~ba~(R+) then

12.30) 1 R~. ~ - ~ Vu(x, y) . V dp(x, y) dx dy= - R~ ~ V(x, y) ~b(x, y) dx dy.

In particular, if v = v~ + v 2 is the decomposit ion (2,23) of v, then Lemma 2.6 and (2,30) imply

1 (2.31) - - A u = v j in Lz(Qj) , j = l , 2.

Pj

Moreover, for all ~bE~(R+) and j = 1, 2 . . . . . n - 1,

02u ~u ~q~ (2.32) a,~ ~ ~b dx de= - o~ ~ ~xj ~xj dx dy.

To see this, note that if ~(y)E~(I1) then Lemma 2.6 gives

c "2 u(x, y) (2.33) ~,'~ ~x~ ~b(x,y)~(y)dxdy=-r, , (" gu(x,Y)oxj ~b(x,Y)c~xj ~(y)dxdy.

266 C. WILcox

Making 0(Y) tend boundedly to 1 on 11 gives (2.32). Next, note that (2.24) implies that

g2u d c~u ~y gu(x,h-) 49(x,h)dx. (2.34) 5 ~-y2 49dx y = - 5 ~yy dxdy+ ~ 8y Yh f h R ~ - 1

This may be verified by using the fact that C2(O1)c~LZz(f21) is dense in L2(f21). (2.34) is obvious if ue C2(~1).

Equations (2.31), (2.32) and (2.34) imply that

v, 49 dxdy= ~ ~(Aut49 dx dy r~ Pl

(2.35) ~' 1 c?u(x, h - )

=-- ~ ~ V u . V49dxdy+ ~ , P l R " - ' Pl Oy 49(x,h)dxdy.

A similar argument gives

v 249dxdy= ~ 1 (Au) 4)dxdy (2.36) m ~2 P2

=--~ 1 Vu. V49dxdy- I 1 u(x,h+) ez P2 ~,-' P2 Oy 49(x,h)dxdy.

Adding these equations and using (2.30) we get

(2.37) R n-' {lp~ •u(x,h+)oy Pll~?u(x'h-)} 4 9 ( x ' h ) d x d y = O O y

for all 49e~(R"~). This implies (2.28) because the class {49(x,h): 49e~(R~)} is dense in L2(R" 1).

3. Improper Eigenfunctions of A

It will be shown in Section 5 that A has a purely continuous spectrum. Thus A has no genuine eigenfunctions in W. The eigenfunction expansion of Section 6 is based on a set of improper eigenfunctions 0: R] ~ C having the following properties:

(3.1) 0(x, y) is locally in D(A); i.e., for each 49e~(R") we have 49 O~D(A).

(3.2) A 0 =2 0 for some 2eR+.

(3.3) 0 (x, y) is bounded on R+.

These properties are used in this section to construct a set of improper eigenfunctions.

Equation (3.2) is the differential equation

(3.4) c2(y)p(y)V �9 ~ V O + 2 0 - - 0 .

On the subdomains f21 = {(x, y): x ~ R" ~ 1 and 0 < y < h} and ~2 = {(x, y): x e R" - 1 and y > h} it is clear that (3.4) is an elliptic equation with constant coefficients.


Hence standard elliptic theory implies that any (distributional) solution of (3.4) defines smooth (class C ~ functions on these domains which can be extended by continuity to be smooth functions on their closures. Combining this with Corollary 2.7 gives the boundary condition

(3.5) 0(x, 0)=0 for all xER "-l

and the interface conditions

(3.6) O(x,h+)=O(x,h-) , 1 ~30(x,h+)_ 1 •O(x,h-) foral lx~R,_X. P2 ~Y Pl ~Y

It is an interesting fact that equation (3.4) has the unique continuation property in the class of solutions which satisfy (3.6). This follows from the classical theorem that solutions of the equation A 0 +/ t 0 = 0 are uniquely determined by the values of 0 and its normal derivative on a surface. This is illustrated in the construction of the improper eigenfunctions below.

The resolvent and spectral family of A will be constructed in Sections 4-5 by using Fourier analysis on the x-variables to reduce A to an ordinary differential operator. This procedure leads to representations in terms of functions of the form

(3.7) 0 (x, y) = e ~v'x 0 (Y),

where p = (PI, P2 . . . . , P,- 1) ~ Rn- 1 and p- x = PI xl + " " + Pn- 1 x,_ 1. In this section all the improper eigenfunctions which satisfy (3.1)-(3.3) and (3.7) are constructed. The functions 0 (Y) will be called reduced eigenfunctions.

Application of conditions (3.4), (3.5), (3.6) to the function (3.7) leads to the following conditions for the reduced eigenfunction.

d2 0(Y)+- (2-2 - [p[2) 0 (y )=0 for (3.8) d ~ 0 < y < h

a 2 O ly) + (3.9) ~-2 - [ p [ O(y)=O for y>h

(3.10) 0 (0)=0

1 , /0 (h+) 1 d 0 ( h - ) (3.11) 0 ( h + ) = 0 ( h - ) , -

P2 dy Pl dy '

where IPl2=p~+p~+...+p2,_~. It will be shown that conditions (3.8)-(3.11) determine 0(Y) to within a constant multiplier. Moreover values of 2 and p will be sought for which (3.7) defines a bounded function; that is,

0.12) 0(Y) is bounded for y~R+.

To construct solutions of (3.8)-(3.12) the following notation and conventions are used.

[ /~ \1/2 / /~ \1/2

r t -i l , . = t -ipl ) ,

268 C. WILCOX

(3.14) 4 > 0 for ,~c2lpl 2, ~=i~' with ~ '>0 for 2<c2lp[ 2,

(3.15) t />0 for 2 > c 2 [p]2, t /=it / ' with t / '>0 for )~<c 2 [pl 2.

For any peR "-~ and 2eR+ such that 2 ~ c 2 [p]2 and 2 4 c 2 ]p[2 (hence ~ 0 and t/=~0) the general solution of (3.8), (3.9), (3.10) can be written

j_.fct sin t/y, 0 < y < h , (3.16) 0 (Y)=~_fl eir 7 e -iCy, y>h.

Moreover, this function satisfies (3.11) if and only if

e ir fl+e -ir 7= (sin t/h)

(3.17) ei~hfl_e_ien,~=p2 r/ (cosr/h)~.

This system determines fl and 7 uniquely in terms of c( (unique continuation). The solution is

1 ( p2 it/ ) (3.18) f l = ~ - sint/h - c o s t / h e-iChor,

1 ( p2 it/ ) (3.19) 7 = ~ - s in t /h+ - - c o s t / h ei4ho~,

Thus the solutions of (3.8)-(3.11) are given by (3.16), (3.18), (3.19) with a an arbitrary constant (that is independent of y). Note that the two cases ).=c~ Ip[ 2 (~-----0) and ,,~---c 2 [pl 2 ( t / ~ 0 ) are not covered by the analysis above. These cases correspond to a Lebesgue null set in the set of all p~R "-1 and can be ignored in the Fourier analysis of A below. Hence, they need not be considered here.

To complete the determination of the reduced eigenfunctions it is necessary to determine the values of 2 and p for which (3.16), (3.18), (3.19) give a bounded function of yeR+. The following observations are helpful:

(a) If ~ is real (2 > c2[p l 2) then ~9 (y) is bounded.

(b) If ~= i 4' is pure imaginary (2<c 2 Ip[ 2) then 0(y) is bounded if and only if 7 = 0 or

(3.20) ~' = - - /02 r/ctn t/h. Pl

(c) If ~ = i~' is pure imaginary and t/ is real then (3.20) has solutions. These are determined below. If both ~ = i~' and t/= it/' are pure imaginary then (3.20) has no solutions.

To verify the last remark note that (3.20) may be written

(3.21) 4 ' - P2 t/, ctnh t/' h. Pl

This has no solutions with ~ '>0 and t/'> 0.


Remarks (a), (b), (c) show that two qualitatively different cases arise in con- structing the improper eigenfunctions of A, depending on whether c~ > c 2 or c 1 < c 2. In the first case c 2 [p[2 > c 2 [p[2. Hence there are improper eigenfunctions for every 2 > c 2 IPl 2 and none for 2 < c 2 [p[ 2. In the second case c 2 IPlZ<c 2 Ipl 2 and there are eigenfunctions for every 2 > c 2 IPl 2 and none for 2<c~ IPl 2. In addition there are eigenfunctions for the values of 2 and p which satisfy c 2 [pl2< 2 < c22 IP 12 and equation (3.20). These values are determined next.

When cl < c 2 equation (3.20) and the equations

(3.22) 4'=([pl2-(2/c2)) 1/2, ~=((2/c2)-IP12) 1/z

determine a functional relation between 2 and p. To analyze it note that (3.22) implies

(3.23) 4'2-1- ~2 = (r 2-- r 2) ~ ,

(3.24) c 2 4,2 q- c2 ~/2 =(c2-c2) IPl 2 .

Next, consider the graph F of (3.20) in the (q, 4') plane. Only the portion in the first quadrant is needed. It consists of a sequence of disjoint branches Fk, k = 1, 2, 3, ..., on which 4' is a monotone increasing function of r/. F k begins at the point (rl,r and has the line q=k~z/h as asymptote. Through each point (r/, r k there passes a unique circle (3.23) and a unique ellipse (3.24). Thus for each branch F k the four variables 2, Ipl, 7, 4' can be expressed as functions of a single parameter. In particular, any one of the four variables may be selected as the parameter. Here it will be convenient to choose IPl as the parameter. Thus F k determines functions

c, (3.25) / 2=2k( IPD | , IPl>Pk=(2k--1)Pa, Pl-- (c2_c~)1/2 2h"

(r = 4k(lPl)J

These functions are used below to construct improper eigenfunctions for A. An equation for the branch F k is

(3.26) rl h=(k - �89 pl ~' ) Pz ~1

where [tan -1 z l < re/2. This suggests a parametric representation of the relation 2=2k(IPl) from which the main properties of 2k(IPl) can be derived easily. The results will be described in terms of the equivalent function

(3.27) 09= ~k (IPl)= ~.k (IP l)l/ 2 ~ O.

Introduce the parameter "c > 0 by

(3.28) P2 #

270 C. WILCOX

Then r /h=(k - �89 z and it follows from (3.23), (3.24) that a parametric representation of (3.27) is given by

I(D-- ClC2 (/32 12 \1/2 h(c~_c~)l/2 (1+ \~-1! z2) [ (k- �89 z]

(3.29) "C~0. {1 + 1 (p t '~ 1/2 C 1 C 2

IPl = h(c2 c~)i/2 \~T-c2 ~Cl Z 2) [(k--�89 + tan -1 z! \ p - l !

The following properties of co k (IpD can be derived from (3.29).

(3.30) eOk(lPl) is analytic and o)~(Ip[)>0 for [pl>=pk.

(3.31) Cl IP[<Ogk(IP[)<c2 [Pl for all IPl>Pk .

(3.32) ~ (Pk) = C2 Pk, ~'k (Pk) = C2"

(3.33) ~Ok(lPl)~Cl IPl for IPl ~ ~ .

The information derived above makes it possible to define the reduced eigenfunctions of A as follows.

Case 1. c 1 > % . In this case there is an improper eigenfunction of the form (3.7) for each pER n-1 and 2 > c 2 Ipl 2 and none for 2<c2 z IPl 2. (Recall that the case 2 = c 2 [pl 2 will not be treated here.) The eigenfunction is given by (3.16), (3.18), (3.19) where ~ is an arbitrary constant. It will be convenient to write the reduced eigenfunctions in the following equivalent form:

[sin r/y, 0 < y < h ,

(3.34) ~Oo(y,p,)O=a(p, 2) iR e ( s i n r / h - i P2rl cosrlh) ei~tr-h), y>h.

For y > h an equivalent form is

(3.35) q/o(y,p, 2)=a(p, 2 ) { s i n q h c o s ~ ( y - h ) + p2rl cosr lhs in~(y-h)} .

Here a (p, 2) is a real positive function of p and 2, which will be defined in Section 4 so as to normalize ~k o (y, p, 2) in a convenient way. Note that Oo(Y, P, 2) is real- valued.

Case 2. c I < % . Here, as in Case 1, there is a reduced eigenfunction given by (3.34) for each 2>c~ Ip[ z. There are no improper eigenfunctions for 2<c~ Ipl 2. For c~ [pt e < )~ < c22 I P 12 there are improper eigenfunctions for those pairs (IP 1, 2) that satisfy (3.20). These are precisely the pairs that satisfy 2=2k(lpL). Note that for each fixed p e r "-1 there are a finite number N(p) of these pairs. In fact,

(3.36) N(p)=k for pk<lp[<pk+l.

The corresponding reduced eigenfunctions are given by (3.16), (3.18), (3.19) with 2=2k([p[), and hence ~--i~,([p[), ~/=qk([P[) and ~=0. It will be convenient to


write them in the following equivalent form:

(sin qk ([Pl) Y, 0 < y < h , (3 .37) Ck(Y, P)=ak(P) ~sin ?]k ([Pl) h e - ~`(Ipl) (r-h), y > h.

Here a k (p) is a real positive function of p and k which is defined in Section 4 so as to normalize ~k k (y, p) in a convenient way. Note that ~O k (y, p) is real-valued.

The improper eigenfunctions of A corresponding to (3.34) and (3.37) will be defined by

(3.38) ~ko(X,y,p, 2)=(2~) (-"+1)/2 eiP'Xq/o(y,p, 2), (2>c~ [p[2),

and

(3.39) ~k(x,y,P)=(2~Z)(-"+x)/ZeiP~'Og(Y,P), ([P[>Pk, Case 2 only).

The factor (27t) I-"+1)/2 is included for convenience in the Fourier analysis in Sections 4-5.

The eigenfunctions have an interesting interpretation in terms of electromagnetic theory. The~ may be interpreted as describing electromagnetic fields with frequency o9 = 1/2 propagating in a layered medium with refractive indices n 1 = c [ 1 ( 0 < y < h ) and n2=ci x (y>h) and with a reflecting boundary at y = 0 . Moreover, ~00 (x, y, p, 2) has the form

Sa(ei(p~'-'~r)-ei(px+'m), 0 < y < h , (3.40) O~ y>h,

where a and b are constants. This may be interpreted as a plane wave which propagates in the direction (Pl, . . . , P,-a, - q ) toward the boundary, is refracted at the interface y=h, propagates in the direction (Pl . . . . . p , _ ~ , - ~), is reflected at y = 0 , propagates in the direction (p~ . . . . ,p,_~,~), is refracted again at y=h and finally propagates in the direction (pa . . . . , P,-1, q)- The changes in direction at y = h can be shown to obey SneU's law n~ sin 0 t = n 2 sin 02, where 0j is the angle between the normal and the propagation vector in medium j. If q > c 2 then all of the improper eigenfunctions are of this type. If c a < c 2 there are, in addition, the functions ~/k (X, y, p), k > 1, which have the form

~a(ei(p'x-"Y)--ei(p'x+'lY)), 0 < y < h , 0.41) i/,~(x, Y, P)= [b e 'p'' e -~'r, y>h

where a and b are constants and 2 = 2 k (IP I). In the layer 0 < y < h this still represents a plane wave and its reflection at y = 0. However, for y > h it must be interpreted as a wave that propagates in the direction (p~ . . . . , p,_ ~, 0), parallel to the interface, and is exponentially damped in the direction normal to the interface.

In optics it is known that if a plane wave in a medium with index n~ strikes an interface with a medium of index n 2 < n~ it will suffer total reflection if the angle of incidence 0 satisfies 0> 0o= sin -~ (n2/nl). A simple calculation shows that the angles of incidence for the plane waves (3.41) are greater than 0 o, while the angles for the plane waves (3.40)are less than 00 . Thus the eigenfunctions (3.38) represent plane waves in both layers while the functions (3.39) represent plane waves in the layer 0 < y < h which are trapped by reflection at y = 0 and total reflection at the interface y = h.

272 C. WILCOX

4. Fourier Analysis of A and the Operator A~

In this section Fourier analysis in the tangential variables x ~ R"-1 is used to reduce A to an ordinary differential operator Ap, pER" 1. The Green's function for Ap is derived and used to construct an eigenfunction expansion. These results are used in Sections 5-6 to derive corresponding results for A.

Let u(x,y)E~. Then Fubini's theorem implies that u(. ,y)EL2(R "-1) for almost every yER+. Thus if if: L 2 (R n- 1)__. L2 (R,-1) denotes the Fourier transform in L 2 (R" 1) then the Plancherel theory implies that

1 (4.1) h(p,y)=(o~u)(p,y)= lim ~ e-lp'*u(x,y)dx

M ~ (2nff -~)/2 i~1__< M

exists in L2(R "-~) for a.e. yER+ and

(4.2) ~ I~(p,y)lZdp= ~ lu(x,y)t2dx fora.e, yER+. R n I R n ~ t

Another application of Fubini's theorem gives

Lemma 4.1. u~3f if and only if hE~. Moreover,

(4.3) IluH~r = I[~llje for all ue~.

It follows that ~ defines a unitary mapping of dr onto itself.

Lemma 4.2. The domain of A may be characterized as the set

(R~_) n (L2 (01) OL2 (O2)) n fu: u( ' , 0)=0, u( ' , h + ) = u ( - , h - ) D(A)=L�89 and

(4.4) 1 t3u(.,h+) 1 ~3u(.,h-) t P2 ~Y - - Pl ~y in L2(R n-1)J.

Proof. It was shown in Corollaries 2.6 and 2.7 that D(A) is a subset of the right- hand side of (4.4). To prove the opposite inclusion note that if u~L�89 and u( ' , 0)=0 in L2(R "-1) then u~L�89176 [3]. Moreover, if ueL~(O 0 ~LZ(f22) and 1/p2 0u(., h+)/~y= 1/p: ~u(., h-)/dy in L2(R "-~) then V. (UP(Y) Vu)~L2(R~). The proof may be based on equations (2.35) and (2.36) which hold under these hypotheses. Adding these equations and using (2.28) gives (2.30), which completes the proof.

The Fourier transform of A will be denoted by/1. Thus

(4.5) A = ~ - A ~ -1 , D(AA)=ffD(A).

Lemma 4.3. The domain of A may be characterized as the set

D(A) = {fi(p, y): p~ Dr fie Jr for ]c~l + j < 1,

p~ D~ fiEL 2 (f21) OL 2 (02) for I~l + j < 2 , (4.6)

fi(', 0)=0, fi(', h + ) - - u ( ' , h - ) and

l~Drfi("h+)= P l in L2(R"-I)}.


(4.9)

and

Proof. This result follows from Lemma 4.2 by the Plancherel theory and the result that (D" De h) (p, y) = (i p)~ De ~ (p, y).

Lemma 4.4. The action of .4 is given by

This follows immediately from the Plancherel theory. It will be convenient to consider the ordinary differential operator obtained by fixing p s R "-1 in (4.7). It defines an operator Ap on the Hilbert space

(4.8) ~r (R + )= Lz (R + , c- 2 (y) p - l (y) d y) .

The domain and action of Ap will be defined by

( 1 1 \ ) (4.10) Apu(y)=--cZ(y)~p(y)Dy~pTy)Dyu)--,pl2u~ for all u~.O(Zp),

in analogy with (2.6), (2.7).

Lemma 4.5. The operator Ap is selfadjoint on ~(R+).

The proof is exactly like that of Theorem 2.1 and will not be given here. In what follows the notation

(4.11) Rr -a

will be used to denote the resolvent of an operator T. Note that

(4.12) u=Rr if and only if ~=Rr

Moreover, if (4.12) holds then

(4.13) (Ap-~)h(p,y)=f(p,y) for a.e. peR "-1.

In this section the resolvent R~(Ap), the Green's function (resolvent kernel) for Ap and the spectral family of Ap are constructed by integrating (4.13). These results will be used in Section 5 to construct the resolvent Rr and the spectral family of A. The following result is needed for the solution of (4.13).

Lemma 4.6. The domain of Ap may be characterized as the set

D(Ap) = L~ (R +) ca (L~ (11) GL~ (12))

ca u: u(0)=0, u(h+)=u(h- ) and DyU(h+)= Dyu(h- ) . P2

The embedding C 1 (ij)c L~ (Ij) holds, cf. [8, p. 3683, and hence the boundary and interface conditions in (4.14) are meaningful. The details of the proof are essentially the same as those for Lemma 4.2, and therefore are omitted.

274 C. WILCOX

To construct u = R; (Ap)f note that (4.13) and Lemma 4.6 imply

2 u(y)+(ci-2 (_lpl2)u(y)= _ c { 2 f ( y ) , 0 < y < h , (4.15) ~,y u(y)+(c2 z ff-[p[Z) u(y)= -c~2 f(y), y>h.

To construct the solutions of these equations let

(4.16) ~/= ]//ci- 2 ( - - Ipl 2 , ~=~/c2 2 ( - I p l 2

denote the analytic functions in the (-plane cut from c 2 Ip[ 2 to + ~ (for q) and from c 2 Ip[ 2 to + co (for ~) which satisfy

(4.17) I m r / > 0 , I m r foral l ( e C .

Note that (4.17) is consistent with the convention of Section 3 for real values of if the upper edges of the branch cuts are included in the domains of definition

of r /and 3. The method of variation of parameters may be used to construct the solution of (4.15). For simplicity it will be assumed that s u p p f is compact. Then the general solution of (4.15) can be written

y

Bt ei,,r+B2e_i,,y - 1 ~of(Y,)sinrl(y_y,)dy,, 0 < y < h , (4.18) u(y)= r t c~

[ Cl eicY nt_ C2 e_i~y l ' c2 ~ f(y ')sinr y>h.

where B1, B2, C t and C 2 are independent of y. Note that if Im (4=0 then Im ( > 0 and hence the function e-i~ r is exponentially large when y --. + or. Thus u e Yg (R+) if and only if C 2 =0. Also the boundary condition u(0)=0 implies that B 1 + B 2 = 0 or B 2 = - B 1. The remaining constants, B 1 and C1, are determined by the interface conditions

(4.19) u(h+)=u(h-) , 1 Dru(h+)= 1 Dru(h_). P2 ,01

These conditions are equivalent to the equations

(4.20)

where

[ e i~h C1-(2i sin r / h ) B l = g 1

i ~ i~h 1 . I ~ e C l - ~ ( 2 ' r l c ~

" 1 h , h t �9 t t 1 ! r [ g l = ~ X - S f(Y ) sin ~ ( h - y ) d y -~-TC-._ ~ f(Y ) cos ~l (h-y ' )dy

C 2 g ~c C 1 ~ ] 0

(4.21) / 1 h | h

| g 2 = 2 ~f(y')cos?,(h-y')dy' 1 ~ f ( y , ) c o s r l ( h _ y , ) d y , . t P2 c2 ~ Pl cl o

Equations (4.20) are a linear system for C1, B 1. The determinant is

(4.22) d(~'P)=-2ieiCh {~l C~ i?"


Note that the equation A ((, p) = 0 is equivalent to the equation 7 -- 0 of Section 3 (see (3.19)). Hence the results of Section 3 imply

Lemma 4.7. I f c a > c 2 then A ((, p) has no real zeros. I f c 1 < c 2 the real zeros o f A((, p) lie in the interval c 2 [pl 2 <2 < c 2 Ipl 2 at the points 2a(lp[), 22 (rpl) . . . . . 2mv)(lpl).

If A (~, p) ~= 0 then (4.20), (4.21) determine unique coefficients Ca, B~. Substituting these into (4.18) with C2=0 and B2= - B 1 gives u(y)=R;(Ap) f . It is easy to see that it has the form

(4.23) u(y)= ~ G~(y, y', p) f ( y ' )dy ' R +

where G;(y, y', p) is the resolvent kernel, or Green's function, for Ap. A straight- forward calculation gives an explicit formula for G~(y,y',p). The following notation will be used:

1 1 i~ (4.24) A o (~, P) = cos ~/h - - - sin r/h

Pl P2 q

(4.25) Z(,,b) (Y)= characteristic function of (a, b).

With these conventions the Green's function can be written

1 sin q y c- 2 (y,) p - 1 (y,) G~(y, y', p) = Ao (~, P) '7

.~Z,O,h)(y,)[COS~l(y, h)+ Pl i~ (4.26) P2 ~/

1 - Z~0,y) (Y') ~ sin r / (y - y'),

car/

and

G~ (y, y', p) =

sin ~l (y' -- h) ) + Z(h, ~) (y') ei~(r'- h) t

O < y < h ,

1 ei~r h) C- 2 (y,) f l - 1 (y') A o (~, P)

f , sinqy' , ( s i n r lhcos~(y ' -h ) (4.27) "~Z~o,n)(Y ) - + Z~h ~)(Y ) I

q , \ rl

1 +Z~y,~)(y')c~2~ s i n ~ ( y - y ' ) , y > h .

The selfadjoint operator Ap on ~ ( R + ) has a representation

Ct3

(4.28) A, = ~ 2 dFl, (2), 0

where Hp(2) is a spectral family of orthogonal projections in ~ (R+) . G~(y,y',p) can be used to derive an explicit construction for Hp (2). The derivation is outlined here as an aid for the more complicated derivation of the spectral family of A in Section 5. The construction of Hp(2) is based on the well-known theorem of

276 C. WILCOX

.rlpl I

Fig. 1. The C o n t o u r F(a, b, (r, 6).

Stone: lim - 1 b

, ,40+ 7z i S ( [ R ~ + i , , ( A v ) - R ~ - i , , ( A p ) ] f , g) d2 (4.29) a

= ([Flp (b) + Flp (b - ) - Flp (a) - Flp (a - )] f g).

The left-hand side of (4.29) can be regarded as a contour integral in the ~-plane. It can be evaluated by representing R~(Ap) by means of G;(y, y', p) and using Cauchy's theorem. To this end, assume that a < 21(Ipl), b > a and

b ~ . R - {21(]pD, 22@1) . . . . ,2mp)([pD}

and consider the contour F(a , b, a, 6) indicated in Figure 1. The figure illustrates the case where q < c 2 and b > c ~ [pl 2. If b < c ] [pl 2 the contour

b - i 6 --, cZ21p i2 - 6 - i (5 --, c~ ]p [ 2 - fi + i fi --, b + i 6

should be replaced by the segment b - i 6 ~ b + i b. The integral

(4.30) ~ ( R ~ ( A p ) f , g ) d ~ F(a,b,a,,~)

can be evaluated by determining the singularities of the integrand, using (4.26) and (4.27), and then applying Cauchy's theorem. Note that ifa is small enough then A o (~, p) has no zeros inside F(a , b, a, 6) except for those points 2 l([pl), 22 (IPl), ..., )~N(p)(IpD (if any) that lie inside F(a , b, a, 3). Note also that the values o f t / and ~ on opposite edges of their branch cuts differ only in sign. Hence if F(z) is an entire analytic function which is even then F(t/) and F(~) are analytic functions of in the entire (uncut) ~-plane. By the same argument A o (~, p) is analytic on the Rie- mann surface for ~ (with branch cut from c 2 [pl 2 to + oo). These remarks show that G; (y, y', p) has two kinds of singularities near the real axis:

(a) Poles at the real zeros of A o (~, p). (b) A branch cut from c 2 Jpl 2 to + o0.

Application of Cauchy's theorem to (4.30) gives

(4.31) ~ ( R ~ ( A p ) f , g ) d ~ = - 2 r H Z Res ( R ~ ( A p ) f , g ) , F(a,b,a, ~) ~Mlpl) < b ~'= 2k(lPD

Spectral Analysis of Wave Propagat ion 277

where the summation extends over the values of k for which s On the other hand

b

j" (R; (Ap) f, g) d~ r = - S (ERA + i~ (Ap)- R z_,~ (Ap)] f, g) d2 F(a,b,a ,6) a

b

+ H ( b _ c 2 [p[2) f ([Ra+,,~(Ap)-ga-i~(Ap)]f'g)d2+~ c:~lpl ~

where H(2) is the Heaviside function and 0 (1) represents the integral of (R; (Ap)f, g) along the segments

a + i a - - , a - i a , c ~ l p [ 2 - 6 - i 6 ~ c 2 [ p [ 2 - 5 + i 6 , b - i a ~ b - i 6 and

b+i6- -*b+ia .

Using (4.26), (4.27) we easily verify that these terms tend to zero with 8 and a. Combining (4.31) and (4.32), and making 6 ~ 0 and then o---* 0, gives

b

lim ~ ([Rx+~(Ap)-Ra_~(Ap)] f, g) dR ~ 0 +

N (p)

(4.33) = 27r ik~=tH(b-2k(lpl))~=R~pl)(R~(Ap)f, g)

b

+ H ( b - c 2 tp[ 2) j" ([Rz+,o(Ap)-Ra_,o(Ap)]fg)d2.

Thus if we put u~ = R~ (Ap)f, Stone's theorem and (4.33) imply

([Hp (b) + lip (b - ) - Hp (a) - llp(a - )] f, g) N(p)

(4.34) = - 2 ~ H(b-Zk(IPl)) ~ R~pl)(u;, g) k = l =

1 b .H(b -c~Ip l 2) ~ (Ua+,o-ua-io,g) d2"

7"/: l c21pl 2

Since get'if(R+) is arbitrary in (4.34), it follows that

[Hp (b) + Flp (b - ) - I Ip (a) - lip (a - )] f N(p)

(4.35) = --2k ~= 1 H(b--)'k([Pl)) ; Re~pbur

1 b +--~H(b -c~ lP l 2) ~ (Ua+io-U;~_io)d2.

7Z I @lplZ

The terms in (4.35) will now be calculated from (4.26) and (4.27). The residue of Gr (y, y', p) at ~ = 2 k (IPl) can be calculated by the formula

(4.36) Res Gr lim (~--2k(IPl)) ar [.= Ak([p[) ;~Ak(lp[)

278 C. WILCOX

if the pole ~k (IP [) is simple. The poles occur at the zeros of A o ((, p). The simplicity of these zeros follows from

Lemma 4.8. Let I PI# Pk. Then (k-�89 7z < h rlk (IP I)< k rc for k = 1, 2 . . . . . N(p) and

(4.37) d ~= Ak(Ipl) d ( A ~

1 {c h q + p~ c~ s in 2_t/_h.~ . -2plc2r12 ost /h sint/h p~ c~ costlh )~=z~{lpl)

(4.38)

then

In particular, this quantity is real and positive when k is even and real and negative when k is odd.

Equation (3.26) implies that rlk(Ipl ) lies between (k-�89 and k re. Formula (4.37) follows by direct calculation from (4.24) and (3.20).

The residues (4.36) can now be calculated from (4.26), (4.27). Recall that F(~/) 1

and F(~) are entire functions of ( if F(z) is entire and even. Moreover, - - sin r/h = P2 c o s q h r/

when ( = 2 k (Ip [). It follows that if

(4.39)

d ~= zkr #k(P)=-d~- Ao(~, P)

1 f , s int/y ' Res G;(y, y',p)= p~-~ [Z(O,h)(y )

~= ,t~(Ipl) s inqh

+Z~h,~)(Y') ei~r sin ~/y c_2(y,) p-l(y,) ) I"/ ~= zk(Ipl)

for 0 < y < h and

(4.40)

,, 1 f , Res G~ (y, y p) = ~ ~)~(0, h)(Y ) sin q y' ~ : zk(Ivl) I"/

sin ~/h_ eir eir c Z(y,) p- l (y,) -t- Z(h, ~) (y ') r/ J ~ = ~ktlpl)

for y > h. Now, by (4.23)

(4.41) Res #;(y)= ~ Res G~(y,y',p)f(y')dy'. ~= ),k([Pl) R+ ~= Zk(IPl)

If this is combined with (4.39), (4.40) it is found that the residue of u~ (y) at ~ = 2k(lP]) can be expressed in terms of the reduced eigenfunction Ok (Y, P) of Section 3. The result is

Res ur (y) = Ok(Y, P) ~ Ok (Y', P) f (Y') C- 2 (y,) p- l (y,) d y' ~= ,~k0pl) " ak(P) 2 Pk(P) ~/k(IPl) sin qk(lPl) h R+

for all y > 0.


The coefficients ak (P) have not yet been defined. They will be chosen so that the coefficient in (4.42) is - 1; thus

- 1 (4.43) ak(P) 2 =

lAg(P) qk([Pl) sin t/k ([p [ ) h"

Note that Lemma 4.8 implies that this quantity is always real and positive. This choice of a k (p) normalizes 0k (Y, P) in ~ (R+) . Note also that Ok (Y, P) is real-valued. Thus (4.42) can be written

(4.44) Res u~ (y) = - (Ok (' , P), fLe(,+) Ok (Y, P)" ~= 2k(lpl)

If we return to equation (4.35) for/ / ,(2), the term uz+~o-ux_ io must be calculated for 2 > c~ Ip[ 2. Recall that

lim ~= +(c~ 2 2-1pl2) 1/2 ~ ) . •

where the square root is positive when 2>c22 ]pl 2. It follows from (4.26), (4.27), the definition of 0o (Y, P, 2), and equation (3.34), that

i Gz+io (y, y', p ) - i Gx_io(y, y', p)= 2 Re {i Gz+io(y, y', p)}

(4.45) _ - 2p2 0o(Y, P, 2) 0o(Y', P, 2) c- 2(y,) p-l(y,) ( \sin 2 t/h +

for all y, y' in R+. Now, by (4.23)

(4.46) ua+ io (Y)- ua-io (Y) = S {Gx+ ,o (Y, Y', P) - Gz_ ,o (Y, Y', P)] f(Y') dy'. R+

Combining this and (4.45) gives

1 (u~+io(y)_u~_io(Y)) = P2 (4.47) 27ti ( p~ ~/2 ) ~ sin2~/h-~ pxZ ~2 c~ h a(P, 2)2

�9 0o (Y, P, 2) J 0o (Y', P, 2) f(x') c- 2 (y,) p-X (y,) dy'. R+

The coefficient a (p, 2) has not yet been defined. It will be chosen so that the coefficient in (4.47) is 1; thus

(4.48) a (p, 2) 2 - P2 rc ~ (sin 2 q h + p!p~2 ~-22 cos2 t/h)"

With this choice (4.47) can be written

1 (uz+io(y)_ux_io(y))_(Oo(., P, 2),f)ae~R+)0o(Y, P, 2) (4.49) 2 rc i

because r (Y, P, 2) is real-valued.

2 8 0 C . W I L C O X

Combining (4.35), (4.44) and (4.49) gives the relation

[IIp (b) + Hp (b - ) - H v (a) - Hp (a - )] f(y)

N(p)

(4.50) = 2 ~ H(b - )~k ([P [))(Ok (*, P), f ) Ok (Y, P) k = l

b

+2H(b-c~[pl 2) (. (Oo(.,p, 2),f)Oo(y,p, 2)d2 c~lpl 2

which is valid for a<21(Ip]) , b>a and beR-{Al(lpD, 22(Ip]) . . . . . 2u<;)([p]) }. The spectrum of Ap may be deduced from this as follows.

1. Take a < b < 21 (IP I). Then the right-hand side of (4.50) is zero. Thus

(4.5i) / / , (2 )=0 for 2<21([p1).

2. Take 2j ([p I) < b < 2j+ 1 ([P I) for j = 1, 2 . . . . . N(p). Then the right-hand of (4.50) is continuous. Thus the point spectrum of Ap is the set {21 ([p]), 2z([pl),... ,2N(p)(Ip])}.

3. Take a<21(lp[), beR-{21(lpl),~.2(Ip]),...,ANtp)(Ipl)} in (4.50) and use 1. and 2. This gives

N(p)

IIp (b) f(y) =- Z H(b - 2 k (IP[))(Ok (', P), f ) Ok (Y, P) k = l

(4.52) b

+H(b-cZlP[ z) [. (Oo(',P, 2),U)Oo(Y,P, 2)d2. c21pl 2

Moreover, this formula extends by continuity to all values of b, since H(,~)= 1 for 2>0.

4. Equation (4.52) implies that Ap has continuous spectrum in the interval [c z Ipl 2, oo).

An eigenfunction expansion for Ap is easily derived from (4.52). It will be written down here because of its close relation to the expansion for A which is derived in Section 5 below. Making b ~ + oe in (4.52) gives

N (p)

(4.53) /(Y)= ~ (Ok(',P),U)Ok(Y,P)+ ~ (Oo(',P, 2),f)Oo(Y,P, 2))d,~., k = 1 c 2 IV[ 2

where the integral converges in ~ (R+) rather than point-wise. The corresponding Parseval formula follows from (4.52) by taking inner products. Thus

(4.54)

In particular,

(4.55)

N(p)

IlIIp(b)f[I 2= ~ H(b--2k([P[))l(ok(',P),f)[ 2 k = l

b

+ H ( b _ c 2 [p[2) ~ [(Oo(.,p, 2),f)12d,~. c2[pl 2

N(p)

Ilfll2= }-" ](Ok(.,p),f)[2 + S ](Oo(',P,2),f)lZ d2. k = 1 ~ rpl ~


Equations (4.52)-(4.55) apply when c 1 < c 2. When c 1 ~_~C 2 the point spectrum of Ap is empty and the sums in (4.52)-(4.55) are absent.

The derivation given above is applicable to functions f eW(R+) having compact support. The results can be extended to all fE 3r (R+) by a limiting procedure. In the general case,

M

(4.56) (~o (', P, 2), f ) = lira| o~ ~o (Y, P, 2) f(y) c- 2 (y) p- ~(y) dy

where the convergence is in L 2 ([c~ f p I z, ~)). Full proofs of these facts are omitted because they are not used in the discussion of A in Section 5. The analogous facts for A are proved in Section 5.

5. The Spectral Family of A

The purpose of this section is to construct the spectral family/7 (2) of A. The construction is based on the improper eigenfunctions of A

(5.1) I/Jo(X,y,p, 2)=(2rQ(-n+l)/2 eip'X~o(y,p, 2 ), (p, 2)Ef2 o,

(5.2) ~k(X,y,p)=(2n)(--"+l)/Z elP'X~kk(y,p), peg2 k, k = l , 2 , . . . ,

where ~ko(y, p, 2) is defined by (3.34) and (4.48), Ok(Y,P) is defined by (3.37) and (4.43),

(5.3) g2o = {(p,2): 2>c~ Iv12},

and

(5.4) f2~ = {p: IPt > Pk}.

The spectral family will be constructed by means of a set of "transforms" of functions f e ~ . Formally the transforms are the ~r products o f f with the eigenfunctions:

(5.5) J~(p, 2)= IOo(X,y,p, 2 ) f (x ,y )c -2(y )p- ' (y )dxdy , (p, 2)EO o,

and

(5.6) j~(p)= ~k (x , y , p ) f ( x , y ) c -2 ( y )p - ' ( y )dxdy , peg2 k, k = l , 2 . . . . .

Of course these expressions are only formal, since the eigenfunctions are not in ~ , and it will be necessary to interpret them as limits as in the Plancherel theory. Recall that the eigenfunctions (5.1), (5.2) are, by construction, bounded functions of (x, y)ER~+ for fixed values of p, 2 and k. Thus if f EL 1 (R~) then the integrals in (5.5), (5.6) converge absolutely for all p, 2 and k and the functions fo(P, 2), fk(P) satisfy

Theorem 5.1. If fELl(R" + ) then fke C (~'~k) for k =0, 1, 2 , . . . .

282 C. WILCOX

Proof. Consider first the case k = 0. Equations (3.34), (3.35) imply that

/ Pz (5.7) [0o (x, y, p, 2)1 <(2n) (-"+ 1)/2 a(p, 2) [1 +

for all (x, y)eR"+ and all (p, 2)e~o. Moreover, the bound in (5.7) is a continuous function on f2 o. To prove that fo e C(f2o) let (Po, 2o)eI2o and let N(Po, 20,6) be a neighborhood of (Po, 20) whose closure is contained in f2 o . Then (5.7) implies that 0o(x, y, p, 2) is bounded on the set R"~ x N(Po, 20,6). It follows that fo(P, 2)~ fo (Po, 2o) when (p, 2 ) ~ (Po, 2o), by Lebesgue's dominated convergence theorem. For the case k > 1, (3.37) implies that

(5.8) [Ok( x, Y, P)I ~ (2 7Z) (-n+ 1)/2 ak(P)

for all (x, y)eR+ and all p e r k. Moreover, ak(P ) is continuous on Qk and hence the continuity follows as in the case k = 0.

Now let f e J f and define

(5.9) fM(x,y)=~f(x_,y) if Ixl_<_M and O<y<M, t o if Ixl> M or y> M.

Clearly fM e. ~f~ Ch L~ (R+) and hence (fM)~ ~- C (Ok) for k = 0, 1, 2 . . . . . Moreover,

(5.10) lira fM=f in ~ . M~oo

The transforms )~ of arbitrary functions f e ~ will be defined by means of

Theorem 5.2. For all f~ovf we have (fM)k~.L2(~k) for k=0, 1, 2, . . . . Moreover, the limits

(5.11) 3~ = L2 (t20-1im (fM)k M~oo

exist for k = O, 1, 2, ....

Theorem 5.2 assigns to each feovf a family of transforms fkeL 2 (t2k). Written in extenso the definitions are

M

(5.12) fo(P, 2)=L2(f2o)-lim j j Oo(x,y,p, 2) f (x ,y)c-a(y)p- l (y)dydy M~oo 0 Ixl<=M

and M

(5.13) j~(p)=L2(f2~)~m ~ ~ Ok(X,y,p)f(x,y)c-2(y)p-X(y)dxdy 0 Ix[<=M

for k = 1, 2, . . . . These functions can be used to construct the spectral family H(2). The construction is described by

Theorem 5.3. For all f, ge~r ~ and all I~eR the spectral family for A satisfies

n (#) f g) = j H ( # - 21fo (P, 2) go (P,)~1 dp d2 (5.14) ao

+ ~ jn(#--2k([Pl))fk(P)gk(P)dP, k = l ~k


and hence

_ _ 2 (5.15) [If[I 2 - ~ IFfkPIL2~,) for all f~9~. k=O

Theorems 5.2 and 5.3 are the principal results of this section. They are formulated for the case where c 1 < c z and the eigenfunctions ~k k with k > 1 exist. In the simpler case where c 1 > c 2 only the eigenfunctions ~'0 occur and the terms in (5.11), (5.14) and (5.15) with k > 1 should be omitted.

Note that the series in (5.14) is actually finite because 2k(IPl)> # for all pet2, when k is large enough. However, the series in (5.15) is infinite when c 1 < c 2 and Theorem 5.3 guarantees its convergence. Note also that Theorem 5.2 provides information about the behavior of ~ (p, 2) near 0f20. The function a(p, 2) is un- bounded in any neighborhood of the set {(p, 2): [p[ =Pk and 2 = c 2 p~}, see (4.48), and hence the square integrability ofj~ on I2 o is not immediately evident.

The proofs of Theorems 5.2 and 5.3 will be developed in a series of lemmas. The method of proof will be to verify the assertions of the theorems for functions f in certain dense subsets of W and then to extend the results by closure, using (5.15). The proofs will be given for the case c 1 < c 2 only. The modifications needed for the simpler case where c 1 > c 2 will be obvious.

The following sets of functions will be used in the proofs of Theorems 5.2 and 5.3.

(5.16) ~vo~ = oeg n { f : supp f is compact}.

(5.17) ~ (R+) = {u: u (x, y) = v (x, y)[R~ where v E ~ (R")}.

(5.18) ~ , = ~ - 1 ~(R--~+) = {/: f = ~ f ~ (~'a+)}.

In (5.18) ~ is the unitary operator in W defined by the Fourier transform (4.1). Note that each of these three sets is contained in LI(R~). Hence the transforms fk are defined by (5.5), (5.6) for these sets and fkEC(f2,) for k=0 , 1,2 . . . . by Lemma 5.1. The following alternative characterization of the transforms will be used below.

Lemma 5.4. I f f is in ~vox or ~ ' then

fo (P, 2) = ~ ~k o (y, p, 2) f(p, y) c- 2 (y) p-~ (y) dy R+

(5.19)

and

(5.20) fk (P) = ~ Ok (Y, P) f(P, Y) C- 2 (y) p - l(y) dy, k = 1, 2 . . . . . R+

Proof. Note that in both cases the integrands in (5.5), (5.6) have supports in a set of the form {(x, y): xER "-1 and O < y < M } and are integrable on this set. Hence (5.19), (5.20) follow from (5.5), (5.6) by Fubini's theorem and the definition (4.1) off(p, y).

The proofs of Theorems 5.2 and 5.3 will be based on Lemmas 5.5, 5.6 and 5.7 below which assert that (5.14) and (5.15) are valid for three different subsets of ~ . To state them let

(5.21) N(/3, 6)= {peR"-1: IP-P[ < 6},

2 8 4 C . W I L C O X

and let

(5.22) 7zl: R+ ~ R "-I

be the projection defined by

(5.23) n~(p, y )=p.

Lemma 5.5. Let p ~R " - i - ~ = l {P: IPI=Pk} �9 Then there exists a 6 = 6 ~ ) > 0 such that (5.14) and (5.15) hold for aU f and g in ~,uf, such that

(5.24) ~z~ (supp f ) c N ~ , 6).

In particular, i f f e ~ ' satisfies (5.24) then j~ELz(Qk) , k = 0 , 1, 2, . . . .

Lemma 5.6. Equations (5.14) and (5.15) hold for all f and g in 9r ~' such that

(5.25) z h ( s u p p f ) ~ U {p: Ipl----pk}=O- k = l

In particular, if f e ~ ' satisfies (5.25), then fkEL2 (f2k) , for k =0, 1, 2, . . . .

Lemma 5.7. I f f c jfvox then fk E L 2 (Ok) for k = O, 1,2, ... . Moreover, (5.14) and (5.15) hold for all f and g in ~vox.

The remainder of this section is organized as follows. First, Theorems 5.2 and 5.3 will be shown to follow from Lemma 5.7. Then Lemma 5.5 will be proved. The proof of this lemma contains the main portion of the proof of Theorems 5.2 and 5.3. Next, Lemma 5.6 will be derived from Lemma 5.5. Finally, Lemma 5.7 will be derived from Lemma 5.6.

Proof of Theorem 5.2. Obviously f M E ~ v~ and hence (fM)keL2(~'2k) for k=0 , 1, 2 . . . . by Lemma 5.7. Moreover, Lemma 5.7 implies that (5.15) holds with fM--f~r in place off , where M and M' are any positive constants. Now {fM} is a Cauchy sequence in ~ by (5.10). Hence {(fM)k} is a Cauchy sequence in L2 ((2k) for k=0 , 1, 2 . . . . . and (5.11) follows from the completeness of L 2 (~2k).

Proof of Theorem 5.3. Lemma 5.7 implies that (5.14) holds with fM, gM in place of f, g. Making M ~ ~ and using (5.10) and (5.11) gives (5.14) for arbitrary f, g in ~ . Finally, setting g - - f in (5.14) and making # ~ ~ gives (5.15).

The proof of Lemma 5.5 will be developed in a series of lemmas. The first is

Lemma 5.8. Let f E Jt ~' and write u; = R~ (A) f so that ~ = R~ (]1)f for all ~ ~ C - R . Then

(5.26) h~(p, y)= ~ G~(y, y', p) f(p, y') dy' R +

for all (p, y)~R+ where G~(y, y',p) is the Green's function defined by (4.26), (4.27).

Proof. It is evident__ from (4.26) and (4.27) that (5.26) with f e ~ ( ~ + ) defines a function of (p, y)e R+. To prove the lemma it is enough to show that this function . . A

is m D(A) and satisfies

(5.27) (i] - ( ) h~(p, y )=f (p , y)


for all (p, y)eR"_. The characterization of D(A) given by Lemma 4.3 will be used. Now (5.26), (4.26) and (4.27) imply that

C 1 sin r / y - sin~l(y-y')f(p,y')dy', 0 < y < h ,

(5.28) h~(p, y)=

IC2e'e~'+ l~-~ ~sin~(y-y')f(p,y')dy', y>h, t C2~ y

where C 1 and C 2 are independent of y. It follows that fi~eC2(t2a)~)C2(f22) and

.~_xz_lple ) ~ f(P,Y) 0 < y < h ,

(5.29) f (P, Y)

( i Ipl2) ~ c 2 7 - , y>h.

Moreover, the integral (5.26) satisfies

1 Orgt~(p,h+)=+DrCt~(p,h_ ) (5.30) r]~(p, 0)=0, ~r P2

by the construction given in Section 4. Also, note that if

s u p p f c {(p, y): [p[<M and O<y<M} then

supp ~ c {(p, y): IPl < M and y > 0}.

Moreover, (4.27) implies that

(5.31) [Dr162162 j = 0 , 1,2,

where C 3 is a suitable constant. It follows that p'Dr ~r for all and j = 0, 1, 2. In addition, the condition

p'~;(p,h+)=p'~r - ) and p~Dy~l~Lz(O1)(~)L2(f22)

imply that p~ Dr fi~e~, where Dy~ denotes the distributional derivative. This completes the proof that fit e D (,4). Finally, (5.29) is equivalent to (5.27) for functions ~ D ( A ) . This completes the proof of Lemma 5.8.

The next step in the proof of Lemma 5.5 is the choice of the constant 6 = 6 (/3). Note that if IPl ~=Pk for k = 1, 2, 3 . . . . then

(5.32) 21 (IP[) < 22 (IP l) < " : < 2rap)(IPl)< c 2 I pl 2.

Moreover, N(p) is constant for all peN(~, fi) if6 is small enough. Also, the functions 2j ([P 1) and cz z [P 12 are continuous. Hence there is a 6 = 6 (/5)> 0 such that the image sets

(5.33) 2x(IN(/3, 6)1), 22 (IN(/3, 6)1), ..., 2m#)(IN(/3, 6)1), c 2 Ig(~, 6)12

are disjoint in R. Thus, there exist disjoint neighborhoods

(5.34) Nj = N(2j (IP I), 6j) c R, j = 1, 2 . . . . , N(/3)

286 C. WILCOX

such that

(5.35) 2j(]NO, 6)[)~ N i, j = 1, 2 . . . . , N(p).

(The bar in (5.35) denotes the closure of the set.) Moreover, it can be assumed that

(5.36) NN(p) c~ c 2 IN(/3, 5)12 = 0.

In the remainder of the proof of Lemma 5.5 it will be assumed that 6 = 6 (13)> 0 has been chosen so that (5.35) and (5.36) hold. The next step in the proof of Lemma 5.5 is formulated as

Lemma 5.9. Let f and g satisfy the hypotheses of Lemma 5,4 with 6=5(/3) chosen as above, and let ~ = ) , + i a with a +O. Then for any # e R

,u

(g, [R~(A)- R~(A)] f ) d2 --1

(5.37)

= ~ ~ ~g(P,Y)[f i~(P,Y)- f i ((P,Y)]C-2(Y)P-I(y)dyd2dp �9 N(p,~) -1 R+

Proof. Parseval's theory and Lemma 5.8 imply

(g, R~ (A) f ) = (~, Re (-d) f ) = (~, fit) (5.38)

= ~ ~g(P, Yi f ir N(p,6) R§

because s_upp fi~(p, y )~ N05 , 6) x R+. Equation (5.37) follows by writing (5.38) for ~ and ~, subtracting and integrating with respect to 2 from - 1 to #, and using Fubini's theorem.

The left-hand side of (5.37) has the limit (g, /-/ (/z) f ) when o-~0. Equation (5.14) will be verified, under the hypotheses of Lemma 5.5, by calculating the limit of the right-hand side of (5,37). It will be convenient to use the notation

(5.39) F(p,)~,a)= ~(p ,y ) [ f i~ (p ,y ) - f i~ (p ,y ) ]c -2 (y )p - l (y )dy . R+

Note that the results of Section 4 and the resolvent identity imply that

F (p, 2, a) = (~ (p,'), [R~ (Ap)- R~ (Av)] f(p,'))~e(R+)

(5.40) = (~ -- ~)(~ (p, "), R: (A p) R~ (Ap) f(p," ))ae (n+)

= (~ - O(R~ (Ap) ~ (p,.), R~ (Ap) f(p, "))ar(R +)-

The following notations will also be used:

(5.41) Nj(#)= Nj c~ ( - 1, #), j = l , 2 .. . . , N(/5), N(#)

(5.42) N ( # ) = ( - 1, # ) - (.J Nj(/~), j=l

(5.431 I(p, #, a )= ~ F(p, 2, a) d2, - 1


(5.44) Ij(p,p, tr)= ~ F(p, 2, a)d2, j = l , 2 . . . . ,N(/3), Ns (u)

(5.45) i(p,p,~r)= ~ F(p, 2, cr)d2. NOt)

These definitions imply that

(5.46) I(p, #, or)= ~.. lj(p, p, a) + i (p, I.t, or). j = l

The limit as r ~ 0 of the right-hand side of (5.37) will be calculated by means of the next two lemmas.

Lemma 5.10. Under the hypotheses of Lemma 5.9,

(5.47) lim F(p, 2, or) = 2 ~ i g,o (P, ~ fo (iv, 2) a~O+

uniformly for (p, 2)eN(p, 6) x _N (#); hence

(5.48) lim/~(p, kt,~r)=2~ti S H ( p - 2 ) ~ o ( P , 2)fo(p, 2)d2 a~O+ c2lpl:

uniformly for p~N(~, 6).

Proof. Formulas (4.26) and (4.27) imply that Gz~ir y', p) is continuous for y e R+, y'e R---+, pc N(/3, 6), 2~ N(bt) and 0 < tr < ao, if o" o is small enough. Note that the last three conditions imply that Ao(P, 2+ia)4=0. It follows that F(p, 2, a) is continuous on the set N(/3,6)• N(p)• [0, ao]. The calculation of the limit of F(p, 2, a) as ~r ~ 0 + is the same as the one leading to (4.49). This gives

(5.49) lim F(p, 2, tr)=2ni I ~ f o ( p , 2)t[Jo(y,p, 2)c-Z(y)p-l(y)dy, ~r~0+ R+

which by_Lemma 5.4 is the same as (5.47). Since the convergence is uniform on N(/3, 6) x N(#), (5.45) implies

(5.50) l im/(p , tt,~r)=2~zi S go(P, 2) fo(P, 2) d2 a~O+ ~'(u)

uniformly for peN(~, 6). Finally, (c 2 [pl z, p)cN(/~) and the integrand is zero for R < c22 Ipl 2. Hence (5.50) is equivalent to (5.48).

Lemma 5.11. Under the hypotheses of Lemma 5.9 there exists a constant C such that

(5.51) IIj(P,t~,a)[<C for (p, tr)cN(~,c~)• tr0] and j = l, 2, ..., N(/3).

Moreover, for each peN(~, (5)

(5.52) lira Ij(p, it, a)= 2n i H(l~- 2j(lPl)) gj(P) fs(P)" a~O+

288 C. WILCOX

Proof. Equations (5.26), (5.40) and (5.44) imply that

Ij(p,#,cr)= ~ F(p, 2, a)d2 Njtu)

=2 i a ~ (R~+i,(Ap)~(p, "), Ra+,,(Ap)f(p, "))~e(g+)d2 Nj (u)

= 2 i a S S(~G~+i~,(y,y',p)~(p,y')dy') Nj(,a) R+ R+

"( S ~+i~(Y Y ,p) f (p ,y")dy")dyd2. R+

Now the set Nj was defined in such a way that G~ (y, y', p) has exactly one simple pole in Nj at (=)-j([Pl) when peN(p, 6). Thus

H ' , , _ s ( y , y , p , 0 (5.54) G~ (y, y , p ) -

where Hi(y, y', p, 2 + i a) is continuous for all y~.R + , y'e R + , pc N (p, 5), 2e N s and 0-<_ tr_-< %. Moreover,

(5.55) Hj(y, y', p, 2s(]p[) ) = R~spl)G~ (y, y', p)

which was calculated in Section 4, equations (4.39), (4.40). Combining (5.53) and (5.54) gives

2 i a Ij(p, #, (r)= S ~ ( ~ Hj(y,y',p, 2+i(~)~(p,y')dy')

N~(.) ( , ~ - , ~ s ( { p f ) ) 2 + ~ 2 R+ ~+

(5.56) "( ~ Hi(Y' y''' p' 2 +i a)f(p, y") dy") dy d2 R+

=2icr 5 (2_2s([p[))z+a 2 Nj (#)

where (bs(p, 2, a) is continuous for all psN(~, 5), 2eNj and 0 < a < a 0. Let Ms= max {4~s (p, 2, o-)[ on this set. Then

ff (5.57) JI~(p, la, a)] < 2 Msj ( 2 - 2s([pl)) 2 + a z d2 < 2 rc M s

for (p,/~, a) in this set, which proves (5.51) with C=2rr max(M,, ..., MN(p) ). More- over, since 4)s(p, #, a) is continuous at 2=25(1p1), a=0 , a well-known theorem on the Cauchy kernel implies (see, for example, [7, p. 28])

(5.58) lim li(p, I~, a ) = Z x i (Pj(p, 2j(lp]), 0). 0"~0+

Combining this with the definition of 4)j(p, 2, a) in (5.56), with (5.55) and with equations (4.39) and (4.40), gives (5.52).

P r o o f of L e m m a 5.5. C h o o s e 5 = 5 (p )> 0 so that (5.35), (5.36) hold and hence Lemmas 5.9, 5.10, 5.11 are applicable. It follows that

,a \j=I[NU)) ) (5.59) ~(g , [Rx+i . (A) -Rx_ i . (A)] f )d2= ~ | ~ I s ( p , # , o - ) + ~ ( p , # , a ) d p . --1 N(#,6) -


Now A >0. Hence the negative real axis is in the resolvent set of A and the limit of the left-hand side of (5.59) when cr ~ 0 + is

(5.60) rc i(g, [1I (#) + 11 (# - )] f )

by Stone's theorem. The limit of the right-hand side can be evaluated by means of Lemmas 5.10 and 5.11. The result is

2 r c i ( m,~)c~Ipi 2 H ( # - 2 ) ~ ' ~ 1 7 6 (5.61)

N(;~) ) + Z S H(#-2~(IPI) )g~-~f i (p)dp �9

j=l N@,o)

This is a continuous function of #. Hence the equality of (5.60) and (5.61) implies that 11(#) is continuous. Finally, after equating 11(# - ) with 11 (#), the equality of (5.60) and (5.61) is equivalent to (5.14) for functions f, g in ~ ' such that (5.24) holds. This completes the proof of Lemma 5.5.

Proof of Lemma 5.6. (5.25) and Lemma 5.5 imply that each f iEsupp f has a neighborhood N(~, 6(/3)) such that (5.14) holds for functions in ~r which satisfy (5.24). Now n l ( suppf ) is compact. Hence there is a finite collection of such neighborhoods, say N(/3,, 6~), i= 1, 2, ..., K, such that

K (5.62) 7~ l ( suppf ) = U NO, , 6,).

i=1

Let ~K=I q~i(p)-----1 be a partition of unity subordinate to this covering [1]. Then (5.14) holds for f (p , y)= ~b,(p)f(p, y) and ~(p, y). Summing over i then gives (5.14) for f and g. Finally, setting g = f in (5.14) and making # ~ oo gives (5.15); and (5.15) implies that J~L2(~k) for k=0 , l, 2, . . . .

Proof of Lemma 5.7. It is easy to verify that the set of functions f ~ 9r ~' which satisfy (5.25) are dense in ~ . Moreover, if fEovF "~ then there exist sequences of functions f . E ~ ' which satisfy (5.25) such that f ,--*f in ~vt ~ and j~ and f have their supports in a fixed set { (p, y): 0 =< y _-< M}. Recall that iffE ~ ~ then by Lemmas 5.1 and 5.4

fo (P, 2) = S ~k o (x, y, p, 2) f ( x , y) c- 2 (y) p-1 (y) dx dy (5.63) R~

= ~ ~O o (y, p, 2) f (p , y) C -~ (y) p -~ (y) dy R+

is a well-defined continuous function in f2 o . The problem is to prove that f oe L z (f2o). Now (5.13) holds for the functions f, and their differences by Lemma 5.6. It follows that {(f,)o} is a Cauchy sequence in L2(Oo) and hence

(5.64) lim (f,)o = go n ~ o o

exists in L2(f2o). Hence, to prove that foeL2(f2 o) it will suffice to show that fo(p, 2)=go(p, 2) for almost every (p, 2)eO o. To this end let K c f 2 o be compact. Then (5.64) implies that

(5.65) II f - go II L~ ( r~ = ,-o~lim II fo - (f,)o II L2 (K).

290 C. WILCOX

Now, by (5.63) and Lemma 5.4, applied to f , , M

(5.66) fo (P, 2) - (f,)o (P, 2) = ~ ~k o (y, p, 2) If(p, 2) -J~ (p, y)] c- 2 (y) p-1 (y) dy. 0

Hence the estimate (5.7) for ~o implies

. p2rl\ M Ifo(P, 2 ) - (f,)o(P, 2)1 <=a(p, 2) 1 + p ~ - ) o ~ If(P,Y)-f,(P,Y)I c-Z(Y) p-l(y) dy

(5.67) =<a(p, 2) l + p ~ - c-2(y)p-l(y)dy)

/ M \1/2 I(y) dy)

for all (p, 2)Ef2 o. Now a(p, 2) .(l+~-~-)P2 t# is continuous and hence bounded for

(p, 2)eK. Thus there exists a constant C such that M

Ifo(p, =< c If(p, y) -L(p , 2)i 2 c - 2 ( Y ) R - I ( y ) d y ) 12 (5.68)

for all (p, 2)sK. Squaring and integrating over K gives

(5.69) lifo - (f,)o 11~2(~) < C z A II f - f ,N / ,

where A is the maximum of 2 on K. Since f , --+f in Jvg it follows from (5.69) that the limit in (5.65) is zero. Thus Ilfo--gollL2~K)=O for every compact K c f 2 o which implies that fo (P, 2) = go (P, 2) for almost all (p, 2) e ~o. This proves that foe L 2 (f2o). The proof that fkeL 2 (Ok) for k = 1, 2 ... . is entirely similar. Finally, the correctness of (5.14) and (5.15) for all f, g in ~/fvox follows from Lemma 5.6 and the fact that the functions in ovf' which satisfy (5.25) are dense in ~ .

6. The Eigenfunction Expansion for A

In this section the representation of the spectral family of A developed in Section 5 is used to derive an eigenfunction expansion for A. The main result, Theorem 6.12, shows that the improper eigenfunctions constructed in Section 3 form a complete set. Theorem6.13 shows that the eigenfunction expansion provides a spectral representation for A. The section begins with a discussion of the function spaces and operators needed to formulate and prove the expansion theorem.

Theorems 5.2 and 5.3 associate with each f E ~ a sequence (fo,f~,f2, ...) such that each ~ ~ L 2 (Ok) and Parseval's relation holds:

(6.1) I l f l t~= ~ llfkl122.2k) for all fs~vf. k=0

This suggests the introduction of the direct sum of the Hilbert spaces L2(Ok) ,

k = 0, 1, 2 . . . . . It will be denoted by

(6.2) ~ = L2 (~o) | L2 (~21) | Lz (f22) |


Thus ~ is the set of all sequences

(6.3) f =(fo, f l , f2 . . . . ), fkeLz(f2k)

such that

(6.4) ~ Iq fkllZL2(~k) < GO. k=0

is a Hilbert space with inner product

(6.5) ( f g)~ = ~ (fk, gk)L2(ak)" k=0

The subspace of o~r 7~ consisting of those sequences for which only the k th member is not zero will be denoted by ~tVk. Thus

(6.6) ~ = ~ n { f = (0, ..., 0, f , , 0,...)}.

It is clear that ~ is isomorphic to L 2 (Ok). The notation

(6.7) ~ = {)~ (P, Y) = ~k (Y, P) fk (P): fk ~ L2 (I2k)}

will be used for k = 1, 2 . . . . . Note that ~k(Y,P) is defined for pEf2k= {p: [PI>Pk} and y~R+. If the definition is extended by

(6.8) ~9k(y,p)=O for y~R+, [P[<Pk

then q~k(Y, P) is defined on R+ and j~e 9r This follows from Fubini's theorem and the normalization

(6.9) ][~k(', P)[[~(~+) =1 , [P[>Pk.

Moreover, the orthonormality of {ffk(', P)} in Yg(R+) implies

Lemma 6.1. For each k>l , ~ is a closed subspace of .g~ and L(p)--'J~(p, y) is an isomorphism of L2(12k) onto ~ . Moreover, ~ffk-l-~ in ~ for all k ~ l, l> l such that k =~ I.

Parseval's relation (6.1) shows that the correspondence

(6.10) f ~,Cg ~ ~ f =(fo,f~,f2 ... . ),

where J~EL z (Ok) are defined by Theorem 5.2, defines a linear operator

(6.11) ~: ~ J t ~.

The following operators will also be needed. (6.12) ~ : the orthogonal projection of Jg onto ~ .

(6.13)

(6.14)

(6.15)

(6.16)

Jk : the unitary operator from ~ to L 2 (Ok) defined by

J~(o, . . . , o , A , o . . . . )=f~.

292 C. WILCOX

where ~ : ~ ~ ~ is the unitary operator defined by the Fourier transform (4.1). Note that ~k and ~k are the operators whose action is defined by

(6.17) ~k f = ~ k f = J ~ , k=0 , 1, 2, ...

where f = f f f . Parseval's relation (6.1) is equivalent to the statement that $ is an isometric

operator:

(6.18) ll4~fll~= Ilftl~e for all f e ~

or equivalently

(6.19) cb* ~ _-_ 1.

In particular ~b is a partial isometry [2] and it follows that ~ ~* =/3 is the ortho- g onal projection in o@ onto ~ ~ , the range of 4,. It is an interesting fact that 4) ~ coincides with ~ . Indeed, we have

Theorem 6.2. tb is unitary; that is, it satisfies both (6.18), (6.19) and ~ = ~ 7 ~ or, equivalently,

(6.20) $ ~* = 1.

The proof of Theorem 6.2 requires several steps, which will be presented as a series of lemmas.

Lemma 6.3. For all gk~L2 (Ok) , k >1,

(6.21) (~* gk)(P, Y) = ~bk (Y, P) gk (/9).

Hence c~* L 2 (f2k) = ~k , k = 1, k, ... .

Proof. Let f s J r ' so that f = ~ f E ~(R--~+). Then by (6.16) and Lemma 5.4,

(6.22) ~k f (P) = S ~k (Y, P) f (P, Y) C-2 (y) p - l (y) d y . R+

Thus

(6.23) = S g ~ ~ k ( y , p ) f ( P , y ) c - 2 ( y ) p - ~ ( y ) d y d p .

~k R +

Using (6.8) and Fubini's theorem gives

(6.24) (q3* gk, f ) je = ~qjk(y,p) g k ( p ) f ( p , y ) c - 2 ( y ) p - ~ ( y ) d p d y

since qJk(Y, P) is real-valued. This proves (6.21) since ~(R-"+) is dense in ~ .

Lemma 6.4. For all gk~L2(f2k) , k >_l,

(6.25) ~* gk(x, y ) = ~ - l i m ~ ~bk(X,y,p) gk(P)d p. M ~ o o pk<lpl<_ M

Proof. Note that (6.16) implies

(6.26) ~k = ~k ~ and hence ~ ' = ~-* ~3~,.

Hence (6.25) follows from (6.21) and the Plancherel theory.


Lemma 6.5. For k > 1

(6.27) 4k 4k* = ~k ~ ' = 1.

Proof. The relations (6.26) imply the first equation of (6.27) because ~ - * = 1. The second equation follows from (6.21), (6.22) and the fact that ~Ok(', p) is a unit vector in ~ ( R + ) .

Corollary 6.~ For k > 1, 4 k is a partial isometry. Hence

(6.28) ~ = 4" L z ((2k) = H

is a closed subspace and

(6.29) Pk = 4ff 4 k

is the orthogonal projection of ~ onto ~ .

Proof. (6.27) implies that 4~' is a partial isometry. It follows that 4 k = 4** is a partial isometry I-2, p. 258].

Corollary 6.7. For all k, l> 1

(6.30) ~k _1_ ~ for k 4: I.

Proof. Let f = 4 * f k e ~ k , g=4t* g~E~et~t. Then by (6.26)

(6.31) (f, g)ae-- (4~' A, 4~* gt)ae---(~*fk, 43~* gt)ae"

But the last expression is zero by (6.21) and the orthogonality of ~Ok(',p) and ~kt(', p) in ~r +) for almost every pER "-1.

Corollary 6.8. For k > 1

(6.32) 4 k ~ = L 2 (f2k)

and

(6.33) 4k ( ~ @ Jet"k) = {0}.

Proof. (6.32) follows from (6.27), (6.28) and (6.33) is a standard property of a partial isometry with initial set ~k 1-2, p. 258].

Corollaries 6.6 and 6.7 imply that the orthogonal direct sum

(6.34) ~ O) ~2 | Jga @ ...

is a closed linear subspace of ~ . Its orthogonal complement in ~ will be denoted by ~0:

(6.35) ~o = ~ ~ ( ~ @ ~2 0 ) " ").

The orthogonal projection of ~ onto Jego will be denoted by Po:

(6.36) Po = 1 - (/1 + P2 + ' " ) -

The final result needed to prove Theorem 6.2 is

Theorem 6.9. 4 o is a partial isometry with initial set ~o and final set L2(f2o); that is,

(6.37) 4~ 4 o= Po and 4 o 4 ~ = 1 .

294 C. WILCOX

The proof of (6.37) is based on constructions of ~o and 4" similar to those of Lemmas 6.3 and 6.4.

Lemma 6.10. For all goEL2 ((20) we have M

(6.38) q3~'go(p,y)=~et~-lim ~ ~bo(y,p, 2)go(p, 2)d2. M~oo c~[p]Z

Proof. Suppose first that supp go is compact in f2 o and let f~ ~f'. Then f e ~ (~+) and by Lemma 5.4

(t~* go, f)Jr = (go, ~o f)ga(t2o)= (go, 4o f)L=~ao) (6.39)

= S go (P, 2) ~ 0o (Y, P, 2) f(p, y) c- 2 (y) p-a (y) ay ap d2. Do R +

Now ~,o(y,p, 2 ) i s defined for yeR+ and (p, 2)ef2o={(p, 2): )~>c221p12}. If the definition is extended by

(6.40) Oo(y,p, 2 )=0 for yeR+, (p, 2 ) e R + - O o ,

then Fubini's theorem implies

(6.41) (43*go,f) je= ~( ~9o(y,p, 2)go(p, 2 )d2) f (p ,y )c -Z(y )p- l (y )dpdy R~ R+

since supp l a n d supp go are compact and Oo (Y, P, 2) is real-valued. Equation (6.41) implies (6.38) when supp go is compact. To prove the general case let ZM be the characteristic function of the set f2oC~{(p, 2): 2 ~ M } and gO,M=ZMgO. Then supp gO,M is compact and gO,M ~ go in L 2 (Oo) when M ~ oo. Hence (6.38) follows from the special case by the continuity of 4*.

Lemma 6.11. For all goeL2(f2o) M

(6.42) q~* go(x , y )=~- l im ~ ~ ~o(X,y,p,2)go(p,2)dpd2. M ~ 0 c2[p1<1/2

Proof. This follows from (6.38), (6.26) and the Plancherel theory.

Proof of Theorem 6.9. Note that for any f e

(6.43) gb f = (fo ,f~ ,f2 . . . . ) = (40 f, cbt f; ,I) 2 f, ...).

If f=Po fe~fo then f l ~ for k = 1, 2 . . . . ; hence by (6.33) we have fk = ~k f = 0 for k = 1, 2 . . . . . Combining this with (6.1) gives

(6.44) tl~0fllm2~o) = IIP0fll~ for all fe~gr

which proves that 4i o is a partial isometry and cb~' q~o =Po. It follows that 40 Ag o is closed in L 2 (s Hence to prove that q~o ~ = L2 (12o) it is enough to show that q~o ~ contains a dense subset of L2(Do). It will be shown that ~(12o)=Cbo ~ . To this end let f(p, 2)~@(~2o) so that, by Lemma 6.11,

(6.45) ~P* f (x , y) = S Oo (x, y, p, 2) f(p, 2) dp d2. 12o


It is easy to verify that if f e ~(f20) then g (x, y)= ~ f ( x , y)e L 1 (R~+), and hence

(6.46) ~o ~* f(P, 4)= q~o g(P, 2)~ C(f2o)

by Theorem 5.1. Note that (6.45) implies

1 (6.47) crP~f(x, y) (27z)(n- t)/2 11,,-1 c~lpl 2

by Fubini's theorem. Therefore

(6.48) ~(p, y ) = ~ ~ ' f(p, y)= c~ IV[ 2

~9 o (y, p, 2') f(p, 2') d2'.

It follows from (6.46), (6.48) and Lemma 5.4 that

M

4~ o ~* f(p, 4) = 430 ~ (p, 4) = lim S 0o (y, p, 4) ~ (p, y) c- 2 (y) p_X (y) d y M ~ a o 0

(6.49) = lira [ Oo(y,p,2) Oo(y,p,Z)f(p,Z)d2' c-2(y)p-l(y)dy M~oo 6 c~lv[ 2

(; ) = lim ~ Oo(y,p,2)Oo(y,p,Z)c-2(y)p-l(y)dy f(p,Z)dZ. M~o~ ~lvl2

Now a simple but lengthy calculation shows that

M

S Oo (y, p, 4) Oo (y, p, 23 c- ~ (y) p - 1 (y) dy 0

[2 c~p~_z ,~ . P2 t/t/ sin {(~ - ~') = a(p,2) a(p, 2') 1 sin t/h sin q n-t-p2-~cOstlhcostl'h ~-i

+ ~ ~2g~-~ sin q h cos t/' h - cos tl h sin q' h cos {(~- ~ ' ) (M-h)} 2cz Pt ~-~'

(6.50) 1( ) + ~ 2 ~ - ~ s in t lhs inq 'h P~tlq' costlhcosffh s in{(~+~') (M-h)}

l (~sinqhcosq'h+-~cosqhsin~l'h) c~ 2c~ pl 4+4'

=a(p, 4) a(p, 2') F(M, p, 2, 2')

where

(6.51)

(2 \1/2 -,pl 2) ,

[ 2 \1/2

296 C. WILCOX

and F(M, p, 2, 2') is defined by (6.50). Substitute (6.50) into (6.49) and change the integration variable from 2' to 4'. The result is

�9 oq~*f(p, 2)= lim S a(p, 2)a(p, 2')F(m,p, 2,2')f(p, 2')d2' (6.52) oc)

= M~lim ~oa(P,2)a(p,2')F(M,p,2,2')f(p,2')ZcZ~'d~ '.

The function F(M,p,)~, Z) consists of four terms. The contribution of the first term to the limit (6.52) can be evaluated by the well-known formula of Dirichlet:

(6.53) lim ; q~(~') sin m ( ~ - r

M~oo - ~ ~ _ r d ~ ' = ~ ~(~)

which holds for all ~b ~ C(R) (as well as for larger classes). By the Riemann-Lebesgue theorem the remaining three terms of F(M, p, 2, 2') contribute nothing to the limit in (6.52). Thus the limit (6.52) is

1 (sin 2 r/ p2 1,12 h + ~ cos 2 tl h). 2c 2 ~ .f(p, 2). (6.54) q~o ~ f(P'2)=zta(P'2)2 " 2c~ P2 Pl

Combining this with the definition (4.48)of a(p, )0 gives

(6.55) q~o cp~f(p, ).)=f(p, 2) for all (p, 2)~ f2 o .

This shows that fecP o Ag and hence ~(~2o)Cq~ o ~ which completes the proof of Theorem 6.9.

Proof of Theorem 6.2. ~ is an isometry by (6.18), and hence $Ar ~ is a closed subspace of ~ . It remains to show that ~ ~vf= ~ . To prove this it will be enough to show that if f l ~ 3 e f then f = 0 . Now, by (6.43) the vector f=(fo,fl,f2 . . . . ) in

is orthogonal to ~ ~ if and only if

(6.56) ( f S g ) ~ = ~, (fk,~bkg)LZ(r2k)=0 for all gEOfi. k=0

Take g E ~ , so that qi k g=bkt ~z g by (6.33). Then (6.56) implies

(6.57) (f~, q~l g)L~r~,) = 0 for all g c ~ .

This implies that f; = 0 because ~l ~ = Lz (Ol)- Since this holds for l= 0, 1, 2,... it follows that (6.56) implies f = 0 in ~ .

The eigenfunction expansion for A can now be formulated. Recall that ~ has the orthogonal direct sum decomposition

(6.58) 3eg= ~ Ggcgk, k=0

see (6.35). The corresponding representation of the identity is

(6.59) 1 = ~ Pk k=0


where Pk is the orthogonal projection of ~ onto ~k' Moreover, Pk = ~ ~k for k=0 , 1, 2 . . . . . where ~k and ~* are defined by the eigenfunctions by (6.17), (5.12), (5.13) and (6.25), (6.42). The equation

(6.60) 1 = ~ ~)ff ~k k=O

is the eigenfunction expansion in abstract form. It implies that every f e ~ has a representation

(6.61) f= ~ +: +kf = ~ +:fk" k~O k=O

The result may be formulated as follows.

Theorem 6.12. Every f ~ ~ has a representation

(6.62) f(x, y)= ~ fk(X, y) (convergent in ~ ) , k=O

where fk~kk is given by M

(6.63) fo(x,Y)=~M-_li~m ~ ~ S ~o(X,y,P, 2)fo(P, 2)dP d2 c2lpl< V-2

and, for k = 1, 2, ..., by

(6.64) fk (X, y) = ~e-lim S fig (X, y, p) fk (P) dp, M~oo pk<iPl<= M

and where fke L2 (f2k) , k=0 , 1, 2 . . . . , are defined by (5.12) and (5.13).

Conversely, f f=(fO,f l , f2 .... )e~,~ is any vector in ~,~ then (6.62), (6.63), (6.64) define a vector fe~/t ~ such that fk is related to f by (5.12) and (5.13).

Proof. The relations (6.62), (6.63) and (6.64) follow immediately from (6.61), Lemma 6.4 and Lemma 6.11. The converse result follows from Theorem 6.2.

The final result in this section states that the eigenfunction expansion provides a spectral representation for A. It may be formulated as follows.

Theorem 6.13. The unitary operator

(6.65) ~ ~--" ~ ~)~k k=O

defines a spectral representation for A in the sense that for all feD(A) we have

(6.66) ~o Af(p, 2)= 2 ~o f(P, 2) = 2 ~ (p, 2)

and

(6.67) q~kAf(p)=2k(lPl)~f(P)=2k(IPl)~fP), k_>-l.

Proof. Only the proof of (6.67) will be given. The proof of (6.66) is entirely similar. Note first that Lemma 6.3 implies for k > 1

(~k f, g)L2(ak)= (f, ~ g)se = ( f ~ g)ae (6.68)

= y) g(p) dp dy R"+

298 C. WILCOX

for all f e ~ and all geL2(~2k). Now let feD(A) and ge~(f2k). Then ~'k(Y, P) g (p)~.D(/]) and

(6.69)

Now

(6.70) A~gk(Y, P)= )'k(IPl) ~bk(Y, P)

(~k A f, g)L2(ak)= ((A f) A, ~* g)g

= (A~ ~ g)~ = (L A ~* g)~.

'b~' g =

by construction. Thus (6.69) implies

((~)k A f, g)L2(ak)=( f, •k([" [) ~ g))ft* (6.71)

--(2k O" [) ~k J~ g)L~(ak)---- (2k (1"[) qbk f, g)L2 (a~,

which gives (6.67) because ~(~2k) is dense in L 2 ((2k).

Corollary 6.14. {Pk}~ is a complete family of orthogonal projections which reduce A; that is

(6.72) Pk A ~ APk, k = O, t, 2 . . . . .

Proof. By [2, p. 530] the relations (6.72) are equivalent to

(6.73) Pk//(#)=H(#)Pk for k=0 , 1,2, ... and all # e R .

This equation is an immediate consequence of Theorem 5.3, which implies

(6.74) (Pk Fl(#) f,g)=(Fl(#)Pkf, g)= ~ fk(p)~,k(P)dp ~k(iv[)_-< u

for k > 1 and an analogous equation for k = 0.

Corollary 6.15. The subspaces {~fk}~ form a complete family of reducing subspaces for A; that is,

APk=PkAPk=--Ak, k=0, 1,2, ... (6.75)

and

(6.76) A= ~ A k. k=O

This result follows immediately from Corollary6.14; see for example [-5, pp. 301-303]. The operator A k may be called the part of A in ~ .

7. Concluding Remarks

The eigenfunction expansion of Theorems 6.12 and 6.13 can be used to construct functions of the operator A. This property can be used to solve boundary value problems involving A. For example, the wave equation (1.6) can be written

(7.1) D~ u+Au=O.

The solution of this equation with initial values

(7.2) u(O,x ,y )=f (x , y )e~ , Dtu(O,x ,y)=g(x,y)e~


is

(7.3) u (t, x, y) = (COS t A 1/2) f ( x , y) + (A- a/2 sin t A 1/2) g (x, y).

The coefficients of f and g in (7.3) are bounded operators in ~ which can be defined by the spectral theorem. Combining this and Theorems 6.12 and 6.13 gives the representation

(7.4) u (t, x, y) = ~ u k (t, x, y) (convergent in ~f~), k=O

where the functions

(7.5) uk(t , ., " ) ~

are given by

forall tER and k=0 ,1 ,2 . . . . .

M

u 0 (t, x, y) = ~r ~ ~ Oo (x, y, p, 2) M ~ 6 c21Pl<v~

(7.6) . { f o (p ,2 )cos t21 /2+~o(p , 2) sin t 21/2 } 2x/2 - d p d 2

and, for k = 1, 2, 3, . . . , by

u k (t, x, y) = ~ff-lim ~ ffk (X, y, p) (7.7) U~ oo pk< Ip[< M

sin t o9 k (p) �9 {J~(p)cos togk(p)+~(p ) -~-k-~ . j dp .

The relation $* tb = 1 of Theorem 6.2 guarantees that every solution of (7.1) with initial values f, gE3tf is given by (7.4), (7.6), (7.7). The relation $ $ * = 1 of Theorem 6.2 implies that (7.4), (7.6), (7.7) gives a parameterization of the solutions of (7.1) by the elements f = (f0, J], f2 . . . . ) and ~ = (go, gl, g2,--.) of ~7~; that is, each choice o f f , ~3r ~ gives a unique solution of (7.1) in ~ .

The fact that the subspaces W k reduce A implies that the solutions of (7.1) with initial values fk, g k ~ remain in ~k for all t. Moreover, (7.4) is a decomposition of the general solution into such partial waves in ~ . Solutions in ~ w i t h k > 1 may be interpreted as waves which are trapped by total reflection at the interface y = h. Recall that the eigenfunctions ffk (X, y, p) decay exponentially as y ~ + ~ . By contrast solutions in ~o may be interpreted as free waves which ultimately propagate away from the interface. These remarks are developed more fully in [9] where the eigenfunction expansion for A is applied to the study of the propagation of electromagnetic waves along a dielectric-clad conducting plane.

The techniques developed in this paper can be used to construct eigenfunction expansions for other operators with piece-wise constant coefficients. It is evident that if c = c (x,) and p = p (x,) are piece-wise constant and have several discontinuities the same techniques can be used. Another operator that can be treated in this way is

/ c~2u U u \ (7.8) A = . . . , x , ) + . . . )

300 C. WILCOX

where

2 < h 2 ' (7.9) c ( x l , . . . , x , ) = c l ' xE + ' " + x"

I c z, x 2 + . . . + x ~ > h 2.

In this case the improper eigenfunctions involve Bessel functions instead of exponentials. However, the spectral structure of A is qualitatively the same as in the case treated above. Details of this case will be given elsewhere.

This research was sponsored by the Office of Naval Research. Reproduction in whole or part is permitted for any purpose of the United States Government.

References

1. DERHAM, G., Vari6t6s diff6rentiables, 2nd Ed. Paris: Hermann 1960. 2. KA'ro, T., Perturbation Theory for Linear Operators. New York: Springer 1966. 3. LIONS, J.L., & E. MAGENES, Non-Homogeneous Boundary Value Problems and Applications.

New York: Springer 1972. 4. PEKERIS, C. L., Theory of propagation of explosive sound in shallow water. Geol. Soc. Am., Memoir

27 (i 948). 5. RIESZ, F., &: B. Sz.-NAGY, Functional Analysis. New York: Ungar Publishing Co., 1955. 6. SCHWARTZ, L., Th6orie des Distributions, vol. I. Paris: Hermann i950. 7. TITCHMARSH, E.C., Introduction to the Theory of Fourier Integrals. Oxford: University Press 1937. 8. WILCOX, C.H., Initial-boundary value problems for linear hyperbolic partial differential equations

of the second order. Arch. Rational Mech. Anal. 10, 361-400 (1962). 9. WILCOX, C. H., Transient electromagnetic wave propagation in a dielectric waveguide. Proceedings,

Conference on the Mathematical Theory of Electromagnetism, Istituto Nazionale di Alta Matematica, Rome 1974 (to appear).

Department of Mathematics University of Utah

Salt Lake City, Utah

(Received December 30, 1974)

Documents

Spectral analysis of the Pekeris operator in the theory of acoustic wave propagation in shallow water