the limiting absorption principle and spectral theory for...

The Limiting Absorption Principle and Spectral Theory for Steady-State Wave Propagation

in Inhomogeneous Anisotropic Media JOHN R. SCI UL NBEROER & CALVIN H. WILCOX

Communicated by J. SERmN

Abstract

Many wave propagation phenomena of classical physics are governed by systems of the Schr6dinger form

- i D t u + A u = f ( x , t) where A = - i E ( x ) -1 ~ A j D j , (1) j = l

E(x) and the A i are Hermitian matrices, E(x) is positive definite and the A i are constants. If f (x, t )= e- ix t f'(x ) then a corresponding steady-state solution has the form u(x, t )= e- ~ ~ t v(x) where v(x) satisfies

(A - 2) v = f ( x ) , x ~R n . (2)

This equation does not have a unique solution for 2 ~ R 1 - {0} and it is necessary to add a radiation condition for ] x[---~ oo which ensures that v (x) behaves like an outgoing wave.

The limiting absorption principle provides one way to construct the correct outgoing solution of (2). It is based on the fact that A defines a self-adjoint operator on the Hilbert space .~ defined by the energy inner product (u, v )= f u*Ev dx. It follows that if ~ = 2 + i a and or=l=0

Rn

then (A- -O v = f has a unique solution v(., O=R;(A)f~.Ct ~ where R~(A)=(A--O -1 is the resolvent for A on .r The limiting absorption principle states that

v ( . , 2) = l im v ( . , 2 + i a ) (3) a ' ' ) 0

exists, locally on R n, and defines the outgoing solution of (2). This paper presents a proof of the limiting absorption principle, under suitable hypotheses

on E(x) and the Aj. The proof is based on a uniqueness theorem for the steady-state problem and a coerciveness theorem for nonelliptic operators A of the form (1) which were recently proved by the authors.

The coerciveness theorem and limiting absorption principle also provide information about the spectrum of A. It is proved in this paper that the point spectrum of A is discrete (that is, there are finitely many eigenvalues in any interval) and that the continuous spectrum of A is absolutely continuous.

1. Introduction

I t is a f u n d a m e n t a l p r o b l e m of c lass ical phys ics to descr ibe the waves p r o d u c e d

in a m e d i u m by the ac t i on of p r e sc r ibed sources . T h e sources a re f r e q u e n t l y osc i l l a to r s ; t h a t is, t h e y h a v e a s inuso ida l t i m e dependence . T h e resu l t ing waves

Wave Propagation in Anisotropic Media 47

may then be expected to have the same time dependence, apart from a transient wave. The problem of determining this steady-state response of the medium is usually called the steady-state wave propagation problem.

This paper deals with the steady-state wave propagation problem for a class of inhomogeneous anisotropic media. The work is applicable to a variety of wave propagation phenomena of classical physics, including electromagnetic waves, acoustic waves, seismic waves, electric waves on transmission lines, etc. A unified discussion of these phenomena is possible because they are all governed by first order systems of partial differential equations that can be written in the Schr6- dinger form

- i D t u + A u = f ( x , 0 (1.1) where

A= - iE (x ) -1 ~ AjDj. (1.2) j = l

Here t ~ R 1 (time), x = (x 1, x 2 , . . . . x~) e R n (space), D, = ~/8 t, Dj = a/~ x~, u = u (x, t) = t ( U l ( X , t ) , U 2 ( X , t ) . . . . . Urn(X , t)) is a function whose values are m x l(column) matrices* over the complex number field C which describe the state of the medium at position x and time t, and E(x), At, A 2, ..., A~ are m x m matrices over C with the properties

E(x) is Hermitian and positive definite for x~R n, and (1.3)

A1, A2, . . . , An are Hermitian and constant. (1.4)

The function f (x , t )= ' ( f l(x, t ) , f 2(x, t) . . . . . fro(x, t)) is a prescribed function which specifies the sources acting in the medium.

Systems of the above form belong to the class of symmetric hyperbolic systems introduced by K. O. FRIEDRICHS in 1954 [6]. Some of the wave equations of classical physics are exhibited in this form in [18].

The steady-state wave propagation problem for (1. I) deals with the case where the source term has the form

f(x, t)=e-fa' f(x), 2eR 1 - { 0 } , (1.5)

and a solution of the same form is sought:

u(x, t)=e-Z ~' v(x, 2) . (1.6)

Thus v(x, ).) must satisfy the partial differential equation

A v - 2 v = f ( x ) , xeR n. (1.7)

The function v(x, 2) is not uniquely determined by equation (1.7) alone and it is necessary to add auxiliary conditions. Physically, a condition is needed which guarantees that v(x, ~) behaves like an outgoing wave for lxl--, oo. The authors have shown in [13] that this may be achieved by means of a radiation condition which is a generalization for anisotropic media of Sommerfeld's radiation condition.

* tf M is a matrix then tM denotes the transpose of M.

48 J . R . SCHULENBERGER • C. H. WILCOX:

A second method for solving the steady-state propagation problem is provided by the limiting absorption principle. This method is based on the fact that A defines a selfadjoint operator on the Hilbert space ,,~ defined by the energy inner product*

(u, v)= ~ u(x)*E(x)v(x)dx. (1.8) Rn

It follows that the spectrum of A is real. Hence if the frequency 2 in (1.7) is replaced by a complex frequency ( = 2 + icy with a . 0 , then (1.7) has a unique solution in

for each f ~ ~ given by

v (x, ~) = R; (A) f (x ) (1.9)

where Rg(A)= ( A - ~)-1 is the resolvent of A. Physically v (x, 2 + i t r) is the solution of the steady-state propagation problem for an absorptive (or dissipative) medium. The limiting absorption principle states that the steady-state solution of (1.7) is the limit of the corresponding solution for an absorptive medium:

v(x, 2)= lim v(x, 2 + i a ) = lira R~+~,(A)f(x). ~ 0 + a~O+

(1.10)

Of course, the limiting absorption principle is not obvious and the existence of the limit (1.10) must be proved. The purpose of this paper is to prove the limiting absorption principle for a class of inhomogeneous anisotropic media. An existence theorem for the steady-state propagation problem follows as a corollary. The limiting absorption principle also provides information about the spectrum of A. It is shown below that the point spectrum of A is discrete; that is, it consists of isolated eigenvalues having finite multiplicity. The limiting absorption principle is then used to show that the continuous spectrum of A is absolutely continuous. This result is important in the theory of scattering for systems (1.1) [14].

This paper is a sequel to another paper by the authors [13] in which a precise formulation of the steady-state propagation problem was given and the uniqueness of the solution was proved. The uniqueness theorem plays a key role in the proof of the limiting absorption principle given below. Thus, this paper provides another example of the principle that uniqueness implies existence, which holds for so many linear problems of mathematical physics.

A precise formulation of the steady-state propagation problem was given in [13]. The definitions and notations of that paper are used here without repetition. The remainder of this paper is organized as follows. The remainder of this section contains a description of the class of media that will be considered and a review of a coerciveness theorem for the operators A, proved by the authors in [15], which plays a central role in the proof of the limiting absorption principle and in the discussion of the spectrum of A. The discreteness of the point spectrum of A is proved in w 2. w 3 contains a proof that the continuous spectrum of A is absolutely continuous. A brief discussion of related literature and unsolved problems is given in w 5.

* If M is a matrix then M* denotes the Hermitian conjugate of M. Thus M*=t~ where is the complex conjugate of M.


Throughout this paper the systems (1.1), and the media described by them, are assumed to have the following properties:

l) E(x) is Hermitian for all xeR". (1.11)

2) A 1, A2 . . . . . A, are Hermitian and constant. (1.12)

3) E(x) is uniformly positive-definite on R"; that is, there exists a constant c > 0 such that ~*E(x)~>c~*~ for all x~R ~ and ~ c m = C x (1.13) C x "" x C (m factors).

4) The medium is homogeneous outside a bounded set; that is, there is a Hermitian positive definite matrix E o and a constant ct>0 such that* (1.14) supp{e(x)-Eo}=B~,={x: Ixl_-<~}.

n

5) The homogeneous medium described by Ao = - iEff 1 ~ AjDy is uniformly propagative, j= 1 (1.15)

6) The sheets Sk of the slowness surface for Ao are all convex. (1.16)

7) E(x) and its first derivatives DyE(x) ( j= I, 2 . . . . . n) are continuous (1.17) and bounded on R".

Hypotheses 1) through 7) are equivalent to the hypotheses of [13], together with 7) as an additional hypotheses. Hypothesis 7) was not needed for the uniqueness theorem in [13]. It is needed here to ensure the validity of a coerviceness theorem for the operator A.

A is an elliptic operator in R a if and only if the symbol

A(p, x)=E(x) -1 ~ Ajp.i (1.18) j = l

is a nonsingular matrix for all xr and all p ~ R " - { 0 } ; that is, rank A(p, x)=m. This is true in Case 1 of [13], when A has no zero propagation speeds. In this case the coerciveness of A follows from the theory of elliptic operators [1, 5, 15]. In Case 2 of [13], A is not elliptic. However,

rank A (p, x) = rank A o (p) = m - k (1.19)

for all x~R" and all p e R " - { 0 } where k is the multiplicity of the zero root of the characteristic polynomial for Ao(p). Such operators will be said to have constant deficit k in R". A generalized coerciveness inequality for nonellipfic operators of constant deficit was proved by the authors in [15] for operators satisfying hypotheses somewhat weaker than 1) through 7) above. For the purposes of this paper the result may be stated as follows.

Theorem 1.1. Let the operator A satisfy hypotheses 1) through 7). Then u ~ D ( A ) n N(A) • implies that D ju~ ,~ for j = 1, 2, ..., n, and there exists a constant K > 0 such that

IlOjull2<K(llAull2+ Ilull 2) (1.20) j = l

* The notation suppf(x) is used to denote the support of a function f(x).

4 Arch. Rational Mech. Anal., Vol. 41

50 J.R. SCHULENBERGER • C. H. WILCOX:

for all ueD(A) c~ N(A) • Here N(A) • is the orthogonal complement in ~ of N(A), the null space of A.

Theorem 1.1 was proved in [15] by constructing an augmented operator A" which is coercive on its whole domain. The construction of A" and the coerciveness inequality for it are reviewed here because they are needed below in the proof of the limiting absorption principle.

Hypotheses 3) and 7) imply that there exist constants c > 0 and c'__> c such that

c~* ~* E( x ) ~<c '~* ~ foral l x~R" and ~ C m. (1.21)

It follows that u e ~ if and only if the components u s of u are in the Lebesgue space Xe 2 = s 2 (R"). The same remark applies to ~ 0 , the Hilbert space associated with Eo. Thus ~ , ~0 and *La2,,,=L~v 2 ~.Sa 2 ~ -.- �9 s (m terms) coincide as linear spaces and the norms in the three spaces are equivalent. The Sobolev space

s D~u~L:'2,,,(G)for j = 1, 2, . . . , n} (1.22)

is also used below. (G is a domain in R".) It is a Hilbert space with norm

llull[G= Ilull~+ ~ tlOjullg. (1.23) j = l

The augmented operator A" constructed in [15] has the structure

A"= (1.24)

where A' is an operator on o~. A' was chosen in such a way that A" is coercive and

N(A) ~ = N(A'). (1.25)

The operator A' is defined as follows. The scalar operator

( -A)~: ~ o ~ o (1.26)

is defined by means of the Plancherel theory of the Fourier transform. Thus if fi(p)= ~u(p) denotes the Fourier transform of u e ~ o (el. [15, (1.19)]) then

O ( ( - a ) ~ ) = ~ o n {u: Ipl fi(p)Ea~o} and (1.27)

((- , t) ~ u)^(p) = I pl fi(p). It follows that

D((- A)~)= s and (1.28)

Jl(-A)~uj[2= ~ [[D~ulJ 2 for all u~D((-A)~). j = l

The operator Po: ~o ~ ~Vfo (1.29)


is defined to be the orthogonal projection in ~o onto N(Ao), the null space of A0. It has the structure

Po = 4~*/30 4~ (1.30)

where ~o(P) is a matrix-valued function of p which is homogeneous of degree zero. An explicit construction of Po (P) is given in [20].

The operator J is the identification map of ~o into ~f:

J: .YYo ---, of', J u = u . (1.31) Its adjoint is the operator

g*: ~ - ' * ~ o , J * = E o ~JE (1.32)

where E and E o are the multiplicative operators on ~ and ~o , respectively, defined by the matrices E(x) and Eo.

Finally, A': 9~--*A" (1.33)

is the operator with domain

D (A') = g n {u : Po J* u e La~, m} (1.34) such that

A' = J ( - A)§ Po J* . (1.35)

Property (1.25) is verified in [15]. The operator

A": J/e--, Jet~ @ ~ (1.36) is defined by (1.24) and

D(A") = D(A) nD(A ' ) . (1.37)

The following coerciveness theorem for A" is proved in [15, Theorem 5.36].

Theorem 1.2. Let the operator A satisfy hypotheses 1)through 7). Then D ( A " ) = -~ , m (R n) and there exists a constant K > 0 such that *

Ilult~<=K(IIAulI2+IIA'ulI2+Ilull 2) for all ueD(A") . (1.38)

Note that Theorem 1.1 is a corollary of Theorem 1.2 and (1.24).

2. The Discreteness of the Point Spectrum of A

Let ap (A) denote the point spectrum of A, that is, the set of all the eigenvalues of A. The purpose of this section is to prove

Theorem 2.1. ap(A) is discrete, that is, there are only a finite number o f eigenvalues o f A in any finite interval o f R 1. Moreover, each nonzero eigenvalue o f A has finite multiplicity.

This theorem is a direct consequence of the coerciveness theorem for A (Theorem 1.1 above) and the following theorem which was proved by SCHULEN- BERGEg in [11, p. 398].

* The notation It u[[ 1, R -= I[ ulh is used. 4*

52 J . R . SCHULENBERGER ~r C. H . WILCOX:

Theorem 2.2. Let 2eap(A)-{0} and let ue ~ be a corresponding eigenfunction. Then supp u c supp { E - Eo} c B~.

Proof of Theorem 2.1. The proof is by contradiction. Moreover, both parts of the theorem can be proved by the same argument. Suppose that there are an infinite number of (distinct) eigenvalues in a finite interval of R 1 or that there exists a nonzero eigenvalue with infinite multiplicity. In either case there will exist a sequence of orthonormal eigenfunctions Ul, u2 . . . . and corresponding eigenvalues 21, 22 . . . . (not necessarily distinct) such that 2n~:0 for n = l , 2, ... and the sequence {2n} is bounded:

I2 ,1<M, n = l , 2 . . . . . (2.1)

Let v e ~ . Then Bessel's inequality implies that

lim (u~, v) = 0; (2.2) n---~ oo

that is, the sequence {u,} tends to zero weakly in ~ . Now, Theorem 2.2 implies that

suppuncB~, n = l , 2, . . . . (2.3)

Moreover, each uneN(A) • since each 2,-~e0. Thus Theorem 1.1 is applicable to the u n and implies, since IluH1,B._- < llull.

IJu, JI2,n < K ( I I A u , I I 2 + J l u n I I 2 ) = K ( I 2 , j 2 + I ) < K ( M 2 + I ) (2.4)

for n = 1, 2, .... It follows by Rellich's compactness theorem [2] that there exists a subsequence {u'} which converges in ~r u" ~ u in ~ . Since u is also the weak limit of the u~ and weak limits are unique, (2.2) implies that u = 0 ; that is, u~ ~ 0 strongly in ~ . This implies that lim Ilu~ll =0, contrary to the assumption that

n--~ ao

tluLII = 1 for all n. This contradiction proves the theorem.

The point 4 = 0 may be an eigenvalue of infinite multiplicity. Indeed, it was shown in [14] that

N(A) = J)V(Ao) = JPo ~eo (2.5)

and it was shown in [20] that dim N(Ao)=OO or dim N(Ao)=0 depending on whether zero is or is not a propagation speed for A o. It is interesting to note where the proof of Theorem 2.1 fails in this case. If Ul, u2, ... is a sequence of null vectors for A, then un~N(A) ~ and hence Theorem 1.1 cannot be applied to this sequence.

3. The Limiting Absorption Principle The purpose of this section is to prove the limiting absorption principle and

an existence theorem for the steady-state propagation problem formulated in w 1. First it is necessary to give a precise formulation of the principle. It was shown in [13] that

v(x, ~)=Rr f ( x ) (3.1)

cannot be expected to converge in ~ when ~ - o 2 e R 1 because a ( A ) = R 1. The solution of the steady-state problem was defined in [13] to be a solution in the


Fr6chet space ~I~r l~ Accordingly, the convergence of v(., 2 + i a ) to v• 2) will be demonstrated in j~,loo. This means that

lim Ilv(', 2+ i a ) - v• (. , 2)[Ix =0 (3.2) a - * O +

for every compact KcRn . With this understanding the principal results of this section may be stated as follows:

Theorem 3.1. Let 2 ~ R 1 _ (ap (A) u {0}) and l e t f e Clo (Rn). Then the limits

lim v(., 2 + i a) = v• (. , 2) (3.3) a - * O +

exist in ~,~flo~. Moreover, v• 2) is the solution o f the steady-state propagation problem for the frequency 2, the source function f and the +_ radiation condition.

Theorem 3.2. The steady-state propagation problem has a unique solution for each frequency 2 ~ R 1 - (r (A) u {0}) and source function f ~ C~ (Rn).

Theorem 3.3. Let A = [a, b] c R 1 - (% (A) u {0}). Then v (., 2 +_ itr) ~ v• (., 2) in ~,~flo~ uniformly for 2cA.

The theorems are proved below by means of a series of lemmas and subsidiary theorems. Theorem 3.2 may be regarded as a corollary of Theorem 3.1.

The proof of Theorem 3.1 makes use of the Hilbert space ~ defined by

Ilull 2= S u ( x ) * E ( x ) u ( x ) O + l x l ) - x - 6 d x (3.4) Rn

and ~ = { u : u is Lebesgue measurable in R n and Ilull~< oo} (3.5)

where 6 is a fixed positive number. Spaces of this type were introduced into the study of the limiting absorption principle by D. M. EIDUS [4]. It is easy to verify that

IlullgR <(1 +R) x +~ Ilull~ (3.6)

and hence ~ c ~1or A first step toward proving Theorem 3.1 is

Theorem3.4. Let 2~Rl - (ap (A)w(O}) and let feClo(R"). Let ( , = 2 , + i a , define a sequence such that a , > 0 and ( , ~ 2 when n ~ ~ . Then the sequence v( . , ~) converges weakly in :~ to a limit v• ( . , 2 ) e ~ . Moreover, v• (., 2) is the solution of the steady-state propagation problem for the frequency 2, the source function f and the + radiation condition.

Note that Theorem 3.2 is an immediate corollary of Theorem 3.4. The proof of Theorem 3.4 is based on a series of lemmas and makes use of the concept of weak convergence in ~o r

Definition. A sequence of functions u, eg~ ~~176 is said to converge weakly in ~o~ to a function ue ~loo if and only if for each f ~ ~ o r and each compact K = R n

lim](u,, f)K = (U, f )K. (3.7) n.--~ oo

54 J. R:ScHULENBERGER & C. H. WILCOX:

Lemma 3.5. I f u. ~ u weakly in ~ then u.--* u weakly in ~loc.

Proof. To prove that u,~u weakly in ~loc, it is clearly sufficient to verify (3.7) with K = B N and N=I , 2 . . . . . The conclusion follows from the identity

(Un, f)B,, = ~ u.(x)* E(x) {)~N(X) f ( x ) (1 +Ix 1)' +6} (1 +Ix 1)- 1-6 d x R. (3 .8 )

= (u., g)6

where Zu is the characteristic function of BN and

g (x) = ZN (X) f ( x ) (1 + I X l) 1 + ~ (3.9)

Lemma 3.6. Let f ~ ;r and suppf= supp {E- Eo} = B p . Let ~b ~ C ~ ( R") satisfy O< tO(x)<__ 1 and

~k(x)=f0 for , x l < p , (3.10) ll for I x l > p + l .

Then v( . , ~)= R~(A) f satisfies the identity

~bv(., ~)= G(-, ~). (Ao~b)v(., ~), Im(4:0 (3.11)

where G(. , ~) is the Green's matrix for the operator A o [19].

Proof. It is easy to verify that if v~D(A) and qb~C~176 and if q5 and Djc~ ( j= 1 . . . . . n) are bounded, then q~ v ~ D (A) and

A(dpv)=q~Av+(A cp)v. (3.12) Thus

A (O v(-, ~))=O(Av(., ~)) + (A ~b) v(., ~)=~kv( . , ( )+ (A ~b)v(., ~) (3.13)

because O f = 0 since supp O c~suppf=$. Equation (3.13) implies that*

(Ao-~) Or(. , ~)=(AoO) v(., ~)+~(Eo ~ E - l ) Or(. , ~), (3.14)

and the last term vanishes because supp{E-Eo}nsupp~=q~. Hence, (3.14) implies (3.11) because R~(Ao) is the convolution operator G(., ~)* [19].

Lemma 3.7. Let 2eR-{0}. Let ~+_n=2n+ ia. define a sequence such that a , > 0 and ~• when n ~ o o . Let gnEa~ a define a sequence such that suppgn~B~ for some c > 0 and all n and g,--* g in ~ when n --* oo. Define

w + . = R ; . . ( A ) g. (3.15)

and assume that there exists a constant K such that

IIw• K for all n . (3.16)

Then {we.} converges weakly in ~ to a limit w• e~e'~. Moreover, w• is the solution o f the steady-state propagation problem with frequency 2, source term g and the +_- radiation condition.

* Note that AqJ=Ao~ because E = E o on supp ~u.


Proof. The proof is based on the fact that spheres in a Hilbert space are weakly compact [21]. If this result is applied to the Hilbert space ~ it follows from (3.16) that there exists a subsequence of {w• say {w'~,}, and a w• such that w ~ , ~ w• weakly in ~r It is shown next that w• is the solution of the steady-state propagation problem with source term g.

Lemma 3.5 implies that w'•177 weakly in ar e1~ and, a fortiori, w'~,~w+ in ~ '* . Let ~br C~ (R"). Then w• thought of as an element of ~ ' , satisfies

w• (~b) = (w• ~b)_~,, = lim (w~-., ~b).~,.. (3.17) B--~ CO

?t

Hence if A= - i ~ A~Dj j = t

A w + (tk) = w • (A qg) = lira ( w ~ . , A ~b).~,,.,

= lim(w~:., a $ ) = l im(Aw~., ~b). (3.18) n ---~ co n---~ oo

Now (A - (~.) w~. = g'. (3.19)

where ((~.} and {g~} are the subsequences of {(• and {g.} corresponding to the subsequence {w'• of {w• Equations (3.18) and (3.19) imply that

A w• (40 = l im ((~:. w~:. + g~, ~ ) = (2 w • + g, ~b) "-' ~ (3.20)

= (2Ew• +Eg, q~).~,, --(2Ew• +Eg)(~b). Thus

A w+ = E(2 w+ + g)eo~ l~ (3.21)

Therefore A w• ~ ygtoo and

(A-2)w• in ~J~e I~ (3.22)

To prove that w• satisfies the ___ radiation condition, apply Lemma 3.6 to w~:,. This gives

~b w~:. = G(., ~: . )* (Ao ~) w~:.. (3.23)

Now supp(Ao~b)w'~.=Bp+ 1 and hence

(Ao~)W~.--.(Ao~)W• in 8 ' (3.24)

when n ~ oo. Moreover, G(. , (2 . ) -~ G• (. , 2) in 9 ' [19]. It follows from the continuity of the convolution operator on ~ ' x S ' [16, p. 13] that making n--*oo in (3.23) gives

~bw_+ =G• (., 2)*(A0~b) w• (3.25) In particular,

w• S G+(x-Y, 2)Ao~(y)w• (3.26) B p + l

for Ix I_-__ p + 1. It follows that w• satisfies the -t- radiation condition (see [13, 19]).

* The notations ~ ' = ~ ' ( R n) and d"=g'(R n) are used to denote the Schwartz space of distributions and the subspace of distributions with compact support, respectively.

56 J .R . SCHULENBERGER & C. H. WILCOX:

It has been shown that there is a subsequence {w~:,} which converges weakly in ~ to w• the solution of the steady-state propagation problem with source term g. Suppose that the original sequence {w• does not converge weakly in to w• Then there will be an element heo~, an %>0, and a subsequence {w~,} such that

[(w~s h)~-(w• h)~l>eo for n = l , 2, 3 . . . . . (3.27)

But (3.16) and the weak compactness theorem imply, as above, that there is a subsequence {wg~,} of {Wgn} which converges weakly in ~ to a limit which is a solution of the steady-state propagation problem with source term g. Moreover, the solution of this problem is unique [13] and hence wg, ~ w• weakly in ~ . This contradicts (3.27). Hence, the original sequence converges to w_+ weakly in ~ . This completes the proof of Lemma 3.7.

Lemma 3.8. Let 2~R 1 -{0}. Define

I:• =I:• (y)-- {~: 1(-21 <~, Im(>0} (3.28)

and let ~ be chosen so that Oq~Z • Let fl=(fll . . . . , ft,) be an arbitrary multi-index. Then the Green's matrix G(x, () for Ao satisfies

OaG(x,O=O(Ixl-("-x)/2) , Ixl--,oo, (3.29)

uniformly for x/ I x l ~ ~ and ( ~ X • .

M. MATSUMURA [9] has recently proved an estimate which implies (3.29) under hypotheses which are satisfied by A o. A proof of Lemma 3.8 can also be given by means of the results and methods of the Appendix to the authors' paper [13].

Lenuna 3.9. Let 2~R1-{0} and define X• by (3.28). Choose ~ so that 0~X• Let feClo(R ~) and s u p p f w s u p p { E - E o } ~ B p. Then D j r ( . , ~)~o~fIo~ for j = 1, 2 . . . . , n, and there exists a constant Ro=Ro( p, n, ~, 2) and for each R > Ro a constant C= C(R, p, n, ~, 2) such that (with the notation (1.23))

IIv(.,OIl~.B,~<f[llv(.,OIIZ +l[fll~] for all (~,Y,• (3.30)

Proof. Let CR (x) = r (I x [ - R) ~ C~ (R") and have the properties 0 < 6R (X) < 1, r and {;

Cg(x)= for I x l < R , (3.31) for I x l > R + l .

It will be shown that if R > p + 1 then CRv (., ~)ED (A")= D (A)c~ D (A'). It is clear that CRY(., ~)~D(A) (cf. (3.12)) and

A (r ~)) = ~ ~bR v(-, 0 + f + (A CR) v(., ~), (3.32)

since Cg(x)= 1 on suppf when R > p + 1. This equation implies that Cgv(., ~)e D(A'). Indeed, R ( A ) c N ( A ' ) ~ D ( A ' ) [15] whence the term on the left in (3.32) is in D (A'). Next, Co ~ (R") ~ D (A') whence f ~ D (A'). Finally, supp A Cg ~ Bg + ~ - Bg~B~. Thus (Ao-~) v(., ~)=0 on supp Ar and hence v(., ~)sC ~~ there. It follows that (Ar v( . , O e C ~ ( R O ~ D ( A ' ). Hence, (3.32) implies that r ~)e D(A') and applying A' to (3.32) gives

~A' (r v (., ~)) = -- A ' f - A' [(A r v (., 0 ] . (3.33)


Now application of Theorem 1.2, the coerciveness theorem for A", to ~bRv(., 0 shows that D~(d~Rv(., ~ ) ) e g ( j = l , 2 . . . . . n) and

I1~. ~(., 011~-<g [IIA(~. ~(' , r ~ + IIA'(O. ~(' , r ~ + ItO. v( ' , r (3.34)

It follows that D~v(., ()e.r To derive the estimate (3.30) note that ~b,(x)= 1 on BR, q~R(X)=--O on BR+ I and I4~R(x)l < 1. Also llhlla~< I[hll. Thus (3.34) implies that

ll"(',Oll~,,.,,<=gEllA(4',,v(',O)[I ~ +tIA'(~.~(',r =+IIO(',OIIL~,]. (3.35)

The three terms on the right in (3.35) will be estimated separately. The last term satisfies, by (3.6)*

lit'(-, if)[[zB, +, =< C, IIv(', r (3.36)

The first term on the right can be estimated using (3.32) and the inequality IIf+gll2=< 2(11f112+ Ilg[12). The result is

I IA(~ o(., r z ilv(., ~)11~ + itfl12+ l[(ac~R)v(., ~)112]. (3.37) N o w

t~

where q = x][ x [ and

~b~ (x) = ~ qj D~ q5 R (x) = ~b~ (I x I - R). (3.39) j = l

Thus supp A q~R = BR + l -- Bs and (3.37) implies

IIA(*. v( ' , 0)11'_-< C2 lilY( �9 , r IIsll 2 ] . (3.40)

Finally, the middle term on the right in (3.35) can be estimated using (3.33) as follows:

lffl 2 I[A'(~R v(., 0)]12 <2 IliA'fit2 +[tA' [(A q~R) v(., ~)]112]

= 2 ~ llO~Pofll 2+ ~ IID~Po[(Ad?R)v(., r II z j = l j = l

" ] =2 [ E llPoDjf[[ z+ ~ IIPoDj[(Adp,)v(',C)][[ 2 t j = l j = l

<2 llOiftl2+ ~ IIOj[(aq~R) v(., ~)]11 z '= j = l

j=i /=i

_~c3 [llfll~ + Ilv(., ~)112 +4j=~ t ,I(Ad, R)Dj v(.. ~)H2].

* In what follows the Cj are constants which may depend on R, p, n, y and 2.

(3.41)

58 J . R . SCHULENBERGER • C. H . WILCOX:

Moreover, (3.38) implies that

li (A qgR) Dj v (-, ~)II 2 ~ C4 [I ~b~ Oj v (-, ~)I12. (3.42)

To estimate the last term, apply Lemma 3.6. Note that $(x) = 1 for t x i > p + 1, (Aod/) v(., ()~C~(R"), and supp (ao$) v(., ~)=B,+ 1 - B a. Thus (3.11) implies

v(x,~)=G(.,~).[(ao$)V(.,~)](x) for I x l > p + l (3.43) and hence

O~ v(x, ~) = Dy G (-, ~)* [(Ao $) v(., ~)] (x) (3.44)

= S DiG(x-y, ()(A o $(y)) v(y, ()dy B p + I - - B o

for [ x [ > p + 1. It follows that

(YR(X)D~v(X,()=CYR(X) ~ DjG(x-y,()(Aor (3.45) Bo+ t-- B a

for all xER". Taking norms in (3.45) gives

lff'R(x)Div(x, OI<I~'R(X)I ~ [DiG(x--y,()[ [(Aof(Y))v(y,()idy. (3.46) Bp+I- -BO

Now Lemma 3.8 implies that

[DiG(x-y, 0 ] < C 5 I x - y [ -~"- x)/2 (3.47)

for Tx-y]>c>O and ( e Z + . It follows that

i Oj G (x - y, () l < Cs I R - p - 11 - ("- 1)/2 < C6 (3.48)

for Ix[ >__R>p+ 1, [ y[ < p + 1 and ~27• Moreover

] (A0 ~' (Y)) v (y, r [ < C71 v (y, ~) I on Bp +l - B . (3.49)

Combining (3.46), (3.48) and (3.49) gives

14~'R(x)Dsv(x,~)l<lc)'g(x)lCs .[ Iv(y, Oldy B,~+ ~ - B e,

< ' < I ~bR(X)I C9 fly( ", ()tlB . . . . 4~R(x)C10 iiv(. ()11 ~ (3.50) and hence

II~b~Ojv(', ()11 z= ~ 14/R(x)D~v(x, ~)12dx R ~

< c l o Ilv(., 011~ j" [~'R(x)]2dx<flt IIv(., 011~. (3.51) R n

Combining (3.41), (3.42) and (3.51) gives

tlA'( . r [llfll~ + Ilv(-, r �9 (3.52)

Finally, combining (3.35), (3.36), (3.40) and (3.52) gives (3.30), which completes the proof of Lemma 3.9.


L e m m a 3.10. Under the hypotheses o f Lemma 3.9, for each s > 0 there exists a constant RI=RI(e, p, n, 7, 2) such that

v (x ,O*E(x)v(x ,O( l+lx l ) - l -adx<~l lv ( . , ( ) [ l~ (3.53) Ixl_->R

for all R> R o and all ~e2~s

Proof. Equation (3.43) above and Lemma 3.8 imply that for I xl > p + 1 and ~ •

Iv(x,O[_- < ~ I G ( x - y , ff)l [(Ao~k(y))v(y,O[dy Bp+l-Bp

(3.54) _-<Cla ~ I x -y l - ( " - l ) / 2 l v (y ,~ ) ldy .

B p + l - - B p

Now if I x] > p + 1 and [y[ < p + 1, the triangle inequality gives [ x - y I > I x[ - p - 1. Hence

I v ( x , ~ ) l < C l a ( l x l - p - 1 ) -("-1)/2 ~ Iv(y,~)ldy Bp+ l-Bp

____cj~ I x l - ( , - x)/2 [Iv(., ~)11~+, (3.55)

< c~s I xl -(~- x)/~ IIv(., ~)1[~.

This inequality implies that

$ v(x, ff)*e(x)v(x, ( ) ( l + l x t ) - l - a d x I~I->_R

<C~6l lv ( ' ,0 l l~ ~ I x l - " + * ( l + l x l ) - ~ - ~ d x (3.56) Ixl>=R

< C~7 [Iv(', ~)ll~ R - a < n [Iv(., OIIg provided that R > R o = (C17 e- 1)1/a, which proves (3.53).

Proof of Theorem 3.4. It will be shown that there exists a constant K such that

IIv(-, ~• for all n. (3.57)

The conclusion of Theorem 3.4 then follows from Lemma 3.7.

Inequality (3.57) is proved by contradiction. If there is no constant K for which (3.57) holds, then there exists a subsequence (~:, = 2"_ ia~ such that

lim IIv(-, f f~)l[~: ~ - ;1 ---~ oo

Define w_+. = Ilo(., ff~,)ll; 1 v(. , ~ , ) (3.59)

so that IIw• = 1 for all n (3.60)

and , ~, ) i - ~ r - (h-~•177 5, ~ J = g , . (3.61)

(3.58)

Then s uppg ,= supp f cBp and g , ~ 0 in ~ when n--*oo. Thus Lemma3.7 is applicable and it follows that w • 1 7 7 weakly in ~ and that w• satisfies ( A - 2 ) w • = 0 in gior and the _ radiation condition. Since 2~ap(A) this implies

60 J .R . SCHULENBERGER ~L C. 1~. WILCOX:

that w• =0 by the uniqueness theorem for the steady-state propagation problem [13]. On the other hand, Lemma 3.10 with e= 1/2 implies that there is an R 1 such that

l l ( n ) = _ ~ w • 1 7 7 (3.62) Ixl>_-nl

Moreover, Lemma 3.8 implies that

IIw+,II~,BR<=C[I+IIv(. ' , -2 _ ~• II.fll~]--< 6~8 (3.63)

for n = 1, 2, .... Hence, by Rellich's compactness theorem there exists a subsequence {w~:n} which converges strongly in 5e2,,,(BRI ). The strong limit of w ~ must be w• by the identity of strong and weak limits in .~2,m(BRI). It follows that if

I2 (n ) = ~ w ' • (3.64) Ixl~Nl

then l iml2(n)= j" w • 1 7 7 (3.65)

n- .oo Ix l=<~ l

Now 1 = tl w~:~ I1~ = I1 (n) + I2 (n) (3.66)

whence by (3.62) I 2 ( n ) = l - I 1 ( n ) > l / 2 for n = l , 2 . . . . . (3.67)

Hence (3.65) implies that

w• (x)* E ( x ) w • + l x l ) - l - ~ d x > l/2 (3.68) lxl_-<R

which contradicts the conclusion reached above that w• This completes the proof of Theorem 3.4.

Lemma 3.11. Let 2 e R ~ -(%(A)w{O}) and let feClo(R"). Then there exists a > 0 such that

lim v ( . , O = v • weakly in o~r~. (3.69) ~--,2

Proof. The statement (3.69) means that given any g e ~ and any e>0, there exists a d=d(e) such that if (e~:• and 1 ~ - 2 1 < d then l ( v ( . , ( ) , g ) ~ - (v• 2),g)~l<e. If this is not true then there exists a ge~r an Co>0, and a sequence ~ with ~e27• ( , ~ 2 and

I ( ", r g) , - • ( ' , g) , [ ->- > 0 (3 .70)

for n = l , 2 . . . . . But Theorem 3.4 implies that v(., (~ )~v• 2) weakly in which contradicts (3.70). This contradiction completes the proof of (3.69).

Proof of Theorem 3.1. Lemma 3.11 implies that v ( . , 2 +_ i a ) ~ v • ( . , 2) weakly in Jt~. Hence only the convergence in ~ o ~ needs to be proved. Now the argument used in the proof of Theorem 3.4 implies that there exists a constant Ks such that

Ilv(., 2+ ia)ll6<K1 for 0 < o < t r o . (3.71)


Hence Lemma 3.9 implies that for each R > Ro there exists a/(2 = K2 (R) such that

llv(.,A+i~r)lll,e,,<g for 0< t r< t r o . (3.72)

It follows by Rellich's compactness theorem that there exists a sequence ~• 2++_ia,, such that (• for n ~ o o and l imv( . , ~•177 exists strongly in

n...~ oo

�9 s But v(-, ~•177 (., 2) weakly in :~a and hence also weakly in ~loo, by Lemma3.5. It follows that w•177 Finally, v(. ,2+itr)-- .v• strongly in .oq'z(BR) when a---} 0 in any way. For otherwise there would exist an % > 0 and a sequence ~• such that ( • when n---} oo and

IIv(', G•177 (. , 2)II.z2,,,(B,,>~o for all n. (3.73)

But by the argument given above there would then exist a subsequence (~:, such that v(., (~-~)--.v• 1) in .s which contradicts (3.73). The strong convergence of v(., l-big) to v(., 2) in La2,m(Be) for every R > R o is equivalent to convergence in A, ~176 This completes the proof of Theorem 3.1.

Proof of Theorem 3.3. The proof is based on a compactness argument. The set Rl-(trp(A)u(O}) is open in R 1 by Theorem 2.1. Hence for each l e a there is a ~>0 such that E• is disjoint from %(A)u{O} and v(., ~)--*v• in ~ ~ 1 6 2 when ~--}1 in 2;+ (~). To prove Theorem 3.3 it is sufficient to show that for each R sufficiently large and each e > 0 there is a d = d(R, e) such that

IIv(., ;~+ i a ) - v • ( . , 2)11~ < e (3.74)

for a < d and all 2cA. To prove this note that for each R, e>0, and 2~A, there is a ~=T(R, e, 2) such that

II v ( . , () - v • ( . , ,~)II B~ < ~/2 (3.75)

whenever (e~• e, 2)). Since A is compact there exist a finite number of 2's, say 21, 22 . . . . . 2m such that the intervals 12-2j l <7(R, e, ;tj), j = 1, 2 . . . . . M cover A. Consider the set

M

[.) {~ : I ~ - 2jl < ~' (R, e, 2j), Im ~ <> 0}. (3.76) j = l

Since it is a union of semicircles with their diameters on the real axis it clearly contains a rectangle

{~=2+ itr: 2cA, 0 < a < d } . (3.77)

Let 2~A and 0<tr, tr'<d and let the points 2+__itr and 2++_ia' lie in the semicircle

Then {~: l ( - 2 j l <y (R , ~, 2j), Imff<>0}. (3.78)

IIv(., ~ + i a ) - v(-, 2___ i a')IiBR

<[iv(.,;~+ia)_v• 2.i)lls,,+llv• (3.79)

by the triangle inequality and (3.75). Making t r ' ~ 0 + in (3.79) gives

IIv(., ,~+ itr)-v• (., ,~)11 ~_-__~ (3.80)

provided tr < d. This completes the proof of Theorem 3.3.

62 J.R. SCHULENBERGER & C. H. WILCOX:

4. The Absolute Continuity of the Continuous Spectrum of A

The selfadjoint operator A introduced in w 1 has a spectral representation [8]

A = ~ 2 d / / ( , b (4.1) ~ 0 0

where/7 (2) is a spectral family on ~ . If ~ is the closed subspace of W spanned by the eigenfunctions of A, then

. ~ = ~ ~ af, P~ (4.2)

and it is known that this orthogonal decomposition reduces A. It can be shown that [8, p. 515].

~ = { u e ~ : (H(2)u, u) is continuous on R1}. (4.3)

Accordingly, ~'~c is called the subspace of continuity for A. Another decomposition of ~ can be derived from A as follows [8, p. 516]. Let A=[a, b] and define

m, (A) = ([17 ( b ) - U (a)] u, u). (4.4)

Then m,(A) defines a finitely additive measure on the ring of sets generated by the intervals A. It can be extended in the standard way [7] to a countably additive measure m,(5 a) on the a-ring of Borel subsets 5e of R 1. Define

J(Y~c = {ue ocg: m,(5 e) is absolutely continuous with (4.5)

respect to Lebesgue measure} and

~ = { u e ~ e g : m,(SQ is singular with respect to Lebesgue measure}. (4.6)

Then it can be shown using the Lebesgue decomposition that [8, p. 516] ~ c - L ~ ,

ace = OCP~ ~ aet],, (4.7)

and the decomposition reduces A. The subspaces ~ and a~, are called the subspaces of absolute continuity and singularity, respectively. It is easy to see that ~ = ~ and hence a(g~c=.r

One may ask for criteria which guarantee that J~=acg~ c, so that ~p=a~ , and

= ~ r G aztp. (4.8)

The validity of this representation is of great importance in scattering theory, where ~a~ represents the scattering states and ~ represents the stationary states. The purpose of this section is to prove the representation (4.8) for the wave equations of classical physics (1.1) subject to the hypotheses 1) through 7) of w 1. The result is stated as

Theorem 4.1. ~ = J g ~ c .

The proof of this theorem is based on two lemmas. The first deals with the spectral measure H(SQ on the Borel subsets 5" c R 1 which is defined by extending the measure /7(A) defined for intervals (a, b] by H(A)=FI(b)-I-I(a) [8, p. 516].


Lemma 4.2. Let s ~-(o'p(A)U{O}) and let feC~o(R"). Then

(1-1 (5 p) f , f ) = 1/27r i I (v + (., 2) - v _ (., 2), f ) d 2. (4.9) 5a

Note that Theorem 3.3 implies that (v+ (., 2 ) - v _ (., 2),f) is continuous on Rl-(ap(A)u{O}). Hence the integral in (4.9) is well-defined. To prove (4.9), first let 5"=A = [a, b]. Then/7(2) is continuous at the points a and b since A = R 1 - (trp(A)u{0}). Hence it follows by a well-known theorem of STONE [17, p. 359] that for all fE

(/7(A)f,T)= lim 1/2rciS([Ra+,,(A)-Ra_,,(a)]f , f)d2. (4.10) a ~ 0 + a

If f e C~ (R ") and s u p p f = BN then by Theorem 3.3

(R~+i,(A) f, f )=(v( . , 2++_ ia),f)nN--*(v+ (., 2), f)s~, (4.11)

uniformly for 2cA. Thus passage to the limit under the integral sign in (4.10) is permissible. This proves that

(/7(A)f,f)=l/2zi[. (v+ (., 2 ) -v_ (., 2), f ) d2, (4.12) d

that is, (4.9) for the case that 5 a = A. The general case follows from this immediately because both sides of (4.12) have unique extensions to the a-ring of Borel sets 6 a.

Lemma 4.3. Let A c R x - (ap (A) w {0}) and let f~ C~ (R"). Thenfa =/7 (A)f~ :~a c.

Proof. Let ~ c R 1 be any Borel set. Then/7(5Q is an orthogonal projection on ~ and H(SP)II(A)=1-I(AacaA) [8, p. 516]. Hence

(H (5 a) fa , fa) = (/7 (5 a) H (A) f , / / (A) f )

= (H (S~) 17 (A) f, 17 (5 a)/7 (A) f ) = (II (Sp c3 A ) f, f )

=l/2zci S ( v+( . ,2 ) -v_ ( . ,2 ) , f )d2 $ a n zl

by Lemma 4.2. In particular, if 50 has Lebesgue measure zero then the integral vanishes, because the integrand is continuous. Thus f~ eJf~ c.

Proof of Theorem 4.1. Since ~ c ~ r in all cases, it is sufficient to prove that ~ c . Let f ~ c and let {f,,} be a sequence such that f,~C~(R") and f , ~ f in ~ . Let A c R 1 - (trp(A) u {0}). Then / 7 ( A ) f , e ~ by Lemma 4.3 and H(A)f, ~ /7 (A) f in ~ . Hence H(A)f~,Y~ since ~ is a closed subspace. Now let A=[a, b] be an arbitrary interval whose endpoints are not in %(A)u{0}. Then A c~(trp(A)w{0}) is a finite set by Theorem2.1, say A n(ap(A)u{O})=

M + I

{al, a2 . . . . . aM}. Then A= U Ai where Al=[a, al], Az=[al, a2] .... ,AM= i = 1

[aM- 1, am] and AM+ 1 = [aM, b]. Moreover, H(ak + )f=/- / (ak-- ) f f o r k = 1,2 . . . . . M M + I

because f ~ . Thus I I ( A ) f = ~ H ( A i ) f. Also I I (Ai) f=l im[H(ai-5 ) - i = l ~ 0

I1(a~_ x + 6)]f~C~ac because [ a i - 5, ai- 1 + 51 ~ R 1 - (trp (A) w {0}) and ~ c is closed.

64 J.R. SCHULENBERGER & C. H. WILCOX:

Thus H(A)fe;,~c for any interval A whose endpoints are not in ~rp(A)w{0}. Let A, = [a,, b,] where a , ~ - ~ , bn--' + oo and an and bn are never in trp(A)u (0}. Then f = limlI(An)f~c, which completes the proof.

5. Concluding Remarks

The limiting absorption principle has been used as a heuristic principle for the solution of steady-state wave propagation problems for many years. The first rigorous proof of the principle for a general class of problems is due to D. M. EIDUS [4] who dealt with scalar elliptic operators A of the second order:

A = - ~ O k (akt (X) O, U) + q (X) U (5.1) k , l = l

which had constant isotropic coefficients outside a bounded set:

akt(X)=6kt for Ixl >--p. (5.2)

The work reported in this paper is a generalization of Eidus' approach to nonelliptic systems A of the form (1.2) that describe waves in anisotropic media. There were two main difficulties in generalizing Eidus' work to this case. First, it was necessary to find a suitable radiation condition for anisotropic media and to generalize the Rellich uniqueness theorem to this case. This was done in [13]. Second, it was necessary to find a generalization to nonelliptic operators of the coerciveness inequality for elliptic operators. This was done in [15]. These two results play indispensible roles in Eidus' proof of the limiting absorption principle.

A generalization of Eidus' method to nonelliptic operators was announced by K. MOCmZUKI in [10]. In this work he considered anisotropic media that are isotropic outside a bounded set. He proved a Rellich uniqueness theorem for this case and showed that the limiting absorption principle will hold if a local version of the coerciveness result (1.31) holds. However, no proof of this kind of inequality is known at present.

I t is desirable to extend the results of this paper to more general operators and more general domains in R". This will be possible when suitable coerciveness results are developed for the corresponding nonelliptic boundary value problems.

This research was supported in part by the Office of Naval Research, Grant No. N 00014- 67-A-0394-0002.

References 1. AGMON, S., A. DOUGLIS, & L. NIRENBERG, Estimates near the boundary for solutions of

elliptic partial differential equations satisfying general boundary conditions II. Comm. Pure Appl. Math. 17, 35-92 (1964).

2. AGMON, S., Lectures on Elliptic Boundary Value Problems. New York: Van Nostrand 1965. 3. COURA~rr, R., & D. HILBERT, Methods of Mathematical Physics, V. 2. New York: Inter-

science Publishers 1962. 4. EIDUS, D. M., The principle of limiting absorption. Mat. Sb. (N.S.) 58 (100), 65-86 (1962)

(in Russian). 5. FIGUEIREDO, D. G. DE, The coerciveness problem for forms over vector valued functions.

Comm. Pure Appl. Math. 16, 63-94 (1963).


6. FRmDRICHS, K. O., Symmetric hyperbolic linear differential equations. Comm. Pure Appl. Math. 7, 345-393 (1954).

7. HALMOS, P. R., Measure Theory. New York: Van Nostrand 1950. 8. KATO, T., Perturbation Theory for Linear Operators. New York: Springer-Verlag 1966. 9. MATSV~JRA, M., Uniform estimates of elementary solutions of first order systems of partial

differential equations. Kyoto University (preprint 1970). 10. Mocmzu~, K., Spectral and scattering theory for symmetric hyperbolic systems in an

exterior domain. Publ. RIMS, Kyoto Univ. 5, 219-258 (1969). 11. SC~Ut~NnVJtOER, J. R., The Green's matrix for steady-state wave propagation in a class of

inhomogeneous anisotropic media. Arch. Rational Mech. Anal. 34, 380-402 (1969). 12. SCHULENBERGER, J. R., Uniqueness theorems for a class of wave propagation problems.

Arch. Rational Mech. Anal. 34, 70-84 (1969). 13. SCmJ~NBERaER, J. R., & C. H. WILCOX, A Rellich uniqueness theorem for steady-state

wave propagation in inhomogeneous anisotropic media. Arch. Rational Mech. Anal. 41, 18-45 (1971).

14. SC'mJLE~mFa~G~R, J. R., & C. H. Wmcox, Completeness of the wave operators for pertur- bations of uniformly propagative systems. J. Functional Anal. (to appear).

15. SCHULm,mERG~R, J. R., & C. H. WmcOX, Coerciveness inequalities for nonelliptic systems of partial differential equations. Ann. Mat. pura appl. 87 (1971) (to appear).

16. SCHWARTZ, L., Th6orie des distributions II. Paris: Hermann 1957. 17. STONB, M. H., Linear transformations in Hilbert space and their applications in analysis.

Amer. Math. Soc. Colloquium Publ. XV (1932). 18. WILCOX, C. H., Wave operators and asymptotic solutions of wave propagation problems

of classical physics. Arch. Rational Mech. Anal. 22, 37-78 (1966). 19. WILCOX, C. H., Steady-state wave propagation in homogeneous anisotropic media. Arch.

Rational Mech. Anal. 25, 201-242 (1967). 20. Wmcox, C. H., Transient wave propagation in homogeneous anisotropic media. Arch.

Rational Mech. Anal. 37, 323-343 (1970). 21. YOSlDA, K., Functional Analysis. Berlin-G6ttingen-Heidelberg: Springer 1965.

Department of Mathematics University of Denver

Denver, Colorado

and

Institut de Physique Th6orique Universit6 de Gen6ve

Switzerland

(Received October 16, 1970)

5 Arch. Rational Mech. Anal., Vol. 41

the limiting absorption principle and spectral theory for...

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