The Limiting Absorption Principle
and Spectral Theoryfor Steady-State Wave Propagation
in InhomogeneousAnisotropic Media
JOHN R. SCIULNBEROER & CALVINH. WILCOX
Communicated by J. SERmN
Abstract
Many wave propagation phenomena of classical physics are governed by systems of the
Schr6dinger form
-iDtu+Au=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 tf'(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 i m 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 classical p h y s i c s to d e s c r i b e t h e w a v e s p r o d u c e d
in a m e d i u m b y t h e a c t i o n of p r e s c r i b e d sources. T h e s o u r c e s a r e f r e q u e n t l y
o s c i l l a t o r s ; t h a t is, t h e y h a v e a s i n u s o i d a l t i m e d e p e n d e n c e . T h e r e s u l t i n g w a v e s
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 Schr6dinger form
-iDtu+Au=f(x, 0
(1.1)
where
A= - i E ( 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
A1, A2, ..., An are Hermitian and constant.
(1.3)
(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
Av-2v=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 tr) 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+
(1.10)
a~O+
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 w2. w3 contains a proof that the continuous spectrum of A is absolutely
continuous. A brief discussion of related literature and unsolved problems is
given in w5.
* If M is a matrix then M* denotes the Hermitian conjugate of M. Thus M * = t ~ where
is the complexconjugateof M.
Wave Propagation in Anisotropic Media
49
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 j u ~ , ~ for j = 1, 2, ..., n, and there exists a constant K > 0
such that
IlOjull2<K(llAull2+ Ilull2)
j=l
* The notation suppf(x) is used to denote the support of a function f(x).
4
Arch. Rational Mech. Anal., Vol. 41
(1.20)
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'~*~
forall x~R" and ~ C m.
(1.21)
It follows that u e ~ if and only if the components us of u are in the Lebesgue
space Xe2 = s 2 (R"). The same remark applies to ~ 0 , the Hilbert space associated
with Eo. Thus ~ , ~0 and *La2,,,=L~v2 ~.Sa2 ~ -.- 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) = Ipl fi(p).
It follows that
D((- A)~)= s
and
(1.28)
Jl(-A)~uj[2= ~ [[D~ulJ2
for all u~D((-A)~).
j=l
The operator
Po: ~o ~ ~Vfo
(1.29)
Wave Propagation in Anisotropic Media
51
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)
J * = E o ~JE
(1.32)
Its adjoint is the operator
g*: ~ - ' * ~ o ,
where E and E o are the multiplicative operators on ~
defined by the matrices E(x) and Eo.
Finally,
A': 9~--*A"
is the operator with domain
and ~ o , respectively,
D (A') = g n {u : Po J* u e La~, m}
(1.34)
A' = J ( - A)§ Po J* .
(1.35)
(1.33)
such that
Property (1.25) is verified in [15]. The operator
A": J/e--, Jet~@ ~
(1.36)
D ( A " ) = D(A) n D ( A ' ) .
(1.37)
is defined by (1.24) and
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 u e D ( 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 SCHULENBERGEg 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 2 e a p ( 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 u n e N ( 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 ( A o ) = 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 w1.
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
Wave Propagation in Anisotropic Media
53
Fr6chet space ~I~rl~ Accordingly, the convergence of v(., 2 + i a ) to v•
be demonstrated in j~,loo. This means that
2) will
lim Ilv(', 2 + i a ) - v• (., 2)[Ix = 0
(3.2)
a-*O+
for every compact K c R n . 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 of 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
Ilull2= 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 ~ R l - ( a p ( 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.
n.--~ oo
(3.7)
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 s u p p f = supp { E - Eo} = B p . Let ~b~ C ~ ( R") satisfy
O< tO(x)<__ 1 and
~k(x)=f0
ll
for , x l < p ,
for I x l > p + l .
(3.10)
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
( j = 1. . . . . n) are bounded, then q~v ~ D (A) and
and if q5 and Djc~
A(dpv)=q~Av+(A cp)v.
(3.12)
A (O v(-, ~))=O(Av(., ~)) + (A ~b) v(., ~)=~kv(., ()+ (A ~b)v(., ~)
(3.13)
Thus
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.
* N o t e that A q J = A o ~ because E = E o on supp ~u.
Wave Propagation in Anisotropic Media
55
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 are1~ 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 $ ) = lim(Aw~., ~b).
n ---~co
(3.18)
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 = lim ((~:. w~:. + g~, ~ ) = (2 w • + g, ~b)
"-' ~
= ( 2 E w • +Eg, q~).~,, --(2Ew• +Eg)(~b).
(3.20)
Thus
A w+ = E(2 w+ + g)eo~ l~
(3.21)
(A-2)w•
(3.22)
Therefore A w• ~ ygtoo and
in ~J~eI~
To prove that w• satisfies the ___radiation condition, apply Lemma 3.6 to w~:,.
This gives
~bw~:. = 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)
Bp+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 <~, I m ( > 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/ Ix 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~]
r
for all (~,Y,•
(3.30)
Proof. Let CR(x) = r (I x [ - R) ~ C~ (R") and have the properties 0 < 6R (X) < 1,
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
~)) = ~ ~bRv(-, 0 + f + (A CR) v(., ~),
(3.32)
since Cg(x)= 1 on s u p p f 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)
Wave Propagation in Anisotropic Media
57
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)=--Oon BR+I and I4~R(x)l < 1. Also llhlla~< I[hll. Thus (3.34) implies
that
=+IIO(',OIIL~,].
ll"(',Oll~,,.,,<=gEllA(4',,v(',O)[I ~ +tIA'(~.~(',r
(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
z ilv(., ~)11~+ itfl12+ l[(ac~R)v(., ~)112]. (3.37)
IIA(~ o(., r
Now
t~
where q = x][x [ and
~b~(x) = ~ qj D~ q5R(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 ~
j=l
llO~Pofll2+j ~= l IID~Po[(Ad?R)v(.,r
IIz
" llPoDjf[[z+ ~ IIPoDj[(Adp,)v(',C)][[2]
j=l
=2 [ E
tj=l
<2 '= llOiftl2+j ~= l IIOj[(aq~R) v(., ~)]11z
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 (-, ~)II2 ~ 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)
=
S
DiG(x-y, ()(A o $(y)) v(y, ()dy
(3.44)
Bp+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
for
(3.47)
Tx-y]>c>O and ( e Z + . It follows that
iOj G (x - y, () l < Cs IR - p - 11 - ("- 1)/2< C6
for Ix[ >__R>p+1, [ y[ < p + 1 and ~27•
(3.48)
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(', ()11z= ~ 14/R(x)D~v(x,~)12dx
R~
< c l o Ilv(., 011~ j" [~'R(x)]2dx<flt IIv(., 011~.
(3.51)
Rn
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.
Wave Propagation in Anisotropic Media
59
L e m m a 3.10. Under the hypotheses of 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+lxl)-l-adx<~llv(.,()[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
Ixl > p +
1 and
~•
Iv(x,O[_-<
~
Bp+l-Bp
_-<Cla
I G ( x - y , ff)l [(Ao~k(y))v(y,O[dy
~
Ix-yl-("-l)/2lv(y,~)ldy.
(3.54)
Bp+l--Bp
Now if Ix] > p + 1 and [y[ < p + 1, the triangle inequality gives [ x - y
Hence
I v ( x , ~ ) l < C l a ( l x l - p - 1 ) -("-1)/2
~
I > Ix[ - p - 1.
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~6llv(',0ll~
~ Ixl-"+*(l+lxl)-~-~dx
Ixl>=R
< C~7 [Iv(', ~)ll~ R - a < n [Iv(.,
(3.56)
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(-, ff~)l[~: ~ -
(3.58)
;1 ---~oo
Define
w_+. = Ilo(., ff~,)ll; 1 v(., ~ , )
(3.59)
so that
IIw•
and
,
(h-~•177
=1
for all n
(3.60)
~, ) i - ~ r -
(3.61)
5,
~ J=g,.
Then s u p p g , = s u p p f c B p and g , ~ 0 in ~ when n--*oo. Thus L e m m a 3 . 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
ll(n) =
_ ~ w•177
(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
liml2(n)=
n-.oo
j" w • 1 7 7
(3.65)
Ixl=<~l
Now
1 = tlw~:~I1~= I1 (n) + I2 (n)
(3.66)
whence by (3.62)
I2(n)=l-I1(n)>l/2
for n = l , 2 . . . . .
(3.67)
+ l x l ) - l - ~ d x > l/2
(3.68)
Hence (3.65) implies that
w• (x)* E ( x ) w •
lxl_-<R
which contradicts the conclusion reached above that w•
the proof of Theorem 3.4.
This completes
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)
Wave Propagation in Anisotropic Media
61
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 i m v ( . , ~•177
exists strongly in
n...~ oo
9s
But v(-, ~•177
(., 2) weakly in :~a and hence also weakly in ~loo,
by Lemma3.5. It follows that w • 1 7 7
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 when
2
~--}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
IIv (., () - v • (., ,~)IIB~ < ~/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 1 2 - 2 j 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+__itrand 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~_-__~
provided tr < d. This completes the proof of Theorem 3.3.
(3.80)
62
J.R. SCHULENBERGER& C. H. WILCOX:
4. The Absolute Continuity of the Continuous Spectrum of A
The selfadjoint operator A introduced in w1 has a spectral representation [8]
A= ~ 2d//(,b
(4.1)
~00
where/7 (2) is a spectral family on ~ . If ~
by the eigenfunctions of A, then
is the closed subspace of W spanned
. ~ = ~ ~ af,P~
(4.2)
and it is known that this orthogonal decomposition reduces A. It can be shown
that [8, p. 515].
~={ue~:
(H(2)u, u) is continuous on R1}.
(4.3)
Accordingly, ~'~cis 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
respect to Lebesgue measure}
(4.5)
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 w1.
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].
Wave Propagation in Anisotropic Media
Lemma 4.2. Let s
63
~-(o'p(A)U{O}) and let feC~o(R"). Then
(1-1(5p) 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 f E
(/7(A)f,T)=
lim 1/2rciS([Ra+,,(A)-Ra_,,(a)]f,f)d2.
a~0+
(4.10)
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 5a = A. The general case follows from this immediately
because both sides of (4.12) have unique extensions to the a-ring of Borel sets 6a.
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 (5a) fa, fa) = (/7 (5a) H (A) f , / / ( A ) f )
= (H (S~) 17 (A) f, 17 (5a)/7 (A) f ) = (II (Sp c3 A) f, f )
=l/2zci S ( v + ( . , 2 ) - v _ ( . , 2 ) , f ) d 2
$ 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.
i=l
Also
II(Ai)f=lim[H(ai-5 ) ~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.
It 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 0001467-A-0394-0002.
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4.
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Department of Mathematics
University of Denver
Denver, Colorado
and
Institut de Physique Th6orique
Universit6 de Gen6ve
Switzerland
(Received October 16, 1970)
5 Arch.RationalMech.Anal.,Vol.41