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The Limiting Absorption Principle for the Elasticity Operator in Homogeneous Anisotropic Media

Alejandro Domínguez-Torres1

Abstract The spectral theory is studied and the principle of limiting absorption is proved for the elasticity operator derived from the wave equations in infinite elastic homogeneous anisotropic media.

The Wave Equation for Elastic Homogeneous Anisotropic Media For a medium obeying the generalized Hooke´s Law, the linear stress-strain relations in rectangular

Cartesian coordinates ; 1,2,3;ix i are [Achenbach p. 52 and Fedorov pp. 8-9]

(1) ; , , , 1,2,3.ij ijlm lmC i j l m

Here ij is the stress tensor, lm

is the strain tensor, and ijlmC are the elastic coefficients satisfying the

symmetry conditions [Achenbach p. 52 and Fedorov pp. 12-15]:

(2) .ijlm jilm ijml lmijC C C C

In this way, only 21 independent constants are involved. Moreover, the convention summation for repeated suffixes is assumed.

The strain tensor may be expressed in terms of the displacement vector ; 1,2,3;iu i by

(3)

1.

2

jiij

j i

uu

x x

In the absence of body forces and considering that the density of the medium is constant and equal to one, the equation of motions are [Fedorov pp. 85-86]

(4) 2 2

2.i m

ijlm

j l

u uC

t x x

It is also assumed that the constants ijlmC have numerical values such that the strain energy W is

positive definite for symmetric stress components ij ji , where:

1 This research paper is derived from the thesis dissertation (Spectral Theory for Elasticity Equations) presented

(August 24th

, 1989) by the author for obtaining the degree of Master of Science (Physics) at the School of Sciences in the Universidad Nacional Autónoma de México. This paper never was published.

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(5) 1

2ijlm ij lmW C .

Solutions of Eq. (4) are the monochromatic plane waves of the form

(6) , i p x tu x t Ae ;

where 31 2 3, ,A A A A is a constant vector, 3

1 2 3, , \ 0p p p p , 2 2 2 2

1 2 3 1p p p p , and

1 1 2 2 3 3p x p x p x p x .

From Eqs. (4) and (6) if follows the relation

(7) 2 0ijlm j l im mC p p u .

Let p be a 3 3 matrix defined by

(8) ijlm j lp C p p .

Therefore, Eq. (7) may be expressed as

(9) 20;p I u .

Eq. (9) is known in literature as “Christoffel Equation”. If 3 \ 0u , then Eq. (9) is equivalent to

(10) , 0p p I .

From the above assumption and derivations, the following Lemma is immediate.

Lemma 1. Matrix p holds the following properties:

i. p is symmetric for 3 \ 0u ;

ii. p is positive definite;

iii. All the roots of , 0p are positive.

The Elasticity Operator and Its Spectral Family Let 2,3 3 3,L be the linear space of all 3 1 column matrix defined on 3 and 3 -valued, such that if

2,3 3 3,u L then u is Lebesgue measurable and square integrable. For 3x and 2,3 3 3,u L ,

let 2,3 3 3,L be equipped with the norm

(11) 3 3

32 22

1n

n

u u x dx u x dx

.

Since Eq. (11) satisfies the parallelogram law, then for 2,3 3 3, ,f g L the scalar product is defined by

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(12) 3 3

3

1

, n nn

f g f x g x dx f x g x dx

Here * denotes the complex conjugate operation. The linear space 2,3 3 3,L equipped with scalar

product (12) defines a Hilbert space which is again denoted as 2,3 3 3,L .

For 2,3 3 3,u L , define the Fourier Transform operator as

(13)

2 2

2,3 3 3 2,3 3 3

32

: , , ;

1ˆ lim .

2

ip x

Rx R

F L L

Fu p u p s u x e dx

Here lims denotes the limit in the strong topology of 2,3 3 3,L . In a similar way, the inverse Fourier

Transform is defined as

(14)

2 2

1 2,3 3 3 2,3 3 3

1

32

: , , ;

1ˆ ˆlim .

2

ip x

Rp R

F L L

F u x s u p e dp

Lemma 2. If 2

2,3 3 3, , ,j i j

u uu L

x x x

, where derivatives are taken in distribution sense, then the

Fourier Transform defined by Eq. (13) holds the following properties

(15) 3 3

2 2 2 2ˆu x u x dx u p dp u p (Parseval Identity);

(16) ˆ; 1,2,3j j

j j

u uF p ip Fu ip u j

x x

;

(17) 2 2

ˆ; , 1,2,3i j i j

i j i j

u uF p p p Fu p p u i j

x x x x

.

On the other hand, let be the operator defined in 2,3 3 3,L by

(18)

2,3 3 3 2,3 3 3, / , ;

.

D f L f L

f p p f p

Lemma 3. Operator defined in Eq. (18) is a self-adjoint operator with respect to the (usual) scalar

product of 2,3 3 3,L .

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Proof. Since the set 3 3 3 3 3 30 0 0, , , ,D C C C D it follows that D

is a dense set in 2,3 3 3,L . The proof will be completed if it is proved that and

, from it follows that .

From Lemma 1 it follows that p is symmetric and positive definite, then for all ,u v D

3 3 3

3 3

†† † †

† †

,

, .

u v p u p v p dp u p p v p dp u p p v p dp

u p p v p dp u p p v p dp u v

This result proves that .

For proving that , let v D ; i.e., 2,3 3 3,v L and , ,u v u , for some

2,3 3 3,L and for all u D . Notice that vector is equal to p v p by definition, then it

follows that

3 3

† †, ,u v u p p v p dp u p p dp u .

Since D is dense in 2,3 3 3,L ; v ; i.e., v D . This proves that if v D , then

v D and v v ; this means that .

Eqs. (8), (18) and (13) to (17) allows to express the action of “elasticity operator” 2

ijlm

j l

H Cx x

as

(19) 1Hu F Fu .

Of course, the domain of elasticity operator is defined by

(20) 2,3 3 3 2,3 3 3ˆ, / ,D H u L u L .

Obviously 2 3 3,D H H , the Sobolev space of all 3 1 column matrixes defined in 2,3 3 3,L

such that the first and second partial derivatives belong to 2,3 3 3,L .

Theorem 4. The elasticity operator H defined in 2,3 3 3,L with domain 2 3 3,H is a self-adjoint

operator with respect to the usual scalar product in 2,3 3 3,L .

Proof. The proof follws immediately if it is noticed that the Fourier Transform F is a unitary mapping

from 2,3 3 3,L to itself and that operator is self-adjoint by Lemma 3.

The plane wave solution (6) for Eqs. (4) generates the following problem for H

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(21) H x x .

Here / 0 and for a given constant vector 3A , x is defined as

(22) 2

; 1ip xx Ae p .

From Lemma 1, it may be considered three values of such that

(23) 1 2 3 .

Moreover, it is to prove that for any , p p .

Let n x be the function associated to ; 1,2,3n n ; i.e.,

(24) ; 1,2,3n n ip xx A e n .

Substituting Eq. (24) into Eq. (21), it is obtained

(25) 0nnp I A .

Thus nA is the eigenvector corresponding to eigenvalue n . These eigenvectors may be taken

orthonormal among them if relationship (23) holds., i.e., 0i jA A unless i j .

Without loss of generality, solutions (24) may be written as

(26)

3

32

1, ; \ 0 ; 1,2,3

2

n n ip xx p A e p n

.

Since , 1,2,3n n , do not belong to 2,3 3 3,L , they will be called the “improper eigenfunctions” of

operator H . However, the spectral properties may be obtained by building, in formal sense, a set of

integral transforms of functions 2,3 3 3,f L with the improper eigenfunctions

(27) 3

, ; 1,2,3n

nf p f x x p dx n

.

Lemma 5. For every 2,3 3 3,f L the following limits exist in the strong topology of 2,3 3 3,L

(28) 2 2

lim , ; 1,2,3n

n

Mx M

f p s f x x p dx n

.

Here 3

2 2

1k

k

x x

.

Proof. From relation (26) it follows that

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(29)

2 2 22 2 2

3

3 32 21

1 1,

2 2

n n ip x ip x nj j

jx M x M x M

f x x p dx f x A e dx f x e dx A

.

Since njA are constants for 1,2,3j ; and 2 3 ,jf L , then from the Plancharel’s Theorem [Bochner

and Chandrasekaran pp.112-113] it follows that the integral (29) converges in the norm of 2 3 ,L and

the limit belongs to the same linear space.

For 2,3 3 3,f L , Lemma 5 associates to it a vector 1 2 3, ,f f f , where 2 3 , , 1,2,3nf L n .

Moreover, the following result holds.

Lemma 6. For each 2,3 3 3,f L , it follows the Parseval Identity

(30) 2 3

3 22

,1

n

Ln

f f

.

Here , 1,2,3nf n ; are defined by expression (28).

Proof. From relations (26) and (27) it follows that

(31) 1 2 3ˆ ˆ ˆ ˆ, ,nf f A f A f A fA ;

where A is a 3 3 matrix whose columns are formed by , 1,2,3nA n ; and f Ff . Moreover, A is an

orthogonal matrix since its columns are orthonormal vector, thus 1 †A A .

On the other hand,

2 3 2 3

3 3 3 3

3 3

3 3 3 3 22 2 2 2

, ,1 1 1 1

2 2 2

ˆ,

ˆ .

n n n n n

L Ln n n n

f f f f p dp f p f p dp f p A dp

f p dp f x dx f

The previous to last equality follows from Parseval’s identity for Fourier Transforms [Bochner and Chandrasekaran p. 113].

Define the following linear operator

(32)

2,3 3 3 2,3 3 3

1 2 3

: , ,

, , .

L L

f f f f f

Therefore, from Lemma 6 the next identity holds

(33) 2,3 3 3, ,f f f L .

This means that

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(34) I .

Therefore is a partial isometry and P is the orthogonal projection of 2,3 3 3,L , the

range of .

(35) 1 2 3 2,3 3 3ˆ ˆ ˆ, , ; ,f f A f A f A f L .

This means that the components of f are the projections of f on each ; 1,2,3nA n . Therefore, the

vector base formed by ; 1,2,3nA n ; vector f may be expressed as

(36) 3

2,3 3 3

1

ˆ ; ,n n

n

f f A A f L

From this interpretation and from Plancherel´s Theorem [Bochner and Chandrasekaran pp.112-113] it is obtained the following result.

Lemma 7. For each 2,3 3 3,f L , the following limits exist on For each 2,3 3 3,L .

(37) 2 2

3

1

lim ,nn

Mnp M

f s f p x p dp

.

Here ; 1,2,3nf n ; are the components of f .

From Eq. (37), it may be seen that

(38) 1f F Af .

Theorem 8. The operator defined in (32) is a unitary linear operator; i.e.,

(39) I .

Proof. The first equality of (39) follows from (34). On the other hand, since A is orthonormal, then it is

the matrix of a bijective linear transformation. Thus, if 2,3 3 3,f L then 2,3 3 3,g Af L .

Moreover, since F is a unitary linear transformation from 2,3 3 3,L to itself, then 1 2,3 3 3,F g L .

Therefore, if h f ,

(40) †ˆf f h h A .

From Eq. (38) it follows that

(41) †† †† 1 †h f FF Af Af fA

.

Combination of Eqs. (40) and (41) gives

†f fA A fI f .

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Eq. (39) is the eigenfunction expansion in abstract form; i.e., for each, it follows the next representation

(42) f f .

This eigenfunction expansion may be used to obtain a representation for the elasticity operator H .

Theorem 9. The operator whose action is given by Eqs. (32), (35), and (36) defines a spectral representation for H in the sense

(43) 1 2 31 2 3, , ;Hf f f f f D H .

Proof. Since 3 3 3 3 3 30 0 0, , , ,D C C C is a dense set on 2,3 3 3,L , thus if

f D H and 3 3g D , then

(44)

1 1 1

† † † † †

, , , , ,

, , , , .

Hf g Hf g F Ff F Ag Ff FF Ag Ff Ag

A Ff g A IFf g A AA Ff g A Af g

Matrix †A A is a diagonal matrix whose components are the eigenvalues of :

(45) †i ijA A .

Thus Eq. (44) becomes

(46) †, , ,Hf g A Af g f g .

Eq. (43) follows immediately from Eq (46) since 3 3 3,D is a dense set on 2,3 3 3,L .

Notice that Eq. (43) implies that

(47) H .

This means that operator diagonalizes operator H .

Let nP be the orthogonal projection on the corresponding eigenspace of ; 1,2,3n n . Then nP is

given by [Kato]

(48)

1; 1,2,3

2n

n

C p

dzP p n

i p z

.

Here nC p is a simple closed curve around ; 1,2,3n p n . From Lemma 6, Lemma 7, Theorem 8, and

Theorem 9, the following corollary is proved.

Corollary 10. Operators , H , and nP hold the following properties on 2,3 3 3,L

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(49) 3

3

1

; \ 0nn

P p I p

;

(50) 3

† 3

1

; \ 0n nn

p A p P p A p

;

(51) 3 3

1 † 3

1 1

; \ 0n n n nn n

H F A p P p A p P p p

;

(52) 3

1

;n n nn

P I P P

.

In order to find out more properties of the spectrum of the elasticity operator, return to Christoffel equation (9) and its corresponding associated equation (10). For a fixed , Eq. (10) defines a two-dimensional surface on the vector spaced defined by vector p . Since the second term in Eq. (10) is

proportional to 2 and 31 2 3, , \ 0p p p p , with

2 2 2 21 2 3 1p p p p ; let

12

pq

be the

“slowness vector” [Achenbach p. 126], then it follows that 2 1

q

. In this way and in virtue of Eq. (8),

Eq. (9) becomes

0ijlm l m im ijlm l m im ijlm l m imp I u C p p u C q q u C q q u .

This last equation in turn implies that

(53) 1, 0q q I .

Eq. (53) describes an inverse velocity two-dimensional surface known in literature as “slowness surface”.

Notice that this surface is independent of 1

2 and only depends on the direction of propagation vector q .

An alternative way of describing the slowness surface is by noticing that Eq. (53) is a polynomial of third degree that in turn may be factorized in a unique way as

(54) 1 2 31, 1, 1, 1, 0q Q q Q q Q q .

The locus described by each 1, ; 1,2,3nQ q n ; is given by the following set

(55) 3 / 1 ; 1,2,3n nS n .

Therefore, the slowness surface is given by

(56) 3

1n

n

S S

.

This description for the slowness surface permits defining a system of generalized radial coordinates on 3 :

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(57)

3

12

12

: \ 0 , ;

: , ; 1,2,3.

n

n

n

S

n

Moreover

(58) 1 , ; 1,2,3n n n .

Let dS be the two-dimensional measurable infinitesimal surface on the unitary sphere in 3 , then

(59) 2nd d d .

Here

(60) 3 n

n n

n

d dS

.

Obviously Eq. (60) defines a finite measure on 2 .

Let 2,3 3,nL S be the Hilbert space of all 3 -valued measurable functions taking values on nS and are

square integrable with respect to the measure nd . Define the following Hilbert spaces

(61) 2,3 3, / ; 1,2,3n n n n n nH L S P n ;

(62) 1 2 3H H H H ;

(63) 2 , ,L d H

.

Define the following linear operator

(64)

2,3 3 3

3

1 2 31

: , ;

, ; , , .n n nn

U L

U P

From Theorem 9 and Corollary 10, it follows the next corollary.

Collorary 11.

i. Operator U is a unitary operator.

ii. Operator 1 :UHU is the multiplication operator by :

(65) 1UHU I .

For each Borel set , let be the characteristic function of for each , and let E be

the spectra family of operator H , then Corollary 11 implies

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(66) 1E f U Uf ;

(67) UHE f Uf .

Theorem 12.

i. Operator H is absolutely continuous;

ii. The spectrum of operator H , H , is 0 .

Proof. Let be a Borel set. From Eq. (66) it follows

1, , , ,H

E f f U Uf f Uf Uf Uf Uf d

.

If 0 (the Lebesgue measure of ), then , 0E f f . This proves (i).

Since operator H is absolutely continuous, then the singular spectrum of H is equal to zero. Therefore,

(68) 0H .

In this way, the interval ,0 belongs to the resolvent of operator H .

The Limiting Absorption Principle for the Elasticity Operator For \z H , let

1R z H z

be the resolvent for the elasticity operator H . Let investigate

when R z takes, in some sense, limit values on the positive real axis when these values are

obtained as limits on R z as z for / Im 0z z z .

Since from Theorem 12 H , it follows that those limits do not exist in the uniform topology of

the all bounded operators from 2,3 3 3,L to itself. However, as it will be seen later, those limits exist

if R z is considered as a function taking values on an optimum topology of linear bounded operators.

This result is known in literature as “the limiting absorption principle”.

For , define the following Hilbert space

(69) 2 22 3 2 3 2 3, , / 1 ,L f L x f L

.

For 2 3, ,f g L , define its scalar product as

(70) 2 3

3

2

,, 1

Lf g x f x g x dx

.

For , let 3 ,H be the Hilbert space given by the closure of 30 ,C in the norm

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(71)

3

2 3

1 2 320,

,

1 ; ,H

L

f F p Ff f C

.

Here F and 1F denote the forward Fourier Transform and inverse Fourier Transform operators defined

on 2 3 ,L , respectively.

Theorem 13. For 1 2 , 0 , and nS defined by Eq. (55), there is a trace bounded operator

; 1,2,3nT n ; from space 3 ,H to space 2 ,nL S such that if n is given by Eq. (58), then

(72) 3 30; , ,n n nT C H

Moreover, nT is a Hölder continuous mapping from 0 to the space of bounded linear operators

from 3 ,H to 2 ,nL S with exponent

(73)

1 3, if ;

2 23

1 , if , with 0 arbitrary small;2

31, if .

2

Proof. This theorem is a particular case of the Trace Theorem proved by Weder [Weder].

Define the following linear operator for 1,2,3n ;

(74)

2,3 3 3,

,

: , ;

.

n n

n n n n n

B L H

B f P T f

Here 0 , 1 2 , and 2 21 x

. Thus, from Theorem 13 is Hölder continuous with

exponent given by (73).

Now define the linear operator

(75) 1, 2, 3,B f B f B f B f .

It follows that this last operator is also Hölder continuous with exponent given by (73).

In this way, from Eq. (65) the next results are immediate for f D H ,

(76) , ;U f B f

(77) , .UH f B f

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Moreover, for z and for each compact interval 0I , such that belongs to I , it follows

that if R z is the resolvent operator of H and CI denotes the relative complement of I with respect to

0 , then

(78)

C

I

B BR z d R z P I

z

.

Let 3

2,3 3 3 2 3

1, ,L L . If 2 3 ,H denotes the Sobolev space of all functions belonging to

2 3 ,L such that its first and second generalized derivatives belong to also to 2 3 ,L , let the space

(79) 2 22 3 2 3 2 3, / 1 ,H f H x f H

.

Define the norm of this space as

(80)

2 3

2 3

2 2

,,

1H

H

f x f

.

Finally, define

(81) 3

2,3 3 3 2 3

1, ,H H .

Theorem 14 (The Limiting Absorption Principle for H ). For each 0 , the limits

(82) 0

0 limR i R i

;

exist in the topology of the space of bounded linear operators from 2,3 3 3,L to 2,3 3 3,H for

1 2 . Moreover, the functions

(83)

, if ;

0 , if ;

R z zR z

R z i z

are locally Hölder continuous on the space of bounded linear operators from 2,3 3 3,L to

2,3 3 3,H with exponent if z , and analytic if 1 2 3 and if Im 0z .

Proof. The existence of limits (82) in the topology of the space of bounded linear operators from

2,3 3 3,L to 2,3 3 3,H follows from the fact that B is locally Hölder continuous. Moreover,

0 . . C

I

B BR i p v d i B B R P I

.

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The Hölder continuity of (83) follows from Privalov-Plemelj’ Theorem [Weder]. The analyticity follows from the analyticity of the slowness surface [Weder].

References Achenbach, J. D. (1975). Wave propagation in elastic solids. North Holland Publishing Company.

Bochner, S. and Chandrasekaran, K. (1949). Fourier Transforms. Princeton University Press.

Fedorov, F. I. (1968). Theory of elastic waves in crystals. Plenum Press.

Kato, T. (1976). Perturbation theory for linear operators. Springer Verlag.

Weder, R. (1985). Analyticity of the scattering matrix for elastic waves in crystals. J. Math. Pures et Appl. 64; pp. 121-148.