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a'^p, SKETCH OF PROOF.
(3.9)
/« = P?/^
Since P% does not alter the data inside the sphere \x\ W)
= [v±(Xy z, w), izv±ix, z, «)]
and define 0± = _ and hence (3.11) shares with (3.10) the property of being an isometry in Z>_. Step two. Extend the isometric property of the representation to all translates of D-. A simple integration by parts shows for a n y / in DA that
— /_(/) = izf.it), at
where f(t) = U(t)f. As a consequence /_(/) = e*''/_. This extends the isometry of the map to all of the translates of P - and hence to all of H since the translates of D. are dense in H (see [s]). It also follows that U{t) is represented as multiplication by e*'* in this representation. Step three. The map /—»/_ is onto Lz^— 00, 00 ; JV). In the case of the free space representation of Uoit) it is known that the translates of P _ fill out Z,2(— 00, 00 ; iV) in the representation space. Since D- and translation are represented by the same objects in both the free space and incoming spectral representations, it follows that the map f—^f- is onto. THEOREM
14. The scattering operator is given by
A ( 0 , co) = [S(.)/_(0, .)](co)
= }-{z, co) - 2{2'n-yiHz f
s{-d,«, z)* /_(z, e)de,
where ^_(r^, z, 0») '^ f-^e*"" s{^, . SKETCH OF PROOF. It suffices to determine the behavior of 2>{z) on D- since S commutes with U{t) and since translates of Z>_ are dense
26
142
p. D. LAX AND R. S. PHILLIPS
in H. Now for / in D_, / - = / o so that ${z)f-=}+ simply becomes
s(2)/_ = /_+(/,^_),
fen-,
A straightforward calculation now yields (3.12). This is roughly what one expects from the classical theory and shows in particular that $(z) differs from the identity on N by an operator with a smooth kernel. Much of the foregoing can be generalized to solutions of symmetric hyperbolic equations which satisfy conservative boundary conditions on some obstacle, provided that these boundary conditions are elliptic for the spatial part of the operator. That there are conservative boundary conditions which are not elliptic is somewhat surprising. REFERENCES 1. A. Beurling, On two problems concerning linear transformations in Hilhert space^ Acta Math. 81 (1949), 239-255. 2. P. R. Halmos, Shifts on Hilhert spaces, J. Reine Angew. Math. 108 (1961), 102112. 3. E. Hille and R. Phillips, Functional analysis and semi-groups^ Amer. Math. Soc. CoUoq. Publ. Vol. 31, Amer. Math. Soc, Providence, R. I., 1957. 4. H. Helson and D. Lowdenslager, Prediction theory and Fourier series in several variables, Acta. Math. 99 (1958), 165-202. 5. P. D. Lax, Remark on the preceding paper of Koosis, Comm. Pure Appl. Math. 10(1957), 617-622. 6. , Translation invariant spaces, Acta Math. 101 (1959), 163-178. 7. , Proceedings of the International Symposium on Linear Spaces, Hebrew University, Jerusalem, Pergamon Press, 1961, pp. 299-306. 8. P. D. Lax and R. S. Phillips, The wave equation in exterior domains, Bull. Amer. Math. Soc. 68 (1962), 47-49. 9. P. D. Lax, C. S. Morawetz and R. S. Phillips, The exponential decay of solutions of the wave equation in the exterior of a star-shaped obstacle. Bull. Amer. Math. Soc. 68 (1962), 593-595. 10. J. W. Moeller, On the spectra of some translation invariant spaces, J. Math. Anal. AppL 4 (1962), 276-296. 11. R. S. Phillips, Spectral theory for semigroups of linear operators, Trans. Amer. Math. Soc. 71 (1951), 393-415. 12. A. Ya. Povzner and I. V. Suharevskil, Discontinuities of the Green's function of mixed problems for the wave equation, Mat. Sb. (N.S.) 51 (1960), (93) 3-26. 13. Y. Foures and L E. Segal, Causality and analyticity, Trans. Amer. Math. Soc. 78 (1955), 385-405. N E W YORK UNIVERSITY AND STANFORD UNIVERSITY
27
COMMUNICATIONS ON PURE AND A P P L I E D MATHEMATICS, VOL. XXII, 1?>1-1Q1
(1969)
Decaying Modes for the Wave Equation in the Exterior of an Obstacle* p. D. L A X A N D R. S. P H I L L I P S Abstract In this paper we study the dependence of the set of 'exterior' eigenvalues {A^} of A on the geometry of the obstacle 0, In particular we show that the real eigenvalues, corresponding to purely decaying modes, depend monotonically on the obstacle (9^ both for the Dirichlet and Neumann boundary conditions. From this we deduce, by comparison with spheres— for which the eigenvalues {A;^} can be determined as roots of special functions—upper and lower bounds for the density of the real {A;^}, and upper and lower bounds for Aj , the rate of decay of the fundamental real decaying mode. We also consider the wave equation with a positive potential and establish an analogous monotonicity theorem for such problems. We obtain a second proof for the above Dirichlet problem in the limit as the potential becomes infinite on (P. Finally we derive an integral equation for the decaying modes; this equation bears strong resemblance to one appearing in the transport theory of mono-energetic neutrons in homogeneous media, and can be used to demonstrate the existence of infinitely many modes. 1. I n t r o d u c t i o n We are concerned in this paper with the behavior of solutions of the wave equation in the exterior of an obstacle. T h e differences between the exterior and interior problems are so great that they tend to hide the points of similarity. O u r purpose is to bring out certain analogies between these two problems and to this end we first discuss the relevant, but familiar, facts about the interior problem, that is, the behavior of solutions of the wave equation (1.1)
M,,
- A M
=
0
in some smoothly bounded compact domain (9 on whose boundary u is required to satisfy a boundary condition, say (1.2)
u = 0
or
d^u = 0
on
d(9.
* The first author was partially supported by the U.S. Atomic Energy Commission under Contract AT(30-1)-1480 at the Courant Institute of Mathematical Sciences. The second author was supported partly by the National Science Foundation, grant NSF GP-8857 and the Air Force Office of Scientific Research Contract AF-F44620-68-C-0054. Reproduction in whole or in part is permitted for any purpose of the United States Government. 737
28
738
p . D. LAX AND R. S. PHILLIPS
T h e spectral decomposition of A over & leads to the following representation of the totahty of solutions of ( L I ) , (1.2): 00
(1.3)
u{x, 0 = 2 i^k^'"'' + h^-'^'^Ä^)
;
k=l
here {jul} are the eigenvalues of —A arranged in increasing order with //^ > 0 and {vj^} are the corresponding eigenfunctions: (1.4)
Av, + f,lv, = 0,
v,{x) = 0
on
d(P.
For the interior problem, ( —A)~~^ is a positive compact operator and therefore the {/Uj^} form a sequence of positive numbers tending to oo; each solution of the wave equation is represented in (1.3) as a superposition of harmonic motions with frequencies /Uj^ . These frequencies are functionals of the domain &^ and much effort has gone into studying the dependence of the set of numbers {/Uj^} on the geometrical properties of d?. T h e following results described in Vol. I of [1] are particularly interesting mathematically and significant from the point of view of physics (for more recent results see also [12]): 1. /u^{(P) depends mono tonic ally on G\ that is, ii (9^ c: C)^ , then (1.5)
i,^{(9,) ^fz,{&,)
for
all
k.
2. T h e asymptotic distribution of the jUj^ for large k is / k V/«
(i.6)
"^-'"{nvl
•
where n is the dimension of the x-space, ß the volume of the /z-dimensional unit ball and V the volume of &, 3. T h e isoperimetric inequality. Among all domains (9 with given volume F, the sphere has the smallest fundamental frequency //^ . We turn now to the behavior of solutions of the wave equation in the exterior ^ of (P, subject to the same boundary condition (1.2). In this case, —A has a continuous spectrum (of infinite multiplicity) extending from 0 to oo and one can again express all solutions of the wave equation as a superposition of harmonic motions involving all frequencies. It turns out that such a representation as it stands sheds no light on the asymptotic behavior of w(x, t) for large t with x fixed. T o get some idea of what kind of asymptotic representation to look for, we first recall that the solution to the wave equation in free space of an odd number of dimensions obeys Huyghens' principle; thus for initial data having compact support, say contained in {x: \x\ < 7?}, the solution will vanish in the cone {x: \x\ .^/^Ai ^ ^ ^ ^ 2 ^ • • • - > - 0 0 .
We now give a brief resume of the theoretical basis for (1.7) and (1.8). Let H denote the Hilbert space of all initial data with finite energy, normed by the energy norm
;i.9)
||«|| =^-^[\du\^ + \u,ndx.
integrated over the exterior domain. Let p be chosen large enough so that & lies inside of the ball {x: \x\ < p). We call a solution outgoing if it is zero for 1^1 < ^ + p, ^ ^ 0, and incoming if it is zero for |;c| < p — /, / ^ 0. T h e set of initial data for all outgoing (incoming) solutions we denote by D^ (D^), L e t U{t) denote the operator which maps initial data into data at time t; it is clear that t h e {U{t)} form a one-parameter group of unitary (energy conserving) operators on H. Next we define the operators (1.10)
z ( 0 = Piu{t)p^,
/ ^ 0,
where P^ (P^) is the orthogonal projection onto the orthogonal complement of DP (Z)^). T h e effect of the projection P^ is to remove signals which might be coming in from far away and the effect of P^ is to remove that part of the signal which has already been converted into an outgoing wave and no longer interacts with the obstacle. Since the data in D^ and D^ are zero inside the ball {\x\ < p}, we see that^ for data / with support in this ball, (1.11)
[Z{t)f]{x)=^[U{t)f]{x)
for all \x\ 0, U{t) maps D^ into itself implies that the operators {Z{t)} annihilate D^ and D^ and form a semi-group of operators on the
30
740
p . D. LAX AND R. S. PHILLIPS
subspace (1.12)
K^ = HQ {D^ ®DP_).
It is a well-known fact in the theory of semi-groups of operators that if Z{t) is a compact operator for some t > 0, then for e v e r y / ' i n K^ one can express Z{t)f asymptotically as
(1.13)
Z(/)/^2v'^XW'
where the {X^} are the eigenvalues and the [w^] the eigenfunctions of the infinitesimal generator B of {Z{t)]. Since the operators Z{t) are contractions, 0^^ A;^ < 0 ; and this combined with (1.10) gives the desired relations (1.6) a n d (1.7). T h e parameter p is arbitrary; happily however t h e eigenvalues {X^ do not depend on p, a n d neither do the eigenfunctions for |^| < p. I n fact the w^{x) obtained for various values of p converge as p —> oo to an eigenfunction of A, with X\ as eigenvalue: (1.14)
^w^ = Xlw^
in
^.
These eigenfunctions behave asymptotically like |;^|"~^ exp { — Xj^ \x\) for large \x\ and therefore lie outside the Hilbert space H. They d o however satisfy an outgoing radiation condition. T o connect the eventual compactness of Z{t) with the geometrical properties o{(9^ we introduce the following notation: Consider all rays starting on the sphere of radius p which proceed toward the obstacle a n d are continued according to the law of reflection whenever they impinge on & until they leave the ball {\x\ < p ) . We call (9 confining if there are arbitrarily long rays of this kind; otherwise, (9 is called nonconfining. Surmising that sharp signals propagate along rays we conjectured (see pages 155-157 of [4]) that Z{t) is eventually compact if and only ii 0 is nonconfining. Ralston [10] has shown in a n important special case of confining obstacles that Z{t) is not compact for any t. I n the opposite direction, Ludwig and Morawetz [5] (see also Phillips [8]) have shown that if (9 is convex then Z{t) is eventually compact. In this paper we study the dependence of the set of ''exterior" eigenvalues {X^ on the geometry of the obstacle 6. I n particular, we show that the real eigenvalues, corresponding to purely decaying modes, depend monotonically on the obstacle C?, both for the Dirichlet and Neumann boundary conditions. From this we deduce, by comparison with spheres—for which the eigenvalues [Xj^ can be determined as roots of special functions—upper and lower bounds for the density of the real [Xf^] a n d upper a n d lower bounds for X^ , the rate of decay of the fundamental real decaying mode. W e also consider t h e wave equation with a positive potential a n d establish an analogous monotonicity theorem for such problems. Finally we obtain a second proof for the above Dirichlet problem in the limit as the potential becomes infinite on (9.
31
DECAYING MODES FOR THE WAVE EQUATION
741
O u r results indicate, roughly speaking, that the real exterior eigenvalues are influenced by bulk properties of the obstacle, while the complex ones are sensitive to surface details. This is merely the first step toward discovering the true relation between the shape of the obstacle and the location of the exterior eigenvalues. W e have written our results in an expository style in the frank hope of attracting others to this interesting and important problem. CONTENTS: Section 2 contains the monotonicity theorem for the transmission coefficients. Section 3 applies these results to derive comparison theorems for the imaginary zeros of the scattering matrix. Section 4 contains the proof of a comparison theorem for the eigenvalues of nonsymmetric operators. Section 5 describes an approximation to the transmission coefficient which is used to determine the asymptotic distribution of real eigenvalues for the sphere. Section 6 contains a comparison theorem for transmission coefficients for potentials in place of obstacles, and Section 7 contains a rigorous proof of the folk theorem that the transmission coefficient for an obstacle with Dirichlet boundary condition is the limit of the transmission coefficients for a sequence of potentials which are zero outside the obstacle and tend to infinity on the obstacle. In Section 8 we derive an integral equation for the decaying modes and use it to show that there are infinitely many such modes. In Section 9 we point out a strong similarity of this integral equation to one which appears in the transport theory of mono-energetic neutrons in homogeneous media. In an appendix we give a heuristic derivation of the relation between the transmission coefficient and the scattering matrix, and the relation between the zeros of the scattering matrix and the decaying modes of the exterior problem. 2. T h e T r a n s m i s s i o n
Coefficient
T h e eigenvalue problem (1.14) imposes an outgoing radiation condition on the eigenfunction. We found it more convenient to make use of a different characterization of the eigenvalues {A^} of the generator B of Z(f), namely one which involves the scattering matrix. As we shall see the ''near-symmetry" of the scattering matrix is especially helpful. In this section we merely state this connection; however in the appendix we present a short course on scattering theory in which all of the relations used are derived or at least made plausible. T h e correspondence between the scattering matrix and the eigenvalues of Z{t) is easily stated: A is an eigenvalue of B if and only if il is a zero of the scattering matrix ^{z) and the degree of multiplicity is the same (see [4], Theorem 3.1 of Chapter 3). Moreover, ^ ( z ) is meromorphic having as its poles precisely the points —iX for which il is a zero of .$^(z) (see [4], Theorem 5.1 of Chapter 3). T h e scattering matrix 27r . J-i
O n replacing /g in (2.6) by its limiting value —47rÄ:(a>5 6; er), we obtain (2.5). COROLLARY 2.2. For the Dirichlet and Neumann problems ak{a), 6; oo. Since (2.8) is a positive definite quadratic form and since (2.9) is the Euler equation associated with that quadratic form, we have the following extremal characterization of (2.8): COROLLARY 2.3.
In the case of the Dirichlet problem^ {ka, a) = inf —
[aW^ + (V^)^] dx
4-77 J[R3
over all smooth functions B with compact support in fRg which^ in (9^ are equal to J exp {ax ' (o}a{a))dco. In the case of the Neumann problem^ •--{ka, a) = inf— [a^B^ + (V^)^] dx 4-77 jRg over all smooth functions B in (P ^ ^ which vanish near infinity^ and which are equal to J exp {ax ' co} fl(co) dw in (9 and have a continuous normal derivative {but need not be continuous) across We come now to one of our main results.
36
746
p . D. LAX AND R. S. PHILLIPS
THEOREM 2.4. Denote by k^ and k^ the transmission coefficients for the scattering objects &^ and &2 5 respectively. If ^\ ^ (^2 ^ ^^^^^ considered as operators on L2{S2)^ (2.10)
oik^{a) < oik^ia)
for
all
a > 0;
here a = 1 for the Dirichlet problem and a = •— I for the Neumann problem. Proof: In the case of the Dirichlet problem, the result follows directly from Corollary 2.3 since in going from ^^ to (D2 we decrease the class of admissible functions. T o prove the corresponding assertion in the N e u m a n n case we set exp {ax • o)} a{a)) day
for
x
in
(P^ ,
v^{x, co; a) a{oj) dco
for
x
in
^^ ,
/•
,/• and
C{x) =A2{x)
~^iW.
Then C{x) is smooth and satisfies the differential equation (2.9) in 0^ ^ } is greater than v near infinity. Again by the maximum principle, we conclude that exp {ax • a>} ^ v{x^ co; a) throughout ^ . Suppose that the scatterer (^g contains (P^ ; then it follows from the foregoing that on the boundary of ^ g ^ ^ have Vi{x^ co; a) ^ exp {ax • o)} = v^ix^ co; a) , Since both v^ and V2 are solutions of the reduced wave equation in ^ 2 which vanish at infinity, it follows that Vi{x, co; a) :^ ^2{^y ^ ? ^) throughout ^ 2 ? ^^d so the same is true of the corresponding transmission coefficients: A:i(co, Ö; a) ^ k^icoy 6; a) . Thus, positivity and monotonicity of the transmission coefficients hold not only in the operator sense but also in the pointwise sense for the Dirichlet problem. Of course, as is well known, these two concepts of positivity are quite distinct, neither implying the other; nevertheless there is some interest attached to those kernels which are positive in both senses and it amused us to find that the transmission coeflBcients belong to this class of kernels.
38
748
p . D. LAX AND R. S. PHILLIPS
3. O n t h e P u r e l y I m a g i n a r y Z e r o s o f t h e Scattering M a t r i x We recall that a purely decaying mode of Z{t) with eigenvalue e~^^ corresponds to a purely imaginary zero of t h e scattering matrix ^{z) at z = -—ia with the same degree of multiplicity. Since, in the notation of the previous section, 6/^[—ia) = I + K{(y), this simply means that the purely decaying modes of Z{t) correspond to those positive values of a for which -— 1 is an eigenvalue of K{a)\ the kernel o^K{a) is given by (3.1)
K{co,e;a) = ~ k{aj, -6; a) ,
where k is the transmission coefficient. by W: \WaW)
Denoting reflection through the origin =a{-d),
the relation (3.1) can be written in operator form as (3.2)
K{a) = f
k{a)W .
According to Corollary 2.2 and Theorem 2.4, ak{a) is a symmetric strictly positive Hilbert-Schmidt operator on L2{S2); a = 1 for the Dirichlet problem a n d - - 1 for the N e u m a n n problem. T h e presence of W complicates the problem since K need not be symmetric. Nevertheless, the following comparison theorem for K is valid. T H E O R E M 3.1.
(a)
The eigenvalues of K{a) are real,
(b) Let K-^{a) and K2{(y) be defined as above for obstacles 0^ and 6^ , respectively, and suppose that 0^ ^ &2 - For fixed c > 0, arrange according to size the positive and negative eigenvalues of K^{a) taking multiplicities into account: ^u) ^ r^^) ^ . . . > 0 > • • • ^ K^ ^ K^-i^ ,
z = 1, 2.
Then, for all integers n, (3.3)
0.
39
DECAYING MODES FOR THE WAVE EQUATION
Proof:
749
In this case the solutions of (2.3)j) and (2.3)2^ have the property v{x, - c o ; a) = v{—Xy co; a) ,
so that k{ — cüy 6; a) = A;(co, —Ö; or) . Consequently^ [kWa]{(o) = [k{co, 6; a) a{-d) = (k{-co,
dd = lk{a), - Ö ; a) a{B) dd
6; a) a{d) dd = [ Wka]{co) .
Moreover, iT* = ( ^ { ^ ) * = Wk = kW = K, as asserted. Step 2, For the Dirichlet problem the positive (negative) eigenspaces of K consist of the even (odd) eigenspaces of k^ whereas for the N e u m a n n problem the positive (negative) eigenspaces of K consist of the odd (even) eigenspaces of ^. H
Proof: T h e involution W decomposes L2{S2) into two orthogonal subspaces and H_ consisting of the even and odd functions, respectively: Wa = a
for all a
in
H^
and
Wa = —a
for all a
in
H_.
Since both k and K commute with Wy each eigenspace of k and K decomposes into the orthogonal sum of a subspace in H_^ and one in H_ . Further, if {y, X} is an eigenpair of k with v even, then {v^ aXßTr) is an eigenpair for TT, whereas if V is odd, then {v^ — aXftir} is an eigenpair for K, and conversely. T h e assertion of step 2 now follows from the fact that k is strictly positive for the Dirichlet problem and strictly negative for the N e u m a n n problem. Step 5. T o complete the proof of the theorem in the case of centrally symmetric objects it suffices to combine steps 1 and 2 with Theorem 2.4 and the minimax principle applied to H_ in the Dirichlet case and H^ in the N e u m a n n case. We now have a substantial grip on the problem of determining the purely imaginary zeros of the scattering matrix; we must find positive values of a for which —1 is an eigenvalue o{ K{a), It therefore suffices to study the growth of the negative eigenvalues {K^{a)] of K{a) as a function of or, picking out those values of (T and n for which /C^(Ö') = — 1. It is known that k{(y) is analytic in a for real o' ^ 0; it therefore follows from the relation (3.2) that K{(r) converges to zero as cr —> 0 + . T h u s the smallest purely imaginary zero of the scattering matrix comes from the smallest root CTQ of ACI((T) = — 1 and hence we obtain, as
40
750
p . D. LAX AND R, S. PHILLIPS
an immediate consequence of Theorems 2.4 and 3.1^ THEOREM 3.2. ^ ^ i ^ ^2 ^ ^^^^ ^^^ smallest purely imaginary zero of ^^ is greater than or equal to the smallest purely imaginary zero of 6^2 J ^^^^ ^^J ^0^^ = ^^^^ I n general the negative eigenvalues {A^(ör)} are not monotone decreasing functions of cr. However, the situation is comparatively simple for star-shaped obstacles. LEMMA 3.3. If & is star-shaped^ then the negative eigenvalues of K{&) are monotone decreasing functions of cr. Proof: W e m a y as well suppose that & is star-shaped with respect to the origin. Let rO be the figure obtained from (!) by the transformation: x -^ rx; 6 being star-shaped means that for r > 1 we will have rO '=^ 6. It is easy to verify that the corresponding solution of (2.3)j) (or (2.3)^) for T(9 and (9 are related by v{xy oj; (T5 rO) = v{T^^Xy o); ra^ (9) , so that (3.5)
k{a), d; a, r(9) = Tk{co, d; ra, (9) .
By the relation (3.2), it follows from (3.5) that (3.6)
K{(y, r(9) = K{ra, &) .
Since & c: rO^ Theorem 3.1 applies; we have K,{a,
rO) ^ K,ia, &),
T > 1,
and combining this with (3.6) we obtain
(3.7)
K„(T(r) ^ K„(cr) ,
r > 1 .
This proves that /c„((r) is a non-increasing function of o*. It is known however for an analytic family of compact operators such as K{a) that the eigenvalues are analytic except for crossing points. Since K^(Ö') tends to zero from below as c; —> 0 + , we see that K^{(y) can not remain constant in any interval and therefore must be monotone decreasing. COROLLARY 3.4. If (P is star-shaped and if for a given a^ n of the eigenvalues of K{(y) are less than or equal to —l^ then the scattering matrix has exactly npurely imaginary zeros {—^ö'fc} with a^ ^ a. For a general obstacle i't seems likely that the negative eigenvalues oi K{a) will not be monotone decreasing functions of a. I n this case^, the comparison theorem furnishes us with a lower bound for the n u m b e r of zeros of the scattering matrix in a given interval.
41
DECAYING MODES FOR THE WAVE EQUATION
751
THEOREM 3.5. If 6 0 > • • • ^ /cj^> ^ /cj^>. Then^ for all integers 72, (4.11) Proof:
vl^^^vl^^
and
K^I> ^ /c^^> .
For all w in ^(-^i) it follows from L e m m a 4.3 t h a t (Ww^ w)
(Wu)^ w)
if
{Ww,w)
^ 0 ,
and
iMr'a;r-iM>ip
^ ' '-
T h e assertion of the theorem is now an immediate consequence of L e m m a 4.2. Before starting the second proof of this theorem we introduce some concepts associated with the theory of indefinite forms (see [7]). A subspace N (or P) is called negative (positive) if Q{x^ x) ^ 0 ( ^ 0 ) for all x in N (P), and it is called maximal negative (maximal positive) if it is not the proper subspace of any other negative (positive) subspace. If these inequalities are strict inequalities for all non-zero elements of N (or P ) , then the subspace is called strictly negative (strictly positive). T h e next theorem generalizes the familiar minimax characterization for the eigenvalues of a symmetric operator to operators which are symmetric with respect to an indefinite form Q.
45
DECAYING MODES FOR THE WAVE EQUATION
755
THEOREM 4.5. Assume that K is a compact Q-symmetric, strictly Q-positive operator in H. Order the positive and negative eigenvalues taking into account multiplicities: (4.12)
Vi ^ ^2 ^ • • • > 0 > • • • ^ Kg ^ /Ci .
Let V denote an arbitrary subspace of dimension n — \ and let N and P denote strictly negative and strictly positive maximal sub spaces. Finally set
U{ V, P) =
sup uep
Q{u, u)
Q(w,F)=0
(4.13)
^
Q(Ku,u)
Q(u,F)=0
Then v^ = inf inf P
(4.14)
U{V,P),
VczP
K^ = sup sup A ( F , P) N
VczN
Applying this to Q{x,y)
=
{Wx,y)
and X = A:W^, we obtain a characterization of the eigenvalues of X which shows directly their monotonic dependence on k. A proof of Theorem 4.5 will appear in [9].*
5. E s t i m a t e s f o r t h e D i s t r i b u t i o n o f t h e I m a g i n a r y Z e r o s o f t h e Scattering M a t r i x T H E O R E M 5.1.
(5.1)
Define
k^iw, Ö, cr) = — (1 + a> • 6)^^ 477
exp {ax • (a> + 6)} dx , JQ
where a = 1 for the Dirichlet problem and •— 1 for the Neumann problem. (5.2)
cf.k^{a) ^ (f.k{a) ^ 3aÄ:o((r) ;
the first inequality holds for all cr > 0, the second for all sufficiently large a.
46
Then
756
p . D. LAX AND R. S. PHILLIPS
Proof: T h e lower bound follows directly from Corollary 2.3 by restricting the integration to (9. I n fact, since A{x) in (9 is given by (2.7), we obtain a(^fl, a)^—
\
47T Je
[a^A^ + {VA)^] dx
= JL \\\a\\
+ (o • 0)^^^*^^^^*^ a{oy) a{d) dco dB dx .
0
Carrying out the x integration first, we can rewrite the right side as a(A:Qfl, a) and so obtain (5.3)
a(A:a, a) ^ a(A:oa, a) .
T o obtain an upper bound we need only choose a particular admissible function B. Denoting this function by B^ , we have of necessity B^{x) = A{x) in &. We extend B^ outside of (9 by reflection across the boundary of Ö?, followed by multiplication by a smooth function 99, constructed so that 0 -^ (p{x) ^ 1 and [\in(9
,
(^ 0 at all points whose distance from 6? is > ^ . I n other words, we set [A{x) in (9, (5.4) ^ ^
BAx) ^^ ^
[oi(p{x)A{T{x)) outside (9 ;
here T{x) is the image of ;c under reflection across d(P. Note that B^{x) vanishes at all points whose distance from 0 is greater than (5, t h a t B^ is continuous across d(9 for a = 1 and that d^B^ is continuous across 90? for a = — 1. Let J denote the J a c o b i a n of T; then J is unitary on dO a n d hence almost unitary near (9. I n particular, at all points in the support of 99 b u t outside of (9 the n o r m of J can be estimated by (5.5)
\J\^l
+s,
where e tends to zero with d. Now, V^^ = VcpA +
(pJVA.
If/? denotes an upper bound for |V99I, then (5.5) and a n obvious estimate give {WB,y ^ ß^A^ + 2ß \A\ (1 + s) \VA\ + (1 + (5.6)
.(..g /
/?2\
47
e^iVAy
DECAYING MODES FOR THE WAVE EQUATION
757
Making use of (5.5) once again and of (5.6)5 we get j^la^Bl
+ {VB,)^]dx
^ (1 + ^)jJ[_[o^
+ ß' + j^A^
+ {l+
e)%VA)^] dx .
T h u s by choosing d sufficiently small so that (1 + e)^ ^ 2 and a sufficiently large, we have f [a^Bl + (V5,)2] dx^2[ j^
[a^A^ + (V-^)^] dx . Je
Since B^ is admissible for both the Dirichlet a n d the N e u m a n n problem, Corollary 2.3 shows t h a t
(5.7) ^ —
[crM^ + (VJ)2] dx = 3 a ( V . ^) •
477 J(P
This establishes the upper bound in (5.4). W e recall that the purely imaginary zeros of the scattering matrix correspond to those values of a for which —1 is a n eigenvalue of K{a). According to t h e theory sketched in Section 3, for symmetric obstacles this means that plus 1 is an eigenvalue of ((r/27r)aA:(ö') with a n odd eigenfunction in the case of the Dirichlet problem a n d a n even eigenfunction in the case of the N e u m a n n problem. Denote by Cjy{a) a n d C^{(y) the n u m b e r of eigenvalues of {(yf27T)oLk{a) greater t h a n 1 which correspond to odd, respectively even, eigenfunctions. Since {af27r)oik{a) depends continuously on a a n d vanishes at cr = 0, we conclude: THEOREM 5.2a. For symmetric obstacles the scattering matrix has at least Cjy{a) (CJ^((T)) purely imaginary zeros of absolute value less than o for Dirichlet {Neumann) boundary conditions. Now for star-shaped obstacles we showed in L e m m a 3.3 that increases monotonically with a. This implies
{al27r)ak{a)
THEOREM 5.2b- For symmetric star-shaped obstacles the scattering matrix has exactly ^ D ( ^ ) ( ^ N ( ^ ) ) purely imaginary zeros of absolute value less than a for Dirichlet {Neumann) boundary conditions. According to Theorem 5.1, for large a the operator {aJ27r)oLk{G) is bracketed between ((7/277)0^0(0') a n d {3a/27r)ixkQ{a); furthermore, for symmetric obstacles the subspaces of even a n d odd functions reduce both k{a) and A:o(^)- Therefore,
48
758
p . D. LAX AND R. S. PHILLIPS
again for large c, it follows by the minimax principle that
and
here C^(Ö') and C^(Ö') are the numbers of eigenvalues with odd, respectively even, eigenfunctions of {af2Tr)oLkQ{ö) whose values are greater than 1, and C^{a) and CN(O') are the numbers of eigenvalues with odd, respectively even, eigenfunctions of {aJ2rr)oikQ{a) whose values are greater than \. As will be seen in the example given below, it is reasonable to expect that C{^{(y) will not differ appreciably from C2>(cr), nor C^{(y) from C^(ö'). If this were so, the asymptotic distribution of the purely imaginary zeros of the scattering matrix of a symmetric star-shaped obstacle would be completely determined by the functions Ci){a) and C^(Ö'). T h e kernel A:Q(a>, 6; a) is given explicitly by the rather simple formula (5.3) and depends additively on 0. W e expected no great difficulty in determining the asymptotic distribution of its eigenvalues, but so far we have succeeded in this only for the case of the sphere, which we now present. W e begin by transforming the quadratic form defining A:Q(CT) by Green's formula: Using the fact that A satisfies the equation a^A — AA = 0, we get a{k,a, a) = - i
I [a^A^ + {VA)^] dx == ^
\ A B^A dS ;
substituting for A as defined in (2.7) we get, after interchanging orders of integration, t h a t (5.8)
oik^ico, 0; Ö') = —
[e^^'^d^e^'^'^ + ^^^'^a^^^^'^] dS .
Next we expand exp {ax • co} into spherical harmonics; we start with the expansion of the exponential function: (5.9)
.- = i
cMPnir) ,
where P^ is the n-th Legendre polynomial. T h e coefficients c^ are given by
(5.10)
cM
= ^ ^ ^
\[fPn{r) dr .
I n particular, (5.11)
e-^-''^J,cJ,a)P,{r^'oy).
49
DECAYING MODES FOR THE WAVE EQUATION
759
W e now take (P to be a spherical ball of radius R about the origin; in this case (5.8) can be rewritten as 0Lk,{a>, d;a)=——\
»77 dR J\n\=.\
e^^^'^e^^^'' drj.
Substituting the expansion (5.11) a n d using the orthogonality relations
(5.12)
f P J T ? • co)P^{n 'e)dr] J\ri\=i
= - ^ (3^.,P(co • 6) , 2n + 1
we get (5.13)
c,k,{co, 0; (T) = - 2 r — — - 4{aR)P„{co 2 n=o 2n + I dR
• 0) .
According to the orthogonality relations (5.12) the operator with kernel P^(a> • 6) annihilates all spherical harmonics of order different from n a n d is equal to 47T{2n + 1)~V on the space of n-th order spherical harmonics. This proves PROPOSITION 5.3.
The eigenvalues of {a/27T)oikQ{a) are
with multiplicity 2n + 1. The eigenfunctions in the n-th eigenspace have the parity of n. Next we estimate the number of those eigenvalues (5.14) which a r e greater than 1 (or > i ) - For this we need to know t h e asymptotic behavior of the coefficients c^ defined in (5.10). Making use of Rodrigue's formula for P^ , namely (_l)n
^n
we obtain, after n integrations by parts, that
c^{a) = —
^
— - ^«^(1 - T 2 ) " dr .
Hence, d — cl(a) = da
»(^'=(^JÖTK!—)'-"('--)"('-^)"-' 50
760
p. D. LAX AND R. S. PHILLIPS
Applying the mean value theorem, this can be written as
For a > O5 the exponential factor exp {^(T + 0 } ^^^^ weight the positive values of (T + t) more than the negative values, and therefore f + f > 0. W e set (5.16)
yn = a = aR y
substitute the expression (5.15) into (5.14) and apply Stirling's approximation. W e then get
^„=/
(£[?"'('-^ 4 '
where
A~(27rn)-^/^(^J(? + f + ?) y
We wish to find the value of y, say y^ , for which X^ = c where c is either 1 or ^. T o this end we study the integrand in the above representation for X^ . T h e m a x i m u m of the expression in the square brackets for — 1 ^ r ^ 1 is attained at a root of its derivative: r ( l - T2) - 2 T = 0 ; the root lying in the interval ( — 1, 1) is
Vi +y' y
At this point the value of the square bracket is
exp{VrT7}Vi_+rl:il. 7 I t is not difficult to show that this is an increasing function of y. Denote by / ^ the value for which this expression equals one:
exp {vr+i?} ^^-^yl-^ This equation has but one solution and its value is (5.17)
7o = 0.66274 • • • .
51
=1
DECAYING MODES FOR THE WAVE EQUATION
761
For 7 > 7o the bracket expression has a maximum greater than one and hence A^ becomes infinite with n. O n the other hand^ for y < 7o ^^e bracket expression has a m a x i m u m less than one so that the integral tends to zero very rapidly; in fact, even with the increasing factor/*^ ^ the eigenvalue A„ tends to zero as n becomes infinite. It follows that y^ tends to }/Q as w —> oo. T h u s for a given a the number of indices n for which the n-th eigenvalue A^ is greater t h a n 1 is by (5.16) approximately aRfy^. Since X^ has multiplicity 2 / 2 + 1 , the n u m b e r of eigenvalues of ((r/27r)aA:Q(ö') greater than 1 is
Approximately half of the corresponding eigenfunctions are even, the other half being o d d ; so the n u m b e r of each is 1/^V
2W We have therefore proved PROPOSITION 5.4. Let C{a) denote the number of purely imaginary zeros of the scattering matrix for a sphere of radius R which are less than or equal to a in absolute valuey under either Dirichlet or Neumann boundary conditions. Then
1 (aRf where y^ = 0.66274 • Remark. T h e exact values of the purely imaginary zeros of the scattering matrix for a sphere of radius R can of course be computed from the eigenvalues of {af27r)o(,k{a). These have been determined by Wilcox [13] for the Dirichlet problem; a^ for the n-ih mode occurs at the real zero of K^^^i2{ — aR)y where ^n+i/2 ^^ *^^ modified Hankel function. T h e asymptotic expression for this zero for large n has been found by Olver [ 6 ] ; it is
in agreement with our estimate. T h e exact value for the lowest mode for both the Dirichlet and N e u m a n n problems is easily c o m p u t e d ; it is simply a^ = I JR. W e can now apply the comparison Theorems 3.2 a n d 3.5 to any obstacle which is bracketed between two spheres. THEOREM 5.5. Suppose that the obstacle 0 contains a sphere of radius R^ and is contained in a sphere of radius R^ . Let C{a) denote the number of purely imaginary zeros
52
762
p. D. LAX AND R. S. PHILLIPS
of the scattering matrix for & under either Dirichlet or Neumann boundary conditions which are less than a in absolute value. Then^ Jim ml
- ß .
In what follows we take 0 < cr < | a . Let G denote the inverse of the selfadjoint differential operator — A + Ö^^ acting in 1.2(1^3); since — A + or^ is strictly positive so is its inverse G. Applying G to (6.3) we get (6.5)
V = G{q exp {ax • oy]) — Gqv .
As is well known, G is an integral operator with kernel \
G{x,y;a)
ß—; a)} dy
(6.9) = ^
le{y,
6)q{y){e{y,
co) -
(/ + Gq)-'Gge{a>)} dy
This could be put in a more convenient form if it were not for the fact that e{co) is not square integrable. T o get around this we set e{x^ co)
! 0
for
|x| < n ,
elsewhere,
and define k^ico, Ö; (7) = i - je^iy,
(6.10)
= 1
Q)q{y){e,{y,
[qe^iO), { / - ( / +
55
a>) -
(/ + Gq)-^Gqe^{o,)) dy
Gqr^Gq]e^{oy)]
DECAYING MODES FOR THE WAVE EQUATION
765
Since, by assumption, q[x) decays like e x p { — a l ^ j } , it is easy to see for fixed a K \oL that k^[(x>y Ö; cr) is bounded and converges uniformly to A:(a>, Ö; c) as n -^ CO] a fortiori^ ^n(^) —^ k{a) in the operator sense. Finally, let a{co) be any real square integrable function on the unit sphere S2 ; multiplying (6.10) by a{6)a{(o) and integrating over S2 in 6 and a>, we obtain the following identity after inverting the order of integration: (6.11)
ik^a,a)=[qA^,{I+Gq)-^AJ,
where ( , ) denotes the Z/2('^2) ii^ner product and A^ is the function (6.12)
A^{x) =
je^{x,aj)a{co)da>.
Note that both A^ and qA^ belong to Z2(R3)LEMMA 6.2.
For bounded non-negative potentialsy the operator
(6.13)
q{I+Gq)-^
is (a) symmetric, (b) non-negative and (c) an increasing function of q. Proof: Suppose for the moment that q is also bounded from below by a positive constant; then the following identity holds: (6.14)
q{I + Gqy
= {q-^ + G)~i .
Symmetry and non-negativity follow immediately from this representation of the operator (6.12). T o show that the operator (6.13) increases with q we note that ^~^ decreases and that according to Lemma 4.4 inversion reverses the order relation among positive operators. T o rid ourselves of the restriction that q is bounded from below by a positive constant, we consider the family of potentials q^ = q J^ e. T h e n in the operator topology, q^ - > q, Gq^ - > Gq and (/ + Gq^^^ -> (^ + Gq)'^ as e -> 0. W e therefore obtain the properties (a)-(c) in the limit. LEMMA 6.3. The operator k on L2{S^ whose kernel is k{co, 6; a) is (a) symmetric, (b) non-negative and (c) an increasing function of q. Proof: Since A:^ - > A: as n becomes infinite, it suffices to prove these properties for k^ . \i q is bounded and non-negative, these properties follow for k^ from the corresponding properties of the operator q{I + Gq)"^ listed in L e m m a 6.2 and the formula (6.11). Finally, if ^ merely satisfies the assumptions (i) — (iii) we set {q{x) if q{x) < m , otherwise
56
766
p . D. LAX AND R. S. PHILLIPS
T h e n , as m becomes infinite, ^ ^ ^ ^ —> qA^ in the ^^2(^3) i^orm, Gq^ —> Gq in the operator topology (even in the Hilbert-Schmidt norm) a n d hence (/ + Gq^"^ —> (/ + Gq)'~^ also in t h e operator topology. I n the limit then we obtain the properties (a) — (c) for these general potentials from formula (6.11). As before the scattering matrix is of the form 6^{-ia)
=I + K{a),
where K{(T) is the operator on L^{S^ whose kernel is
K{o),B;a) = ^ k{co,-6;
a) .
Proceeding as in Section 3 we obtain as an immediate corollary of Lemma 6.3 T H E O R E M 6.4.
Suppose that y for all x^
and that q^ ^ 0. Then the smallest purely imaginary zero of ^^ is in absolute value greater than or equal to the smallest imaginary zero of ^^ . At the moment we do not have an analogue for Theorem 3.5. ?• S c a t t e r i n g b y a n O b s t a c l e a s t h e L i m i t o f P o t e n t i a l S c a t t e r i n g : A N e w Proof of the Monotonicity Theorem I t is part of the folklore in scattering theory that the exterior problem under Dirichlet boundary conditions is the same as scattering by a potential which is infinite on the obstacle a n d zero elsewhere. I n this section we give a precise formulation a n d a rigorous proof for this proposition; and in the process we obtain a n amusing new proof for the monotonicity theorem for the case of scattering by an obstacle for the Dirichlet problem. Let (P be a smoothly bounded obstacle. W e choose a smooth non-negative function g{x) which is everywhere positive inside the obstacle 0 a n d which vanishes identically outside (P. For each integer n^ we set 9n = ^5 • Consider a plane wave exp {ax • co} a n d denote by v^ = vj^x^ co; a) the resultant signal scattered by the potential q^ ; that is^ v^ is t h e solution of equation (6.3)^: (7.1)
- At;„ + oH^ + q^v^ = g^e--- R^ we obtain (7.4)
v,{x) = f
[v,{y) dß{x,r, a) - G{x,y; a) d^y)^
where 9^ denotes diflferentiation with respect to |j); |. According to relation (6.6), for j in a compact set and \x\ large.
G K , ; . ) = ^ i r ( . . . . + 0(l));
58
; a) (7.5)
I
f
f
ad'J
= Tv^{y, (X); a) - — 47T J\y\=RL \y\
1
dyV^^iy, co; a) U"^-" dS . J
According to Theorem 7.1, as n - > oo, Vj^{y, o); a) tends to v{y, co; a), uniformly in £0 and on compact subsets of ^ , Since ^„ = 0 in ^ , the differential equation satisfied by the functions y„ in ^ is simply
According to standard interior estimates for solutions of elliptic equations, we can conclude that each derivative of v^ converges to the corresponding derivative of V uniformly in co and on compact subsets of ^ . In particular, lim v^{y, (o;
G)
= v{y, (o; a)
and lim d^v^{y, (o; a) = d^Jy
^y Q-) ,
uniformly for all y on {y : \y\ = R} and all co. In view of (7.5) this implies that lim k^{a)y d; a) = k{cOy 6; a) uniformly for all co and 0, as asserted in Corollary 7.2. T h e monotonicity theorem for the exterior Dirichlet problem is a simple consequence of L e m m a 6.3 and Corollary 7.2. In fact, let (P^ and &2 be two obstacles, ^i ^ (^2^ ^ ^ ^ denote the corresponding transmission coefficients under Dirichlet boundary conditions by k-^{a) and k^icf)- Let g^ and g^ be smooth non-negative auxiliary functions with support (9^ and (9^ y respectively; it is clear that they can be chosen so that, for all x^
Finally, denote by k^^^{a) and k2^^{(y) the transmission coefficients corresponding to the potentials
59
DECAYING MODES FOR THE WAVE EQUATION
769
respectively. Clearly q^ ^{x) ^ q^^ J^x) and therefore^ according to Lemma 6.3^
as operators. Letting n become infinite and applying Corollary 7.2, we deduce that
this is the monotonicity theorem for scattering by obstacles for the Dirichlet problem. We turn now to the proof of Theorem 7.1. Proof:
Each function v^ satisfies equation (7.1) in fRg :
- A . „ + (?„ + (r2).„ = ?„.— , and vanishes at infinity. W e claim that v^ is non-negative throughout R 3 ; for otherwise, v^ would have a negative minimum somewhere and at this point we would have Ai;^ ^ 0 and hence the left side of (7.1) would be negative, the right side non-negative, which is impossible. Let x^ denote the point at which v^ attains its m a x i m u m ; at this point, Az;^ ^ 0 and we deduce from (7.1) that
Since q^ vanishes outside of 0^ we conclude that x^ lies in (9 and that ^ni^n) = max|z;^(^)I ^ ^^^ . where R =
max |;^| .
This shows that the functions v^{xj co; a) are bounded uniformly with respect to n and ca for fixed (T > 0. According to (6.2), u^ is related to the scattered signal v^ by
and it follows from this that the functions W^(A:, a>; a) are also uniformly bounded in n and o> over any compact subset of R3 . Now, u^ satisfies the equation (6.1)^ : (7.6)
-^u^
+ {q^ + a^)u^ = 0
in
R3 .
Outside of (9 these equations do not depend on n so that - A t / ^ + (T^i/^ = 0
60
in
^.
770
p . D . LAX A N D R. S. PHILLIPS
It follows from interior estimates for solutions of elliptic equations that the first derivatives of the u^{x^ co; a) are uniformly bounded in n and co in any compact subset of ^ . Multiplying equation (7.6) by u^ and applying Green's formula in the ball {x : \x\ < R}^ chosen so that the obstacle lies in its interior^ we get (7.7)
f [{Au^r J\x\), (7.11)
d <s{n,d)
61
+ oo? a n d tends to 0 as cr —> — oo. T h e existence of infinitely many eigenvalues follows as before. T h e argument presented above is a crude version of one given by V a n Norton in [12] for the existence of infinitely many exponential modes for a ball. All the eigenfunctions found by him h a d spherical symmetry. V a n Norton's result was extended to arbitrary convex (9 in an interesting paper of Ukai [11].
69
DECAYING MODES FOR THE WAVE EQUATION
779
Appendix A Short C o u r s e i n S c a t t e r i n g T h e o r y M a n y of the intuitive ideas associated with scattering theory for the wave equation are quite distinct from those associated with the more famihar q u a n t u m mechanical theory. It m a y therefore be helpful if we sketch in a heuristic way the connection between the spectrum of the infinitesimal generator of the semigroup {Z{t)} defined in Section 1 and the zeros of the scattering matrix; rigorous proofs can be found in our book [4], T o begin with, scattering theory deals with the asymptotic behavior of solutions for large values of /, more precisely with the relation of the asymptotic behavior of solutions as i—> 00 and as t-^ —oo. For a solution u of the wave equation in the exterior of an obstacle which has finite energy, it can be shown that for large 1^1 almost all of the energy of u is carried out to infinity along rays; in other words, such solutions can be described asymptotically as follows:
/ ^ n - M , —j (A.l)
as
^-^00,
as
i —>. — 00 .
u{x,t)
This asymptotic form shows that, for large |/|, u^ and u^ are approximately equal in absolute value and much larger than spacial derivatives of u in directions orthogonal to the radius vector; and it also follows that the energy outside the shell \t\ — c , oo); it therefore follows from part (c) of L e m m a A.2 that the set of u corresponding to such nice k form a dense subset of the outgoing solutions in the sense of the energy norm. Next we make use of Theorem 2.1 from Chapter V of [4], which asserts that translates of outgoing solutions are dense in the energy norm among all solutions. I t follows that the translates of a dense subset of the outgoing solutions are also dense among all solutions. We have just shown the existence of the limit (A.3)_^ for a dense subset of outgoing solutions; and it can be seen by inspection that if the limit k^ in (A.3) exists for a given solution w, then it also exists for any ttranslate of u^ the value of this limit being the translate of k^ . This proves part (i) of Theorem A . l , that is^ the existence of the limit (A.3)^ for a dense set of w; in the process we have also proved p a r t (iii). T o prove part (ii) we note that, according to part (d) of L e m m a A.2, each k with support in (p, oo) represents an outgoing solution and that, according to part (c), this representation is isometric. Translates of A: represent translates of w, by (iii), and since translates of those k with support in (p, oo) are dense in Z2(R5 'S'2), this proves part (ii). DEFINITION. T h e operator S relating functions k__ and k_^ which represent the same solution u is called the scattering operator: S :k__->k_^. According to T h e o r e m A . l , translating u in t corresponds to a translation of both k_ and k by the same amount. This shows that S commutes with translation; it follows from this that 6* is a convolution operator: (A.5)
k^{s) =S^k_
= [s{r)kjs
-r)dr.
If we use a Fourier transformation on (A.5), denoting the transform of »S* by 6^^ those of A:^ by a^ , and the variable dual to s by z, we then obtain (A.6)
fl^(z)
= ^{z)a_{z)
.
T h e temporary suppression of the variable 9 means that k^^ and a^ are thought of as vector-valued (that is L2{S d_je)
+ ^
Ä;-(a>, 0; z) .
T h e operator ^{z) is of course completely determined by how it acts on ^-functions; in fact, writing
we see that [S^{z)ä]{, - 0 ; z) a(0) dd ,
in other words, (A.22)
^ ( z ) = / + K'^iz) ,
where K^^{z) is an integral operator with kernel 7C^«(a>, e;z)
= ^
75
k^^{to, - 0 ; z) .
DECAYING MODES FOR THE WAVE EQUATION
785
Next we establish the connection between the scattering matrix and the semigroup {Z{t)} defined in Section 1. U p to this point we have regarded the functions k^ and k_ as representing solutions of the wave equation in the exterior domain; from now on we shall regard them as representing the initial data of these solutions. Also, it will be convenient to regard k^{Sy d) as vector-valued functions of s with values in Z2('^2)W e recall that H denotes the space of all initial data with finite energy, and D^ and D^ denote the initial data of outgoings respectively incomings solutions, i.e., solutions which vanish for |^| < p + f, / > 0, respectively |x| < p — ^, i < 0. LEMMA A. 3. (i) In the outgoing translation representation D^ is represented by L^{p, oo). (ii) In the incoming translation representation D^ is represented by L2{— oo^ ~p)« (iii) In the outgoing translation representation D^ is represented by 6*^2(— oo, —•/>). Proof: As in the proof of Theorem A. 1, parts (i) and (ii) are restatements of parts (d) and (e) of L e m m a A.2; and part (iii) is a direct consequence of the definition of the scattering operator. THEOREM A.4. The scattering matrix S^{z) is analytic in the lower half-plane and grows at most like exp {—2/> J^m z). Proof: As stated in Section 1, D^ is orthogonal to D^ ; since the representations we are using are unitary, it follows that the subspaces representing D^ and D^ in the outgoing translation representation are also orthogonal. I n view of parts (i) and (iii) of Lemma A.3, this means that SL2{— oo, —/>) is orthogonal to ^2(^5 ^)i which implies that SL2{— 00, —p) is contained in L2{— 00, p). Now according to (A.5), SL2{— 00, — p) is the convolution ofvS'(r) withZ2(— 00, —/>); since we have shown that the result of these convolutions always has its support in (— 00, p), it follows that the support of S{r) itself lies in (—00, 2 p ) ; this is known as the principle of causality. T h e Fourier transform of a function with support in (—00, 2p) is analytic in the lower half-plane and of exponential growth there, as asserted in Theorem A.4. As in Section 1, the semigroup {Z(f)} is defined by
z(o = P'^u{t)p^_,
f ^ 0,
where P^ and P^ denote the orthogonal projections onto the orthogonal complements of D^ and D^ , respectively. T h e domain of Z{t) is K^: Kf" = He
(DP^® DO
.
It follows from L e m m a A.3 that the outgoing translation representation of ^ ^ is (A.23)
Z 2 ( - o), p) e SL2{-
76
00, - p ) .
786
p . D. LAX AND R. S. PHILLIPS
Since this is a translation representation, U{t) acts as right shift by t units; P ^ , on the other hand, corresponds to multipHcation by the characteristic function of (— 00, /)). So, in this representation, Z{t) acts, on the space of functions (A.23), as a right shift by t units followed by restriction to (— oo, p). If A is an eigenvalue of the infinitesimal generator B of {Z{t)} and w is the corresponding eigenfunction, then Z{t)w = e^^w • Denoting the outgoing translation representer of w by A, it is readily seen from the above description of Z(i) that h has to be an exponential function on (— oo, p):
(
e^^^n
on
(— oo, p) ,
0
on
(p, oo) ,
where n is an element of L2('^2)- According to (A.23), w belongs to K^ only if its representer h is orthogonal to SL^i— oo, — p)y i.e., {Sk, h)
=0
for every k in Z2(—oo, — p). By ParsevaPs formula, this can be rewritten in terms of Fourier transforms as (A.25)
{^ic, A) = 0 .
We can calculate the Fourier transform of h explicitly: 1
h{z)
ß^iz-X)p
=——=——-n. V 277 i z - X
Substituting this into (A.25) we get
(A.26)
^{z)k{z) J-oo
• n -, =- dz = 0 iz + A
for all k in i^2(~ ^ ? ~p)' Now according to Theorem A.4, ^{z) is analytic in the lower half-plane and grows there at most exponentially. Choosing the support of k to lie sufficiently to the left of — p, we can make the integrand in (A.26) tend to zero exponentially as ^ m z -> — oo. We then shift the contour from the real axis to the parallel line ^m z = c and let ^ - > — oo. T h e integral tends to zero, leaving us with only the residue at z = iX. Denoting (A.27)
k{ü)
= m
77
DECAYING MODES FOR THE WAVE EQUATION
787
a n d omitting the exponential factor we get from (A.26) (A.28)
^{il)m
•n = 0
for all m of the form (A.27). Clearly every m in L2{S^ can be represented in this way by taking k{s) = g{s)m^ g some scalar function; hence (A.28) implies that the range of ^{il) is orthogonal to n. Since by (A.22)5 ^{z) is of the form / plus a n integral operator, it follows by the Fredholm alternative t h a t ^{il) annihilates some nonzero function in L2^{S^. This completes the identification of the eigenvalues of the infinitesimal generator of {Z{t)} with the zeros of the scattering m a t r i x . Bibliography [1] Courant, R., and Hubert, D., Methods of Mathematical Physics^ Vol. 2, Interscience Publishers, New York, 1962. [2] Friedländer, F. G., On the radiation field of pulse solutions of the wave equation, 11, Proc. Royal Soc. London, Vol. 279A, 1964, pp. 386-394. [3] Kac, M., Can one hear the shape of a drum'^ Amer. Math. Monthly, Vol. 73, No. 4, 1966. [4] Lax, P. D., and Phillips, R. S., Scattering Theory, Academic Press, New York, 1967. [5] Mora wetz, G. S., and Ludwig, D., The generalized Huyghens principle for reflecting bodies, Gomm. Pure Appl. Math., Vol. 22, 1969, pp. 189-205. [6] Olver, F. W. J., The asymptotic expansion of Bessel functions of large order, Philos. Trans. Royal Soc. London, Ser. A., Vol. 247, 1954, pp. 328-368. [7] Phillips, R. S., The extension of dual subspaces invariant under an algebra, Proc. International Symp, Linear Spaces, Jerusalem, 1960, pp. 366-398. [8] Phillips, R. S., A remark on the preceding paper of C, S, Morawetz and D, Ludwig, Comm. Pure Appl. Math., Vol, 22, 1969, pp. 207-211. [9] Phillips, R. S., A minimax principle for the eigenvalues of symmetric operators in a space with an indefinite metric, J . Functional Anal., to appear. [10] Ralston, J., Energy decay problems in scattering theory. Thesis, Stanford University, 1968. [11] Ukai, S., Real eigenvalues of the mono-energetic transport operator for a homogeneous medium, J . of Nuclear Science and Technology, Vol. 3, No. 7, 1966, pp. 263-266. [12] Van Norton, R., On the real spectrum of a mono-energetic neutron transport operator, Gomm. Pure Appl. Math., Vol. 15, 1962, pp. 149-158. [13] Wilcox, G. H., The initial-boundary value problem for the wave equation in an exterior domain with spherical boundary, Amer. Math. Soc. Notices, Vol. 6, 1959, pp. 869-870. Received J a n u a r y , 1969.
78
COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS, VOL. XXV, 8 5 - 1 0 1
(1972)
On the Scattering Frequencies of the Laplace Operator for Exterior Domains* P E T E R D. L A X Courant Institute^ N. Y. U. AND R A L P H S. P H I L L I P S Stanford University
Abstract I n this paper we show that the so-called scattering frequencies of the Laplace operator over an exterior domain, subject to Robin or Dirichlet boundary condition, cannot lie in certain portions of the upper half-plane. T h e excluded sets depend only on the type of boundary condition and the radius of the smallest sphere containing the scattering obstacle. 1* I n t r o d u c t i o n Let 0 be a compact, smoothly bounded set in R^ called the obstacle; we denote by G the complement of O. T h e scattering eigenfunctions v of the Laplace operator A over the exterior domain G are defined as solutions of (1.1)
Av + r,^ü = 0
in
G,
subject to Dirichlet, N e u m a n n or Robin boundary conditions on 9G, and subject to a radiation condition at infinity: For |;c| large, v{x) is a superposition o^ fundamental solutions:
(1.2)
; \x -
;, y\
Ä ^ > 0 ,
y restricted to some compact set. T h e n u m b e r ^ will be called a scattering frequency \ it is a complex frequency, with positive imaginary part, so that it * This work was supported in part by the United States Atomic Energy Commission under Contract AT(30-1)-1480, by the National Science Foundation imder Grant GP-8867 and by the United States Air Force imder Contract AF-44620-71-C-0037. Reproduction in whole or in part is permitted for any purpose of the United States Government. 85 © 1972 by John Wiley & Sons, Inc^
79
86
PETER D. LAX AND RALPH S. PHILLIPS
corresponds to a mode that decays in time as well as oscillates. T h e quantity ^^ is called a scattering eigenvalue of —A. Note that the incoming fundamental solution (L2) grows exponentially as |;c| —> oo; therefore so do the scattering eigenfunctions. T h e spectrum of A over G is absolutely continuous; the scattering eigenfunctions play no role in the spectral resolution of A. However^ these eigenfunctions, as explained in [2] and [3], enter the asymptotic description for large t of solutions of the wave equation u^^ — IS.U = 0 ,
X in G^ t real;, u subject to prescribed boundary conditions on dG. In two previous papers [3] and [4] we have studied the distribution of the scattering frequencies of the Laplace operator over the domain G; in this paper we continue this study. W e establish the existence of regions of the upper halfplane which are free of scattering frequencies; these regions depend only on the radius p of the smallest sphere which contains the scattering obstacle a n d , in case of the Robin boundary condition
"^ (1.3)
— + KV = 0 dn
on
dG,
on the lower bound KQ of the function K. For the Dirichlet problem, a region empty of scattering frequencies and containing the origin is determined in [7] by an estimate depending on the geometry of the body. In Section 2 we show that the disk of diameter l / p located in the upper halfplane a n d containing the origin is free of scattering frequencies ^. T h e proof relies on the characterization of such ^ values as poles of the scattering matrix, the description of the scattering matrix in terms of the so-called transmission coefficients, a n d on the holomorphic character and exponential boundedness of the scattering matrix in the lower half-plane. I t is known that for a spherical scatterer of radius p with either Dirichlet or N e u m a n n boundary conditions the smallest purely imaginary radiation frequency is f/p. Furthermore, approximate solutions for the Helmholtz resonator (see Morse a n d Feshbach [5]), that is a punctured spherical shell with N e u m a n n boundary conditions, indicate that, as the hole in the shell becomes smaller, two frequencies symmetrically positioned with respect to the imaginary axis tend toward the origin. T h u s the disk is quite sharp as far as the purely imaginary scattering frequencies are concerned. Nevertheless the disk can be improved upon for Dirichlet and Robin boundary conditions. I n fact, we show in Section 3 that there is a n open set in the complex plane containing the interval (—7r/p, Trfp) which is free of scattering frequencies for the Dirichlet case.
80
ON THE SCATTERING FREQUENCIES OF THE LAPLACE OPERATOR
87
T h e intuitive reason for this can be described as follows: Suppose on the contrary that there were a sequence O^ of obstacles contained in the ball of radius p around the origin^ and corresponding to these obstacles a sequence of radiation frequencies t,^ tending to a point ;f in {—rrjp^ ^1 p)- Let us assume that the exterior domains Gj converge; the limit will be an exterior domain G plus possibly a number of interior domains D which get pinched off in the limiting process. T h e sequence of scattering eigenfunctions u^ may tend either to a scattering eigenfunction of G or to an ordinary eigenfunction of — A for one of the interior domains D^ with eigenvalue %^ < {TTJpy. T h e first alternative is ruled out by a theorem of Rellich, which states that there are no real scattering frequencies. T h e second alternative is ruled out also: an ordinary eigenvalue of —A over D must be greater than or equal to the lowest eigenvalue of -— A over the ball of radius p, because of the monotone dependence of the eigenvalue of —A on the underlying domain (see Courant-Hilbert [1]). Since the lowest eigenvalue of —A over the ball of radius p and subject to the Dirichlet condition is {TT/p)^^ we have a contradiction. Of course the reasoning sketched above is only heuristic^ since a sequence of exterior domains can behave in a very complicated fashion; nevertheless a proof, presented in Section 3, can be carried out along these lines. T h e same reasoning applies as well to the Robin boundary condition (1.3) with positive K: (1.4)
AC^/Co>0.
I n this case the role of the theorem on monotonic dependence of the lowest eigenvalue on the underlying interior domain is taken over by THEOREM 1.1. Denote by X the lowest eigenvalue of — IS. over D^ subject to the boundary condition (1.3). Denote by AQ the lowest eigenfunction of —A over a ball B containing D^ subject to the boundary condition
(1.5)
dv —^ + ^0 1^0 = 0 dn
on
dB,
where KQ is a constant as in (1.4). Conclusion: (1.6)
A^Ao.
This result is due to Payne and Weinberger [ 6 ] ; we conclude this section with a new proof of Theorem 1.1, based on the m a x i m u m principle.
PETER D . LAX A N D R A L P H
Proof:
(1.7)
S.
PHILLIPS
T h e lowest eigenfunction I'Q of — A in a ball is of the form
Vo{x) =
sin x^
T h e boundary condition (1.5) imposed at equation (1.8)
tan xp =
r = p leads to the transcendental
XP I -
KQP
T h e smallest positive solution Xo gives the smallest eigenvalue A^ = Xo T h e lowest eigenfunction VQ has the following property: dvo
at every interior point of the ball B. T h e n , in particular, for any unit vector n, dvo
(1.9)
*(;fo)^(Zo + ^^o)r'l ^ [1 - a |»?ol + 0(4)]-'
.
Thus, replacing c in L e m m a 2.1 by [1 — a [rj^] + 0{r)l)], we see that 0 ( Q can have no null vector for ^ in the circle of radius a~^ + 0{\r]Q\) and center iXo y —^~^ + 0{\rjQ\), and passing to the limit as rj^-^O we obtain the desired result.
84
ON THE SCATTERING FREQUENCIES OF THE LAPLACE OPERATOR
91
We shall apply Corollary 2.2 above to find a circular region free of scattering frequencies of the Laplace operator over G. T o accomplish this we need to relate the scattering frequencies of •— A to the poles of the acoustic scattering matrix associated with the exterior domain G in question. T h e relation is the following: ^ is a scattering frequency of —/S. in G if t, is a pole located in the upper half'plane of the scattering matrix ^ ( 0 We shall use the following description of the scattering matrix 6^ (see [2], P- 170). (A) ^{Q is an opera tor-valued meromorphic function of ^, holomorphic in the lower half-plane^ and of the form (2.4)
^ ( 0 = .2^>^(I>(0,
where O is an inner factor. (B) For J^^-^0,6^ is of the form (2.5)
6^{Q = / + K{t)
,
where K is an integral operator^ mapping functions on the unit sphere S^ into S2 y whose kernel K is
(2.6)
K{oy,Q-Q=^k{-d,oya)ZTT
Here k is the transmission coefficient associated with the exterior G of an obstacle in the following fashion: For ^ p.
elliptic estimates
p p of the normalized eigenfunctions Vji V has the following properties: (i) V is locally L^ for (ii) V satisfies (3.10)
|^| > p ;
A2; + ;C'^ = 0
>
90
\A > P \
ON THE SCATTERING FREQUENCIES OF THE LAPLACE OPERATOR
97
(iii) for any d^ p K d 0,
(3.12)
f
\v^-v^
dS = 0
Proof: Part (i) follows from L e m m a 3.1. T o prove part (ii) we note that each v^ satisfies, for \x\ > p, a second order elliptic equation Az;^ + ^^ v^ = 0. Using standard elliptic estimates, we deduce from the uniform boundedness of I
\v^\^ dx that all derivatives of v^- are locally uniformly bounded.
Using
p). Proof: Assuming the contrary, there exists a sequence of scattering eigenfunctions v^ on domains G^ such that the corresponding frequencies t,^ tend to a point x ^^ {~^lp:> ^1 p)- Normalize v^ as before; then, by L e m m a 3.1, a subsequence converges to a limit v and, by L e m m a 3.2, v satisfies (3.10)-(3.12) with ^ real. It follows from (3.11) that for \x\ large v has the asymptotic behavior (3.13) where r = |x|,
v{x)=-
^-^xr
k{d) +
= x/r, a n d
k{d) = f
/Wz^ö-ny - —I dS
J\v\=^d
[
91
dnj
98
PETER D. LAX AND RALPH S. PHILLIPS
Furthermore, the asymptotic behavior of the derivatives is obtained by differentiating (3.13); in particular,
(3.14)
?M=_,-,C!f:,(9)+o(i). or
r
\ry
Substituting the asymptotic description (3.13) a n d (3.14) into (3.12) we get
f
\k{d)\'dd + o{^ = 0 .
J\x\=R
\K/
Letting i? -> oo, w e conclude that the square integral of \k\ is 0 ; since k is continuous, it follows that k{d) = 0. Looking again a t (3.13) we deduce from this that
"(-)=o(^)
as
\x\ —> 00
Such a function in R^ is square integrable; b u t according to Rellich, if ;^ 7^ 0, the only square integrable solution of (3.10) in \x\ > /^ is identically zero. So we conclude that rv = 0 for \x\ > p. If ^ = 0, we have to argue somewhat differently to reach the same conclusion. I n this case, v is h a r m o n i c ; hence its asymptotic behavior is
(3.15)
.W~o(i),
f~0(i).
Consider now relation (3.5) for v^ :
(3.16)
f {\v,f-tU^^'}d^ = J\x\r=R i v.^^dS. on
JGR
As j -> 00, the right side of (3.16) tends to
J\x\^R
dn
which, according to (3.15), is 0(1/72). Since ^ ^ - > % = 0, the second term in (3.16) tends to 0 as j -> 00; thus we see that, for large j ,
I
GR
R
92
ON THE SCATTERING FREQ^UENCIES OF THE LAPLACE OPERATOR
99
Now (vj)^ —> u^ , uniformly for 2p < |;^| < p, a n d we conclude from the above that, as j —> 00, r
I .2 I ^ const.
2p p, we return to the identity (3.16) choosing R = p -j- d with d small but greater than 0. Since on {|x| = p + 6}^ Vj a n d dvjdn tend uniformly to v a n d dv/dn^ respectively, both of which a r e zero, it follows that the right side of (3.16) tends to zero as 7 —> 00. Likewise, ^j —^ X and, since
j
(v^)^ is uniformly bounded, (3.16) implies that, as j —> 00,
(3.17)
f
{l(^.)/-X^k,f}^^->0.
Let (p{\x\) be a C°^ function which is equal to 1 for \x\ 00, v^ a n d {v^)^ tend to zero uniformly in /> + 2^ < k i < p + ^5 it follows that, for j large, w^ differs little from v^ a n d {w>)^ differs little from {v^)^ . Hence, it follows from (3.17) that, as 7 - > 00,
(3.18)
f
{\{w,)J['-x'\w,\'}dx^O.
JCfp+S
Finally, we remark that the normalization a n d the fact that y^ —>-0 for p + J- I a n d therefore that
lim j
\Wj\^dx=l.
T h e domains Gp^^ a r e all contained in t h e ball |A:| ^ p + d. According to the monotonicity principle, the spectrum of — A over G^^.^ lies above the lowest
93
100
PETER D. LAX AND RALPH S. PHILLIPS
eigenvalue X^ of — A over the ball of radius p + ^5 under Dirichlet boundary conditions. Consequently, for any function w which is zero on the boundary of Gpj^^ 5 the Rayleigh quotients of w cannot be less than X^ : (3.20)
f
\wJ'dx^?.J
\w\'dx.
T h e functions w^ are zero on the boundary of G^^^ and hence they satisfy (3.20). If x^ were less than X^ , then (3.18) and (3.19) would contradict (3.20). This proves that \x^ ^ XQ . As is well known, for a ball of radius {p + (3), XQ is (77/(p + ö))^y and therefore, l^l = W ( p + ^)- Since this holds for all ^ > 0, \x\ must be at least 7r//>, as asserted in T h e o r e m 3.3. A similar result holds for the Robin boundary condition (3.1). O n e slight difference is that in this case the left side of (3.5) has to be augmented by the term
f
K\v\'dS;
JdG
the same term has to be added to the left sides of (3.16), (3.17), T h e second difference is that to conclude as before that x^ ^ Theorem 1.1, instead of the monotonicity theorem. Since the Ao of — A over a ball of radius p, subject to the boundary given by the smallest solution v of (3.2), we conclude:
(3.18) and (3.20). ^0 y ^^ appeal to lowest eigenvalue condition (1.5), is
THEOREM 3.4. There is an open set of the complex t^-plane containing the interval { — v/py vfp)^ where v is defined by (3.2), which is free of scattering frequencies of — A under Robin boundary condition (3.1) for all domains exterior to obstacles contained inside the ball {\x\ < p}. 4. R e m a r k s All of our results carry over for odd-dimensional spaces of dimension n > 1. For n > 3 and odd, it is easy to see from the form of S (see [2], p . 170) that S\0) = 0 and it follows from this that Theorem 2.4 remains valid. T h e analogues of Theorems 3.3 and 3.4 are also true, only the interval needs to be adjusted to the fundamental frequency for the ball of radius p in R ^ ; the proofs are the same. T h e even-dimensional case is different. I n this case the scattering matrix is not simply related to an inner factor as in (2.4), so it is unlikely that T h e o r e m 2.4 is valid. O n the other hand, if interpreted properly, Theorems 3.3 a n d 3.4 do carry over. Here the essential difference between the even a n d odddimensional problem is that the incoming fundamental solution e{x^ ^ ) , while
94
ON THE SCATTERING FREQUENCIES OF THE LAPLACE OPERATOR
101
single-valued for n odd, is a multi-valued function of t, for n even. However, if we treat e as single-valued in the ^-plane, cut along the positive imaginary axis. Theorems 3.3 a n d 3.4 do remain valid except a t the origin for n = 2, which is the only case for which e{x^ Q does not converge as ^ —>^ 0 to the fundamental solution for the Laplacian.
Bibliography [1] Courant, R., and Hubert, D., Methods of Mathematical Physics^ Vol. I, Interscience Publ., New York, 1962. [2] Lax, P. D., and Phillips, R. S., Scattering Theory^ Academic Press, New York, 1967. [3] Lax, P. D. and Phillips, R. S., Decaying modes for the wave equation in the exterior of an obstacle^ Comm, Pure and Appl. Math., Vol. 22, 6, 1969, pp. 737-787. [4] Lax, P. D., and Phillips, R. S., A logarithmic bound on the location of the poles of the scattering matrix^ Archive Rat. Mech. and Analysis, Vol. 40, 1971, pp. 268-280. [5] Morse, P. M., and Feshbach, H., Methods of Theoretical Physics^ McGraw-Hill, New York, 1953. [6] Payne, L, E., and \'\'einberger, H. F., Lower bounds for vibration frequencies of elastically supported membranes and plates, ], Soc. Industrial and Appl. Math., Vol. 5, 1957, pp. 171-182. [7] Mora wetz, C. S., O/z the modes of decay for the wave equation in the exterior of a reflecting body, to appear. Received August, 1971,
95
COMMENTARY ON PART V
27, 31, 35, 51, 71 The Lax and Phillips approach to scattering theory was originally developed by them for the Euchdean wave equation in the exterior of a compact obstacle O in R^. The papers included in this volume were critical ones in the development of the theory. In [27] it is shown that any solution of the wave equation as above has some of its energy propagate out to infinity, where it behaves like a solution to the free space wave equation. A number of the ingredients that constitute their abstract functional-analytic setup for scattering theory, such as the incoming and outgoing solutions, are already present in this paper. The paper [31], written jointly with C. Morawetz contains most of the key notions for the general setup of their scattering theory axioms. Specifically, the semigroup Z{t) and its infinitesimal generator B, whose eigenvalues in C correspond to the poles of the scattering operator, are introduced. The main result in [31] is the exponential decay of solutions to the wave equation when n is star-shaped (in dimension 3). The proof relies on the work of Morawetz [Mo]. The paper [35] gives the setup and techniques for what is known today as the Lax-Phillips time-dependent scattering theory. A comprehensive treatment of the theory is given in their well-known book Scattering Theory [L-P]. The revised edition of this book [L-P2] contains an up-to-date epilogue with a wealth of information about recent developments stemming from the book. In [35] one finds fax-reaching insights and conjectures about the relation between the geometry of an obstacle fi and especially the motion of rays in R'^\ri and the distribution of scattering poles, ft is said to be nontrapping if there is a ball C containing Ü and a io > 0 such that any ray entering B and obeying the laws of linear motion and reflection in the complement of Q. exits B within time io- A basic conjecture put forth in [35] is that the semigroup Z{t) is eventually compact (which implies that the imaginary parts of the poles tend to infinity) iff 0 is nontrapping. If Q is trapping, this was established by Ralston [R]. In the other direction Ludwig and Morawetz [L-M] established the conjecture if Ü is convex. Further results were obtained by Morawetz-Ralston-Strauss [M-R-S], and the complete solution (that is, if Q is nontrapping, then Z{t) is eventually compact) was given by Melrose [Mel]. His work maiies use of the modern machinery of propagation of singularities for such wave equations as well as the corresponding microlocal analysis. Paper [51] initiates the finer study of the distribution of the poles of the scattering operator in analogy with the well-studied Weyl law for eigenvalues of the Laplacian on a compact domain. Unlike the latter case, the poles are not restricted to lie on a line, so that counting them asymptotically or studying their possible monotonicity properties under changes of the obstacle are problematic even in their formulation. It is therefore quite surprising that in [51] Lax and PhiUips estabUsh a monotonicity property (in terms of increasing Q) for the purely imaginary poles of the scattering operator. Using this they give asymptotic order of magnitude estimates for the number of such purely imaginary poles for a general fi. More recently there have been a number of works concerning bounds on the number of poles in a disk of radius r as r —> oo. We mention in particular the works of Melrose [Me2] and Zworski [Z]. Another striking result is that of Ikawa [I], who gives a precise description of the poles near the real axis for the case that Ü consists of two disjoint convex bodies containing
25
96
an unstable periodic ray bouncing between them. A closer analogue to Weyl's law in this exterior scattering problem concerns the winding number of the phase of the scattering operator along the real axis. Majda and Ralston [M-R] (see also the references on page 281 of the revised edition of [L-P]) estabhsh an exact analogue of Weyl's law for this winding number. The Lax-Phillips scattering theory as well as the set of problems and ideats that come out of it are still very active areas of research today.
References [I]
itawa, M. On the poles of the scattering matrix for two strictly convex obstacles. J. Math, Kyoto Univ. 23, 127-194 (1983).
[L-M] Ludwig, D.; Morawetz, C. The generahzed Huygens principle for reflecting bodies. CPAM 22, 189-205 (1969). [L-P]
Lax, P.; Phillips, R. Scattering Theory revised edition, A.P. 1989.
[M-R] Majda, A.; Ralston, J. An analogue of WeyFs formula for unbounded domains. Duke Math. J. 45, 183-196 (1978). [Mo]
Morawetz, C. The decay of solutions of the exterior initial-boundary value problem for the wave equation. CPAM 14, 561-568 (1961).
[M-R-S] Morawetz, C ; Ralston, J.; Strauss, W. Decay of solutions of the wave equation outside nontrapping obstacles. CP4M 30, 447-508 (1977). [Mel] Melrose, R. Singularities and energy decay in acoustical scattering. D. M. J. 46, 43-59 (1979). [Me2] Melrose, R. Polynomial bound on the number of scattering poles. J. F. A. 53, 287303 (1983). [R]
Ralston, J. Solutions of the wave equation with locahzed energy CPAM 22, 807-823 (1969).
[Z]
Zworski, M. Sharp polynomial bounds on the number of scattering poles for radical potentials. J. Funct Anal. 82 (1989) 370-403.
P. Sarnak
26
97
PART VI
SCATTERING THEORY FOR AUTOMORPHIC FUNCTIONS
Translation Representations for the Solution of the Non-Euclidean Wave Equation"*" PETER D . LAX Courant Institute AND R A L P H S. PHILLIPS Stanford University
0* Introductioii
In 1972, Pavlov and Faddeev [10] discovered an elegant connection between harmonic analysis for functions automorphic with respect to a discrete subgroup of SL(2, JR) and the Lax-Phillips scattering theory as applied to the non-Euclidean wave equation; we have elaborated their idea in our monograph [8]. Recently, M. A. Semenov-Tian-Shansky [11] has treated harmonic analysis and scattering theory for Riemannian symmetric spaces of negative curvature also along the lines of the Lax-Phillips theory. We have thought it worthwhile in the present paper to go over the same material as Semenov-Tian-Shansky, but in much greater detail, for the special case of non-Euclidean three-space. The main difference between the two approaches derives from the fact that Semenov^Tian-Shansky obtains the translation representations from the asymptotic behavior of solutions to an intrinsic hyperbolic system of partial differential equations, which reduces to the non-Euclidean wave equation for symmetric spaces of rank one. In our work we obtain the translation representations from the Radon transform; this leads to many more detailed properties of these representations. We start with the Cauchy problem for the non-Euclidean wave equation on n = {w = (x, y, 2); z>0}: M« =Lu = z^dz~^du + u^ w(w,0) = / i ,
Mt(w,0) = / 2 .
Let 1/(0 denote the solution operator relating initial date F^if^.f^)
to data
* The work of both authors was supported in part by the National Science Foundation; the first author under Grant No. NSF MCS-76-07039 and the second under Grant No. NSF MCS77-04908. Reproduction in whole or in part is permitted for any purpose of the United States Government. Communications on Pure and Applied Mathematics, Vol. XXXII, 617-667 (1979) © 1979 John WÜey & Sons, Inc. 0010-3640/79/0032.0617$01.00
99
618
p. D. LAX AND R. S. PHIULIPS
at time t : {u(t)yUt(t)}. Since energy is conserved, 1/(0 defines a oneparameter family of unitary operators on the Hubert space ^ of all initial data F, normed by the energy form (0.2)
W!l=-(/i,L/,) + (/„/,);
here the inner product on the right is the L2 inner product with the non-Euclidean volume element dw:
(0.3)
^'^^"I ^(^)8(^^^-
A solution u of (0.1) is called outgoing from the point j if it vanishes in the forward cone with vertex /: (0.4)+
u (w, 0 = 0 for non-Euclidean dist. (w, /) < f,
f = 0,
and u is called incoming from / if (0.4)_
u(w, t) = 0 for non-Euclidean dist. (w, /) < - r ,
r ^ 0.
The set of initial data for all incoming (outgoing) solutions form the incoming (outgoing) subspace of ^ is denoted by Q}-i9)+). We prove that the subspaces 2- and 2f+ have the following properties: (i) 17(02)^ c2>^
^^'^^
for t^O,
(Ü) nw)®-=o,
nt/(02)-,=o,
(iü) U 1/(0^- = ^ ,
U u{t)Q)^ = ^,
(iv)
2tLQit.
Our main tool in proving these properties is the Radon transform introduced by Gelfand et al. [1] and studied in great detail by Helgason in a series of publications. The Radon transform is defined as
(0.6)
f(s,ß)=f
f{y^)dS,
where the domain of integration is the horosphere (s, ß), that is the sphere tangent to the x,y-plane at ß whose non-Euclidean distance from / is s. Setting (0.7)±
R±F=const, ißte'f^ T d.e'f^
100
TRANSLATION REPRESENTATION FOR THE WAVE EQUATION
619
and denoting the reflection s-^ -s by "^^ we show in Theorems 3.4 and 3.7 that (0.8)
Ti = J?^ and T2 = JR^
are translation representations of Ü, that is, T is a unitary mapping of ^ onto L2(IR, N), N some auxiliary Hubert space, with the intertwining property (0.9)
TLr(0=UtT,
where Ut denotes right translation by t units. One obtains a spectral representation from a translation representation by Fourier transformation. The scattering operator S relates the above translation representations: (0.10)
S : TiF~>T2F.
The spectral representer of S is called the scattering matrix. In Theorem 3.7 we prove that (0.11)
R^(2J = L2(H^,N)
and i?^2^=^) =^ L2(IR^, N) .
®_ and 2+ have in common an infinite-dimensional subspace and Theorem 3.7 also furnishes us with an explicit characterization of JR±(2)_nS+). The situation is quite different in the Euclidean space where the incoming and outgoing subspaces are orthogonal; the non-Euclidean analogue of this property is (0.5. iv): 2^f ± ©t, established in Lemma 4.1. As a by-product of our analysis we prove the following Paley-Wiener result (Theorem 3.14): A function / in £2(1!) vanishes at all points outside of a given horosphere (or sphere) X if and only if d^e^fis, ß) vanishes on all horospheres lying outside of S. In Theorem 3.15 we derive a similar result for data in 5if. These results are in contrast to the Euclidean case where the vanishing of / on all hyperplanes not intersecting a given sphere is necessary but not sufficient for / to be equal to 0 outside the sphere. Our result is a somewhat surprising sharpening of a theorem of Helgason in [5]. For completeness we have included a brief description of the geometry of non-Euclidean space in Section 1. In Section 2 the Euclidean analogues of the results quoted above are derived to give the reader a better perspective on the problem; this material can be found in one form oj another in [6], Chapter 4. Finally, an explicit representation for the scattering matrix is derived in Section 4.
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p. D. LAX AND R. S. PHILLIPS
1. Non-Eodidean Space The three standard models of non-Euclidean space U are the upper half-space (l.l)i
(x,y,z),
z>0,
the unit ball (l.l)ü
X^+Y^ + Z ^ < 1 ,
and the semi-hyperboloid
(i.i)iü
T^-(e+n^+e) = i,
T>o.
We shall refer to these as models (i), (ii) and (iii). In this paper we shall be using the IBrst two of these models. The models are entirely equivalent to each other; the mappings relating (i) and (üi) are (1.2)
/, ^J-
^. .s
/x^+y^+z^+1 x^+y^+z^-1 /x^+y^+z^ + 1 X y x^ + y^+z^-l\
(T,en,0=(
Yz
'?!'
Yz
)
and (1.2)'
(x,y,z)=l
-,
\T-r r-L
-,
-).
T-tl
The mappings relating (i) and (ii) are conformal maps between the x, z upper half-plane and the unit disk in the X, Z-plane mapping (0, z) on (0, Z), combined with rotation around the z-(respectively Z-) axes. The Riemannian metric in (i) is (1.3)i
, 2
as
=
dx'^dx'^dz'^ 5 •, z
in (ii) (1.3)ü
, ^ 2 _ . dX^ + dY^+dZ^ (1-(X^+Y^+Z^))^'
Note that (1.3) is conformal with the underl5dng Euclidean metric. In model (iii) we endow the hyperboloid with the Lorentz metric of
102
TRANSLATION REPRESENTATION FOR THE WAVE EQUATION
621
4-space in which the hyperboloid is imbedded: (1.3)iH
do-^ = -dT^ + d^^ + dT|^+d^\
The geodesies in both model (i) and model (ii) are circular arcs perpendicular to the boundary. The length of each geodestic is «>; for this reason the points of the boundary of II—the plane z = 0, including the point at oo^ in model (i), the sphere X ^ + y ^ + Z ^ = l in model (ii)—are called points at infinity of II. We denote this set by B, and its points by ß. II equipped with the metric (1.3) is a space of constant negative curvature, n possesses a six-parameter group of isometrics called non-Euclidean motions and denoted by F. These are most easily described for (iii) as the group of Lorentz transformations, i.e., linear maps of r, |, 17, ^-space into itself which preserves the Lorentz form T^ —(|^ + T}^ + ^^), as well as preserving the direction of time. It is well known that the Lorentz group is isomorphic to SL(2, C) modulo ±L The isometrics in model (i) can be described elegantly^ with the aid of the 2 x 2 matrices (^5) of SL(2, C), (1.4)
ad-bc = l ,
in the following fashion: to each point (x, y, z) of II we assign a quaternion w with three componentSy (1.5)
w = x + yi + z],
where i^ = j^ = —1, ij — '-ji. It is not hard to verify that the mapping defined by (1.6)
TW = w' = (aw + b)(cw + d)'^
maps n onto II, with inverse r"^ given by (1.6)'
w = idw'-b)(-cw' +
ar\
and that the assignment (^5) —> r is an isomorphism. Note that these mappings restricted to the boundary JB of II are the classical Mobius transformations of the complex (x + iy)-plane onto itself. It it easy to verify using (1.6) that the group of motions F is symmetric in the sense that, given any two points Wi and W2 of II, there is a motion r such that TWi = W2, TW2 = Wi; this r is uniquely determined, being reflection across the geodesies through the midpoint w^ on the geodesic arc connecting Wi and W2. The motion r is an involution, i.e., T^ = I. The subgroup of T keeping a point Wo fixed is isomorphic to the group of ^ We thank Professor Werner Fenchel for acquaintmg us with this description.
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p. D. LAX AND R. S. PHILLIPS
rotations SO(3,IR); this is most easily seen in model (ii), taking Wo to be the origin. This subgroup acts transitively on the boundary. It follows that, given two boundary points ßi and ß2, and two interior points» Wi and W2, there exists a motion r such that Tßi = ß2, TWI = W2The isometric character of the mapping (1.6) is easily verified. The mappings r of II induce mappings of functions u defined on II; we define u^ by (1.7)
U"(W) = U ( T " ' W ) .
Since r is isometric, (1.7)'
I
|u-(w)Pdw= f
|i4(w)Pdw,
where dw is the non-Euclidean volume element, which in model (i) is /^ ^x (1.8)i
, dxdydz dw= 1—.
The quantity (1.7)' is called the invariant L2 norm. Denote the differential of the mapping rw = w' by dw'fdw; then 5w' =
5w.
It follows from the isometry of r with respect to the metric (1.3)i that |8tv'|/z' = \Sw\/z; therefore, aw' (1.9) dw
2'_ 1-=-^^ Z
where 17 K an orthogonal transformation. Let u be any C^ function, u^ defined by (1.7), i.e., u^(w') = ii(w). Denote the gradient by
Then u^ = u^{w% = u'iwX' T^. dw
104
TRANSLATION REPRESENTATION FOR THE WAVE EQUATION
623
Hence, by (1.9), \u^\z = \ul'\z'.
Since dw^dw' by definition (1.8)i, we get
The quantity above,
(1.10)
J K P ^ ,
is called the invariant Dirichlet integral. Its first variation is also invariant:
This shows that (1.11)
z^ö-d = Lo, z
called the Laplace-Beltrami operator, is invariant under non-Euclidean motionsNote that Lo, being defined as the first variation of the Dirichlet integral, is symmetric with respect to the invariant L2 norm. Next we define a non-Euclidean equivalent to the Euclidean notion of plane. A plane with normal ß through a point w in Euclidean space can be thought of as the limit of spheres of radius R through w as the center of the sphere tends to = ±distance of / from horosphere through w and ß, where / = (0,0,1) in model (i), / = (0,0,0) in model (ii); the plus sign (1.12) holds if / lies outside the horosphere and the minus sign holds if / lies inside the horosphere. Under isometry <w, ß) changes somewhat like its Euclidean counterpart: (1.13)
(2^^^7r)-^.
jR^. is called the outgoing translation representation. THEOREM 2.2. R^ gives a unitary representation of ^ for the group of operators U(t): (i) Denote R^F by fc+(s, ß), F in ^; then
(2.16)
R^ Uit)F = K{s - 1 , ß) .
(ii) For every F in 5if, (2.17)
\F\l =
\R,F\\
110
TRANSLATION REPRESENTATION FOR THE WAVE EQUATION
629
where \ \ on the right is the L^ norm; (iii) R^. maps W onto L2. Proof: (i) Let u be the solution of (2.12) with initial value F. Apply d^ to (2.14); using the definition (2.15)+ of R^ we can express the resulting relation as (d,+a,)K+F(0 = O. This shows that JR+F(0 is a function of s - t alone. Since F(t)=U{t)F, (2.16) follows. (ii) As remarked in (2.7), / is an even function of s, ß. It follows that d^fi is even and dj2 odd; thus the two components are orthogonal to each other in the L2 norm. Hence, in view of definition (2.15), we have
(2.18)
iK.FP=c^(|afAP+|aJ,P).
By (2.6), c^|dj2p = l/2p; using (2.6)' with g = A/, as well as (2.11), we get - ( / i , A/x) = -c\dj,,
d, A^) = -c\dA,
dtfx) = c^dth, « / i ) •
Definition (2.13) and the last two relations yield
|F||=|/#-(/x; Ah)-cH\dM+\dm^) • Combining this with (2.18) we deduce (2.16). (iii) now follows from (2.17), combined with part (iii) of Theorem 2.1, applied separately to fx and /2. The incoming translation representer k^ of / is defined analogously, as c(^/i + 5s/2)- It follows from the evenness property (2.7) that M5,ß) = M ~ 5 , ~ ß ) . The data / i , /2 are easily expressed in terms of their representer k+:
(2.19)
/i(w) = !c|JMw-ß,ß)dß.
To see this substitute the definition (2.15)+ of fc+ into (2.19); since dj2 is an odd function, its contribution to the integral in (2.19) is zero, while that of ds/i is, according to (2.5), equal to /i(w).
Ill
630
p. D. LAX AND R. S. PHILLIPS
According to part (i) of Theorem 2.2, R^{u(t\u,(t)}^k^is-t, stituting this in (2.19) we obtain (2.20)
ß). Sub-
u(w, t) = \c\ I M w • ß - 1 , ß) dß ;
this is a representation of solutions of the wave equation as a superposition of plane waves. THEOREM 2.3. Let {fi,/2} be initial data in ^ that are equal to 0 for | w | > a . Then the solution u of the wave equation with these initial data is zero at w = 0 for \t\>a.
Proof: Let l^j be greater than a; then every point w of the plane w • ß = s lies in | w | > a , where fi and /2 ^ ^ equal to 0. It follows from (2.3) that /i and /2 are both equal to 0 for | s | > a ; therefore, by (2.15)+ we conclude that M5,ß) = 0
for
|5|>a.
Consequently, for w = 0, |t|>a, the integrand on the right side of (2.20) is equal to 0 for all ß; this proves the theorem. Theorem 2.3 shows that the value of u(0, 0 does not depend on values of the initial data inside the ball |w|r. Denote R^F by k. Then, (Ü) fc(s,ß) = 0/or H > r , (iii) for every non-negative integer n, THEOREM
\ s^k{s,ß)ds is a polynomial in ß of degree at most n. Conversely, if k is in L2 and satisfies (ii) and (iii), then k = JR+F for some F of finite energy, satisfying (i). 3. The Radon Transform in Non-Euclidean Space Helgason in [3] defines the Fourier transform of a function / on n in analogy with the Euclidean case, but replacing w • ß by <w, ß) as defined in (1.12): (3.1)
/(A, ß)-'j2^
I /(>^)^^P W-"^*^) ß>}^^-
Just as the Fourier transform in Euclidean space gives a spectral representation of the Laplace operator, so (3.1) gives a spectral representation of the non^-Euclidean Laplacean LQ defined in (1.11). To see this we note that in the Euclidean case each exponential exp{-iw • $} is an eigenfunction of A: (3.2)
A exp {~iw • 1} = -IIP exp {-fw • | } .
From this we deduce, using the self-adjointness of A, that for every / in C^ (3.3)
A/ = (A/, exp) = (/, A exp) = - | | P / .
A similar equation is satisfied by the non-Euclidean exponentials e = e(w, ß. A) = exp {(1 - iA)<w, ß)}: (3.4)
Loe = - ( l + A^)e,
where LQ is given by (1.11). Since LQ is invariant under r, and since (1.13)
115
634
p. D. LAX AND R S. PHILLIPS
implies e(TW, rß) = const. e(w, ß ) , it follows that it suffices to verify (3.4) for a single value of ß. We choose ß=ßoo in model (i); by (1.14), e(w, ß j = 2^^~^\ and hence (3.4) can be verified by a simple calculation. Formula (3.4) suggests that we replace LQ by L defined as (3.5)
L = Lo+l = z ^ a z - ' d + l ;
L satisfies (3.4)'
Le = -A^e.
Since L is selfadjoint, we get for any / in C^ (3.6)
Lf = (L/, e) = (f.Le) = ^k^f,
where ( , ) denotes the invariant L2 scalar product. This shows that the Fourier transform (3.1) furnishes a spectral representation for L. The inverse of the Fourier transform is quite analogous to the Euclidean case:
(3.7)
/(w)==^^^3^,,^ j j/(A,ß)exp{(l + iA)<w,ß)}A^dAdß, R+
B
where dß is defined in (1.17). Note that only positive values of A are used; of course, an entirely analogous inversion formula can be based on A0 we can choose g in C^(IR) so that (i) g ( 0 ^ 0 for all t, (ii) g(0 = 0 for r ^ 8 . (iii)
I g(dist. (w, /)) dw = 1.
The foregoing implies that (i)' h(z, w ) ^ 0 for all z,w, (ii)' hiz, w) = 0 when dist. (z, w) ^ e, (iii)'
h(z, w) dz = 1 for
all w.
Given any function / in II, continuous and of compact support, we define /e(w)=JMz,w)/(z)dz.
It follows from a standard result in analysis that /e(w)~»/(w) uniformly as 8 --> 0. Since /^ is a superposition of symmetric functions, we are done. This completes the proof of part (ii). We turn now to part (iii) of Theorem 3.4; since ^ is the closure in the energy norm of C^, and since, by part (ii), R+ is isometric from the energy norm into the L2 norm, it suffices to show that there exists a set of h(s, ß) in the range of R+ that is dense in L2. Since in model (ii) the L2 space of functions is spanned by functions of the form h„(5)y„(ß), Y^ some spherical harmonic, it sufiices to display, for each Y„, a set of h{s) which are dense in L2 and for which h{s)Yn{ß) = R+f. For this we shall use LEMMA
(3.32)
3.5. I/, in model (ii), / is of the form f(w) = gyjÖ),
where g depends only on |w|, Ö = w/|w| and Y^ is a spherical harmonic, then the
127
646
p. D. LAX AND R S. PHIIJJPS
Radon and Fourier transforms are likewise of the form ,3 32y
f(s,ß) = k(s)Y^(ß), /(A,ß) = MA)Y„(ß).
Sketch of proof: It is well known that a linear mapping of the space of functions on the sphere S^ into itself which commutes with rotations has each spherical harmonic of order n, n = 0 , 1 , . . . , as eigenfunction. Let p be a rotation around /, i.e., pj = ]; according to relations (3.15), (3.15)' with p in place of T,
ns,ß) = f(s,p-'ß). Consider functions / of the form
/(w) = gp(e),
ö = w/|w|,
and g a fixed function of |w|. It follows from the above relation that for any weight function m(s) the mapping
\fis,ß) dm(s) commutes with rotations p and therefore maps spherical harmonics into spherical harmonics. Relations (3.32), (3.32)' follow. Consider now / of the form (3.32); then / is of the form (3.32)'. Using formula (3.27)' on the data F = ( 0 , / ) we see that R^F is of the form R^F=k(s)Y^(ß). According to part (i) of Theorem 3.4, k(s-t)Y^{ß) also belongs to the range of R^. We claim that, for an appropriate choice of the function g appearing in (3.32), the translates of k span L2(1R)- To see this we recall that translates of an L2 function fc span L2 if and only if the Fourier transform k of k does not vanish on a set of positive measure. According to formula (3.10), the Foiuier transform of e^f with respect to s is the Fourier transform / of /. Thus, for / of form (3.32), 7riA/(A,ß) = fe(A)yjß). Now choose g to have compact support; it follows from the definition (3.1) of the Fourier transform that, for / of compact support, /(A, j3) is entire analytic as a function of A. Such a function cannot vanish on a set of positive measure unless it vanishes for all A. We show now how to choose g so that f^O^ in
128
TRANSLATION REPRESENTATION FOR THE WAVE EQUATION
647
particular so that
By definition (3-1), with f==gY^, (3.33)
f ( - i i , ß) = j ^ ; ^ j g(|w|) exp {|(w, ß}}YM
dw.
Next we use formula (1.16) for (w, ß), and the well-known generating function for the Legendre polynomials: (3.34)
exp§<w,ß>}= ( i _ 2 ~ ! ' ^ C | ^ [ 2 f ' = (l-|>vp)^^^ S IM^P^e • ß) ,
where Pj^ is the fc-th Legendre polynomial. It is well known that, for fixed ß, Pk(ö • ß) is a spherical harmonic of order k; moreover. (3.35)
^
P^(0'ß)YAe)d0
= \^
for
k9^n,
Now we express the integration in (3.33) by polar coordinates: dw^drA{r)de, where r is the non-Euclidean distance from /, and A(r) the non-Euclidean area of the sphere of radius r. Substituting (3.34) into (3.33) and using (3.35) we get R-kU ß) = jrMl - r Y ' h ( r ) A ( r ) dr YJ,ß) . All we have to do is to choose g so that g has compact support and the above integral is not equal to 0. This proves part (iii) and so completes the proof of Theorem 3.4. THEOREM
(3.36)
3.6.
Denote JR+F= k+; then A(w) = \c\ I exp {(w,ß)}fc^«w, ß), ß) dß.
A similar formula can be given for the second component /2 of F. Proof: Substituting the definition of fc+ in (3.27) into the integral, we get
129
648
p. D. LAX AND R. S. PHILLIPS
two terms; the first, ~
I exp {<w, ß)}a^e7iL=<w,ß> dß,
is, according to (3.13), /i(w); thus to prove (3.36) we have to verify that the second term is zero; that is, (3.37)
I exp {<w, ß)}d,e^/2L= s — (w, 7 ) ,
we get (3.37) with ß replaced by 7. This completes the proof of Theorem 3.6. Let 17(0 be the solution operator of the non-Euclidean wave equation, F some element of ^ and R^F=k+. Then, according to (3.28), R^U(t)F = k+{s-t,ß). Setting this into the inversion formula (3.36) we obtain the following formula for the value of the solution of the wave equation u(w, 0 with initial data F: (3.41)^
M(W, 0 = \c\ j exp {<w, ß)}k^«w, ß > - f, ß) dß.
Similarly, we have (3.41)^
M(W, 0 = \c\ I exp{<w, ß)}fc_«w, ß)+1, ß) dß.
We show now that, conversely, if fc+ is any C°° function, (3.41)+ defines a C°^ solution of the wave equation (3.22). To see this let us abbreviate by q the function (3.42)
q(w) = exp{<w,ß)}.
According to (3.4)', q^~'^ satisfies
which implies that (3.42)'
Lq = 0
and L(qlogq) = 0 .
It is easy to deduce from these two relations that (3.42)"
ql+q^y + ql =
qVz\
Using these relations we can verify that u satisfies the wave equation by directly differentiating (3.41)+. The same holds for (3.41)_. Note that if k(s, ß) is a distribution in IRxJB, then (3.41)4. defines u as a distribution that satifies the wave equation. The same holds for (3.41)^. The analogous representation for solutions of the Euclidean wave equation was
131
650
p. D. LAX AND R. S. PHILLIPS
given in (2.20). Just as in the Euclidean case, Huygens' principle is an easy consequence of (3.41)+: THEOREM 3.7. Let F be initial data in ^ whose components are zero at all w whose distance from j exceeds a. Then the solution u of the wave equation (3.17) with initial data F is zero at w = j for |f|>a.
Proof: All points of the horosphere <w, ß) = s have distance at least \s\ from J. Therefore, if F = 0 at points whose distance from / exceeds a, it foUows from the definition (3.11) of the Radon transform that both /i and f2 are equal to 0 for |s|>a; therefore, so is R^f—k^. The solution u of the wave equation with initial value F is given by (3.41)^_; setting w=/, we get
w(j;0 = k l j M - t , ß ) d ß ; clearly the integrand equals 0 for |r|> a. We now draw some further consequences of formula (3.41)^. which are analogues of Theorem 2.4; again it is convenient to work with model (ii). THEOREM 3.8. Let fc+ he any distribution on RxS^, and define u by (3.41),. (i) M(W, 0 = 0 for all w, t if and only if the n-th spherical harmonic components of k^ are of the form
(3.43)
kM ==^k^ y j ß ) dß = X Kn^nC"^.
0 < m ^ n.
(ii) u(w, 0 = 0 in the forward cone: dist. (w,/) 0. (iv) u(w, 0 = 0 for dist. (w,/) d5 dß, where p^ is the transpose of p. Since p is a rotation, p^ = p~^, and hence the above can be rewritten, with p~^ß = y as a new integration variable, in the form Jfe(s,py)y^PN(ds. py)4> ds dy. Comparing this with (3.51) we conclude since k is arbitrary that PN(ds,py)
=
PN(dsyy),
i.e., that P^ has constant coefficients. We make use now of the fact that i;(w) = 0 to conclude from (3.49) that ^k{s,ß)YAß)P^idJdßds = 0 for any spherical harmonic Y^ of order n, since Y^O) is a sum of terms of the form ß^ for \N\ = n. In the notation of (3.43) we can, after carrying out the ß integration, write this as ^K(s)Pn{ds){'s)ds = 0 for all a. Recallmg (3.27):
we see from Corollary 3.10, (i)± that G is orthogonal to both 2+ and 2)° if and only if assertion (iv) holds. This concludes the proof of Theorem 3.12. If in particular we take gi = 0 and set gj = g, we conclude from the above that the following theorem holds. THEOREM 3.13. Let g be in LaCn). Then g = 0 outside the ball BO, a) if and only if ase^g = 0 for s>a.
An analogous characterization can of course be given for 1^(11) fimctions that vanish outside any given ball. A weaker version of Theorem 3.13 can be found in Helgason [5], page 128. Remark. This result is a non-Euclidean analogue of the Paley-Wiener Theorem 2.6, Avith these differences: necessary conditions for g to be supported in the ball B(/, a) are for g to be equal to 0 for \s\>a and for
(3.57)
{e—g{s,ß)ds
to be polynomial in ß of degree at most m. The first of these conditions follows from the observation that the horosphere (w, ß) = s does not intersect the ball Bij,a) when |s|>a. The second follows if we insert formula (1.16)
138
TRANSLATION REPRESENTATION FOR THE WAVE EQUATION
657
for <w, ß) into the definition (3.11) of g. We get
f e-""g(s, ß)ds=
{
J
J_oo J<W,ß> = S
f
exp {-m{w, ß)}g(w) dS ds
which is indeed a polynomial in ß of degree at most m. What is remarkable in Theorem 3.13 is that, in contrast to the Euclidean case, only a part of these necessary conditions already suffice to make g vanish outside the ball JB(/, a). In particular, g is not required to vanish for sa. Since horospheres are limits of spheres one would conjecture THEOREM
3.14. Let g be in L2(Il}; then g = 0 outside a horosphere 2 if
and only if (3.58)
ase^g = 0
on almost all horospheres that lie outside of 2. Remark. We were unable to deduce this from Theorem 3.13; an independent proof is given below. Proof: Since the necessity argument is obvious from the definition of the Radon transform, we treat only the sufficiency. It is convenient to deal with model (i) and to specialize X to a horosphere associated with ßoo> say z = 2a. In this proof we shall reparameterize the horospheres exterior to X by ßGlR^ and the Euclidean radius r of the horosphere. In terms of the non-Euclidean distance s of the horosphere from / = (0, 0,1), it is easy to see from (1.15) that
(3.59)
e^=i±^. 2r
Denoting the Radon transform of f in terms of the above parameters by f(r, ß) and using spherical coordinates about the center of such a horosphere,
139
658
p. D. LAX AND R. S. PHILLIPS
we can write /(r, ß)= I I /(ßt + rsinecos} d4> ZTT Jo
is the Bessel function of order zero. A change of the variable of integration from B to z = r(l—cosö) allows us to write (3.62) as
(3.62)'
- 1 - 5 f exp {-iß • |}/(r, ß) dß = r f ' Jo(lli(2zr- 2^)^'^)/(e z) ^ . (2'7r) J
Jo
2
•
The corresponding expression for r ' dr(l/r)f(r, ß)), obtained by differentiating (3.62)', is then ^
(27r)
| e x p { - i ß • i}r''%(^^f(r, ß)) dß
(3.63)
.^^2r)
f^^ r(^,u-:r
:-^^'-^ '^'^^^^''' f^^'^'Ki:
where Ji = JQ is the Bessel function of order one. We note that the kernel of this integral operator, that is 1/2
JM\(2zr-zY'^)
mrz)
_ -.2\l/2 > (2zr~2^)
is continuous on 0 ^ z ^ 2 r < 2 a . The familiar Fourier transfonn Plancherel relation giveis
(3.64)
f i/(w)Pdw= Jn
n
1/(1, z) d^dz
Jo JIR2
Using (1.17) and (3.59) the Radon transform Plancherel relation (3.14) can be rewritten in terms of the (r, ß) variables as
(3.64)'
f \f(w)\Uw = :^
Jn
f
OTT JR2 JO I
141
nr''%(-f(r,ß))\\rdß,dß, Vr
/I
660
p. D. LAX AND R. S. PHILLIPS
Given / in L2(n), by approximating / by CliTL) functions, it follows that, for almost all |, (i) /(^, z)/z^^^ belongs to L2(H) and (ii) the relation (3.63) holds for almost all n In particular, if / satisfies (3.58)' for r /i locally in L2 and it follows by elliptic theory that L/i = 0 for z < 2a. Conversely, suppose that L/i = 0 for z < 2 a . Then, by (3.56), ( G , H ) E = 0 for all smooth H with compact support in z < 2 a . Taking into account the unit speed of propagation we deduce similarly that for |t|<e (3.68)
(Lr(r)G, H)H = (G, U(-t)H)^ = 0
for all smooth H with compact support in zi''tl) + (ge.2,M
for all such H. It follows that Lg^i = 0 for all z ^) .
By definition, S takes L^iH-) = T^(2L) into T2(S)1). Accordmg to (4.9), T2(2>1) is orthogonal to T2(S>+) = L2(1R+) and hence is contained in L2(IR_) as desired.
145
664
p. D. LAX AND R. S. PHILLIPS LEMMA 4.3.
Set
(4.11)
5ir = 3ife(2iie2>0.
Then (4.12)
5ir = ®_n2J+.
Proof: It is evident that (2)^)^ = 2J± and hence (2)Le2);)-'c:(2l^)-' = 2)i. This proves that (4.13)
(2L®Ql'+yc:Qi_n9f+.
On the other hand. 2 i i = 2)ic:(®_nS>+)-^
and hence SJi©2>;c(S>_n2J+)-^, or by duality (4.13)'
(2J1 © 2>;)-' => 2i_ n 2(+.
The relations (4.13) and (4.13)' imply the assertion of the lemma. It is proved in [6], page 53, that 5^ is completely determined, up to right multiplication by a unitary operator on N, by the properties (i)'-(iii)' and the range of 5^ on 3if2. This indeterminacy is in this case further restricted by the fact that S^(cr) commutes with rotations about /. As in Lemma 3.5, this implies that 5^ takes ^(o-)y„(|3) into a function of the same form. Moreover, since the rotations act transitively on the sperical harmonics of a given order, the right multiplier is a scalar of absolute value one on each such ^ * ^ subspace. The range of 9' in ^€2 is precisely ^2(2»-) and ^2 © ^2(2» 1)=(Lzm Q ^2(20) e ^2(® 1)
146
TRANSLATION REPRESENTATION FOR THE WAVE EQUATION
665
by Lemma 4.3. In terms of model (ii),
is, according to Theorem 3.9, the set of all functions k(5, ß) in L2 whose n-th spherical harmonic components are of the form (4.15)
{t
cVne^(s))Yn(ß),
where
(4.16)
for for
^rnis)=[l^
s>0, si). HereJt suflSces to show that 5^0(^2) J-^"^e^Y;t for 0 < m ^ n . Now for arbitrary ke^2y we have by (4.16)' (4.18)
(^oK^-'e^YJ^^ifl^^^Kia^ da, VZTT-'p==icr —ip cr + im
where K(a)^\
k(cr,ß)yjß)dß
also belongs to the Hardy class. Since 1
FT Q,
(LI)
serves as a model for a non-Euclidean geometry in which the motions are given by the group G of fractional Hnear transformations: ^"^
^^ + b T—7
., _. (1-2)
where a, bj c, d are real and ad-be--
1;
(1.3)
Received by the editors August 1, 1979. AMS (MOS) subject classifications (1970). Primary *The work of both authors was supported in part by the National Science Foundation» the first author under Grant No. MCS-76-07039 and the second under Grant No. MCS-77-04908 A 01. © 1980 American Mathematical Society 0002-9904/80/0000-0100/$09.75
261
150
262
P. D. LAX AND R. S. PHILLIPS
G is isomorphic with SL(2, R)/±L The Riemannian metric (1.4) is invariant under this group of motions. The invariant L2 form is
/ / " ' ^•
CS)
The invariant Dirichlet form is ff(ui
+ t^)dxdy.
(1.6)
The corresponding Laplace-Beltrami operator L,^y'^=^y\^; + ^^')
(1.7)
is then clearly invariant. It turns out that the operator L defined as L = Lo + 1/4,
(1.7)'
also invariant, has more useful analytic properties, as will be seen in what follows. A subgroup r of G is called discrete if the identity is not a Hmit point of F. A fundamental domain F for a discrete subgroup T i s ^ subdomain of 11 such that every point of 11 can be carried into a point of F by a transformation in r and no point of F is carried into another point of F by such a transformation. A function/defined on 11 is called automorphic with respect to F if fiyw)^fiw)
(1.8)
for all Y in F. Because of_(1.8) an automorphic function is completely determined by its values on F. I f / i s continuous then (1.8) imposes a relation between values of / at those pairs of boundary points of F which can be mapped into each other by some y in F. If/is CS then (1.8) imposes a similar relation on the first derivatives of/at such pairs of boundary points. The Laplace-Beltrami operator, being invariant, maps automorphic functions into automorphic functions. Alternatively we can consider automorphic functions as being defined on F: In this case we introduce the space L^iF) of functions on F square integrable with respect to the invariant measure and define L as the selfadjoint extension of the differential operator defined by (1.7)', subject to the above mentioned boundary conditions. We shall study the spectral properties of L by means of the non-EucHdean wave equation u^^ = y^^u + u/4 = Lu.
(1.9)
This turns out to be a convenient analytic tool. As Semenov-Tian-Shansky [11] has recently shown, this equation also has an intrinsic meaning in the Lie algebra framework for problems of this kind. Indeed many of the classical concepts previously introduced in the study of these problems (see Kubota [4]) appear in a natural way in the scattering theory setting for the nonEucHdean wave equation. In particular the translation representations for the wave equation are closely related to the Radon transform associated with 151
SCATTERING THEORY FOR AUTOMORPHIC FUNCTIONS
263
symmetric spaces as well as the theta series employed by Kubota in the study of Eisenstein series. Perhaps the simplest discrete subgroup of G is the modular group, consisting of the fractional linear transformations with 0, 6, c, d integers satisfying (1.3). A convenient fundamental domain for this subgroup is the geodesic triangle F: - 1 / 2 < jc < 1/2,
y >Vl-x^ .
(1.10)
For the sake of simplicity, most of the detailed discussion in this paper will be carried out for this spectral subgroup. We conclude this introduction with a brief description of our abstract scattering theory and its application to the classical wave equation in Euclidean space. For details we refer to our book [5]. The theory concerns a one-parameter group of unitary operators 1/(0 acting on a Hubert space % for which there exist certain subspaces ^_ and ^ ^ , called incoming and outgoing with the following properties:
(ii) (iii)
(i)
1 / ( 0 ^ - C ^^
n
U{t)^^ = {0} - n U{t) 0. In this case the Fourier transform of S(t)y called the scattering matrix and denoted as S (z), is analytic in the lower half plane. It tums out that many important properties of the scattering
152
264
p. D. LAX AND R. S. PHILLIPS
process can be related to the behavior of the meromorphic continuation of 5 (z) into the upper half plane. This connection can be studied most directly through the one-parameter semigroup of operators Z(/), carved out of the group U{t) as follows: Z ( 0 = P^U{t)P^,
/>0,
(1.12)
where P^ and P^ are the orthogonal projections that remove the components of data that belong to ^ ^ and ^_., respectively. It is easy to show that Z(t% t > Oy annihilates ^ ^ and ^_ and acts as a semigroup of operators on % ^ % Q (öj)^ © \. Writing (2.2)' as the sum of integrals over FQ and Fl an integrating by parts over Fj, the energy form becomes
.. .2
a " •1/2
- - V -1/2
+
u{x, a)p dx.
^-\dx^ y' (2.2)"
From now on we omit the subscript F except where it is needed for clarity. Next we introduce a new form which is very close to (2.2)" but which has the advantage of being positive definite: G{ü) = E{u) + 2K{u\
(2.4)
K{u)^jj^dx^ I J
+ ff F
^^dxdy. y
(2.6)
By Rellich's theorem, K is compact with respect to C We denote by %Q the completion in the G-norm of the space of C** automorphic data with compact support in F. Since the A'-form is compact with respect to the G-form it follows that the E and G-forms are equivalent on any closed subspace of %Q on which E is positive. In order to treat u and u^ on an equal footing we rewrite the wave equation in component form as Uf = t?,
Vf =^ Lu;
(2.7)
or in matrix notation as V,=-AV
(2.7)'
where K.(«)
and >. = ( 0
^^).
(2.7)"
We take as D(A)y the domain of ^, the set of data (t/, v) for which both (u, v) and (Ü, LU) belongs to X^, Lu being defined in the weak sense. It can be shown that A generates on %Q a group of bounded operators U{t) which grow exponentially in the G-norm but are unitary with respect to the indefinite energy form E. Since (2.1) is hyperbolic, signals carried by solutions of (2.1) propagate with non-EucUdean speed < 1. Of particular interest to us are certain subspaces of %^ which we call incoming and outgoing spaces and denote by ^_ and ^^, respectively. For the modular group and F of the form (1.10) these subspaces are the initial data for solutions of (2.1) with support in F^ and which are independent of x for / < 0 (/ > 0) in the case of ^^ (^^); see §6 of [6\. A solution of (2.1) which is independent of x satisfies the differential equation u,^ = y^^u + u/4.
(2.8)
The change of variables s = log>^, V = u/Vy , transforms (2.8) into the classsical wave equation:
(2.9) (2.10)
whose general solution is v = l(s + t) + r{s- t). (2.11) The first term on therightcorresponds to a wave traveling to the left, the 155
SCATTERING THEORY FOR AUTOMORPHIC FUNCTIONS
267
second term to a wave traveling to theright.The corresponding incoming and outgoing solutions of (2.1) are of the form u(w,t)=y'/Mye'),
(2.12)_
uiw,t)=^y'/Mye-'); (2.12)^ here
^ < a. We define the incoming and outgoing subspaces ^ ^ as the closure in %Q of the initial data of the above respective incoming and outgoing solutions:
^ ^ =closure{jK^/V(^),V/V(>')), q) in C^, zero fory < a, cp' == 9y^ > a, P^ is orthogonal with respect to both forms. Notice also that since ^ _ and ^^ are orthogonal, P^ P^+P^
(3.1)
is the orthogonal projection on %. Using the relations (2.16) and (2.17) it is easy to verify for a n y / i n %Q that g = Pf can be decribed as follows: g =/
for>^ < a. forj;>a.
(3.2)
We now set Z{t) = P^ U{t)P^
for t > 0.
(3.3)
THEOREM 3.1. The operators Z{t\ / > 0, form a strongly continuous semigroup of operators on %. Z(t) annihilates both ^ _ and ^ ^ .
A proof of this theorem can be based entirely on the properties (2.14),. and (2.14)' (see Theorem 2.7 of [6]). We denote the infinitesimal generator of Z(t) by B. We have noted earher that U grows at most exponentially: \\U{t)\\ < Ce-K Obviously Z satisfies an analogous inequahty:
{3A)u
||Z(0|| < Ce"*. Consequently for/in SC and Re X > « we can write
(3.4)z
i\I-Br'f=^
Ce-^ZiO/dt
•'o
= P^ r
e-^U{t)fdt = P^QJ - AY'f,
(3.5)
Combining this with Theorem 2.1 we can now state THEOREM
3.2 The resolvent of B is compact on %.
COROLLARY 3.3. The resolvent of B is meromorphic in the entire complex plane and B has a pure point spectrum. 157
SCATTERING THEORY FOR AUTOMORPHIC FUNCTIONS
269
Since by (2.14) l/(0^+ C ^ + for / > 0, we see that P^U{t){I-P^)
= 0.
(3.6)
Hence for/orthogonal to ^ _ we have Z(t)P^ / = P^U{t)P^f=
P^UiOf.
If in addition/ belongs to the domain of A, then we can differentiate the above relation at / = 0 and obtain BP^f^P^Af,
(3.7)
a relation we shall use in §4. 4. The Eisenstein series and its analytic continuation. We are now ready to construct the generalized eigenfunctions of A associated with the continuous part of its spectrum. It is readily verified that for any complex z the data h{w) = {y'/^^^'^ izy^/^^'' }
(41)
locally satisfies Ah = izh. Since A is invariant, the function h^{w) = Ä(Y>V) satisfies
(4.2)
Ah^ = izh" (4.2)' for every y in G. Since h is by definition independent of x, it is automorphic with respect to the subgroup Y^ consisting of y: w -> >v + AI, « integer; that is Ä(Y>V) = Ä(>v),
y G r^.
(4.3)
In order to construct an automorphic function out of h for the entire modular group r we have to sum Ä(y>v) over all right cosets of F modulo T^. This sum is the Eisenstein series: e(wy z) =
2
Ky^y
(4-4)
It is well known that this series converges for Im z < - 1 / 2 ; except for the first term h{w), the series converges in the G-norm for such z. Since (A — iz) annihilates each term we conclude that {A - iz)e = 0 (4.5) in the sense of distributions for Im z < - 1 / 2 . It follows from elliptic regularity theory that e is C °^ and satisfies (4.5) pointwise. Recall the definition (2.7)" of A as d i); it follows from this and (4.5) that the first component e^ole satisfies the equation Lei = -z\^ (4-6) Integrating this with respect to x gives the following ordinary differential equation for the zero Fourier coefficient of e^: {y^d} + l/4)eP> = -z^ef\ The solutions to this equation are superpositions of 7*^, where K satisfies K{K
-\)
+ 1/4 = - z ^
158
(4.6)'
270
P. D. LAX AND R. S. PHILLIPS
that is ic = 1/2 ± iz. Since by (4.5), €2 = ize^ we deduce for Im z < - 1 / 2 that the zero Fourier coefficient of e defined by (4.4) is of the form e(^>(7, z) = {j^^/^-^^ /zyV2H-i> } + ^(^){^i/2-/z^ /zyV2~^ )^ (47) where s{z) is some function of z. A straightforward calculation, using (4.4), shows that
^ ^^^^
r(i/2)r(/z)^(2/z) r ( l / 2 + /z)Ul + 2/z)'
^ ^
f being the Riemann f-function. Recalling the definition of P described in (3.2), it is clear that Pe has finite G-norm and hence belongs to %. For our purposes it is more convenient to truncate e in a somewhat different fashion but still bring it into %. We choose \;^/2|>^^^>.V2(^^^i-H/zyj
(4 j i )
where RecaUing the definition of ^ ^ , it is clear that k is orthogonal to ^ _ and ^ ^ and so lies in %; one sees by inspection that k{z) is an entire function of z. Next we operate on (4.10) by P^; since k lies in %y P^k =' k and hence we get P^^/= izP^f+k. (4.12) Set g — P^ f* Since/is orthogonal to ^ « and belongs to D{A)y the relation (3.7) is applicable. Using (4.12) we obtain Bg = izg + k.
(4.13)
For iz in the resolvent set of 5 , the relation (4.13) can be rewritten as g-^'-iizI"
Py^k.
We conclude from this and Corollary 3.3 that 159
(4.14)
SCATTERING THEORY FOR AUTOMORPHIC FUNCTIONS
271
LEMMA 4.1. g{z) can be extended to be meromorphic in the whole complex plane J having poles at most at the points - i times the spectrum of By that is ^ia(B).
We can now prove THEOREM 4.2. The Eisenstein series e can be extended to be meromorphic in the whole complex plane with poles at most at the points -ia(B); the analytic continuation of e is an eigenfunction of A. REMARK. The analytic continuation of Pe(z) is in the %Q topology, that of the zero Fourier coefficient of e is pointwise. PROOF. By construction e(z) = g{z) for Imz < - 1 / 2 except for the zero Fourier coefficient when y > b and there e^^\z) is given by (4.7) which holds for y > a^. We see from (3.2) that the action of P^ effects only the zero Fourier coefficient of data and this only for y > a. Hence it follows from (4.9y and (4.8) that the zero Fourier coefficient of g{z) is also given by (4.7) for aQ 0, we have
T^( U{t)d) = T^( U{t)do)
for / > 0 and s > log a.
Using this relation and (5.11) of Lemma 5.1 and its corollary, we conclude that T{t)T^d = T{t)T^do
for t > 0 and s > log a.
Since this holds for all / > 0, it follows that it must hold for all s. This proves part (a). Substituting (5.19) into the definition (5.10)4. of T^ we obtain r^t/o=VIeV(^0-
(5.22)
The proof of part (b) is now immediate since by definition (2.2)"
= iTyWf
dy = i r
e^|(p'(eOP äs
(5.23)
•'-00
•'O
which by (5.22) •'-00
Combining this with the assertion of part (a) gives part (b). This completes the proof of the lemma. Combining Lemma 5.3 with Lemma 5.1 and its corollary, we deduce the following COROLLARY 5.4. (a) T^{U{t)d^ = T^(U{t)d); (b)E^U{t)d)^r\T^dfds. •'-00
We are now ready to construct the ^^-translation representaion. Recall that ^ ^ = ö ' ^ + > that is the elements d'^ of ^ ^ are obtained by projecting the elements d^oi^^ into OC^: d'^ = d^ +/>- +P^. where/7^ belongs to ^^. According to (5.6), ^ ^ is orthogonal to 6^_ and it follows from this and the relations (5.5) and (5.5)' that/?^, = 0: ^ ; = d^ +/;_.
(5.24)
Using (5.3) and (5.3/ we deduce that EAd^) = EM^y
(5-25)
This shows that Q' defines an isometric map on ^^. to ^ ^ . It follows that the translation representer of d'^ can be obtained directly by means of the operator Tj^ as we now prove: LEMMA 5.5. T^d'^
=
T^d^.
164
276
P, D. LAX AND R. S. PHILLIPS
PROOF. Since d^ and d'^ differ only by an element of ^^ and since ^_ is spanned by the {j^^ ) of (5-2), it suffices to show that
T^ fi = 0.
(5.26)
We note that Uif^ff = e^^Jf. Using this fact and Lemma 5.1 we conclude that T^ Sf = const eV.
(5.27)
As noted in (5.6), ^_ is orthogonal to ^ ^ ; we now show that T^ / = 0 for 5 > log a for any element / in %Q which is orthogonal to ^ ^ . To see this take d in ^ ^ of the form (5.20); Üie restriction ofrfto F is given by (5.19). Hence we have
0 - EM ä) = / / [y[^^)^^ - ^-^]
^, the above equation becomes •'-CX)
Recalling the definition (5.10)^. of T^^ we see that this relation says that T^ f is L2"^rthogonal to 9^