Selected Papers I

21

1964I

SCATTERING THEORY

137

annihilates all fin H which vanish in \x\ <3p. LEMMA

2. Z7o(2p) maps the complement of D- into D+.

Lemma 2 implies that P+Uo{2p)P^=0, whence for f^Zp (3.6)

Z(0 = P+U(t)P- = P+MU(t - 2p)P_.

So Z(/)e-^'J^ = 62Xpp^M I

£/(/ - 2p)e-^'dtP- = P+M(A - X)-»P_,

where ^ denotes the infinitesimal generator of U{t). It is easy to show that (A —X)~^ raises by one the degree of differentiability; since by Lemma 1 the value of Mf does not depend on the values of / outside the sphere | x\ ^Sp, it follows by Rellich's compactness criterion that M(A —X)~* is a completely continuous operator. For the pure initial value problem for hyperbolic equations with variable coefficients it is known that the sharp propagation of signals is along characteristic rays. This generalized Huygens principle can be reformulated as follows: Let G\ and G% be two closed sets in i?» with the property that no characteristic ray starting at time zero in Gi passes through Gi at time t. Let Pi and Pi be operators such that the range of P\ consists of functions which vanish outside of Gx, while Pi annihilates all functions whose support lies outside Gi. Then P2?7o(OPi, is completely continuous. We believe that this principle also holds for the mixed problem as well (for general hyperbolic equations with variable coefficients), provided that rays are interpreted as reflected rays. A demonstration of this for the interior problem for convex domains has been given by Povsner and Suharevskii [l2]. We say that an obstacle has property L if there exists a number / such that any ray starting in the sphere \x\ ^3p leaves the sphere \x\ S3p after time /. REMARK. Star-shaped obstacles have property L. Assuming the generalized Huygens principle to hold we assert: THEOREM

10. Z(f) is eventually compact if and only if the obstacle has

property L, 22

138 PROOF.

(3.6)'

p. D. LAX AND R. S. PHILLIPS

[January

The identity (see [9]) Z{t + 4p) =

P+MUit)MP^

follows similarly as (3.6). Take both Gx and d as the sphere \x\ ^3p; the operator MP- has the property required of P i while M has the property required of P2. So by the generalized Huygens principle Z(Z+4p) is completely continuous. The necessity of property L follows from known properties of propagation of high frequency signals along rays. Theorem 10 implies that Z{t) has a standard discrete spectrum. There can be no eigenvalue of absolute value one since this would correspond to a solution of the wave equation which is a purely imaginary exponential in time, and according to a theorem of Rellich there are no such solutions with finite energy. Thus the spectral radius of Z(t) is less than one; by the Gelfand formula we conclude that |[Z(0|1 decays exponentially. Thus Theorem 10 gives another proof of the result of [9]. Similar reasoning gives the following result: l e t / be any element of K, ^dJi its formal Fourier expansion in terms of the eigenfunctions of Zit); then

is an asymptotic expansion for Z{t)f. Next we wish to characterize the eigenvalues and eigenfunctions of the generator B of Z{t). For this purpose we say that a solution of the reduced wave equation (3.8)

A«-/x2« = 0

in the exterior domain is outgoing if the free space solution of the wave equation with initial d a t a / = [«, —iiu\, in symbols Uo(t)f, vanishes for I :\;| p. This notion is equivalent with the Sommerfeld definition of outgoing when /* is imaginary. Moreover for arbitrary fx in the case n = 3 such a solution of the reduced wave equation can be represented as 1 r / du u{x) = — I [u 4TJT\

dn

du\ V — ) dSy dn/

where v = e'"'/r, r=\x—y\y and T is any smooth surface containing the obstacle but not containing x. The converse is also true. THEOREM 11, IX is an eigenvalue of the generator of Z{t) if and only if there exists an outgoing solution of the reduced wave equation (3.8) satisfying the boundary conditions.

23

1964]

SCATTERING THEORY

139

We consider Za{t) =P^+ U(t)Pt as a-^co ; in the limit this is simply U(t). According to Theorem 4 the eigenvalues ß of the generator of Zo(0 are simply related to the zeros of the scattering operator; thus they are independent of a ^ p . The eigenfunctions depend upon "a" but according to Theorem 6 in a rather trivial fashion. In fact ior b>a'^p, SKETCH OF PROOF.

(3.9)

/« = P?/^

Since P% does not alter the data inside the sphere \x\
exists. Each fa satisfies Za(t)fa = e-"'/«. Since Za(t)f= U(t)f for \x\
Ua a solution of the reduced wave equation Aw« — ß^Ua = 0

which is zero on the obstacle. Since Ua(x) = Ub(x) for \x\ < a , the limit lim Ua = u a—•CO

exists, u is in the exterior a solution of the reduced wave equation and is zero on the obstacle. The d a t a / = [u, IJLU] can be thought of as a generalized eigenfunction of U{t); not only does it not lie in H, but it blows up exponentially in |ic|. /'' is orthogonal to Z>_ and so i s / * for all a ^ p by (3.9). As a consequence the free space solution Uo{t)f vanishes in \x\ p so that u is outgoing. Conversely if u is an outgoing solution of the reduced wave equation (3.8) satisfying the boundary conditions, then e~'" is an eigenvalue of Z{t). To prove this one shows that the free space translation representation o f / = [M, —ßu]isoi the form (0,

5 < -

h{Sf (a) = •{

{ «(«)«"%

Setting 24

s > p.

p,

140

P. a LAX AND R. S. PHILLIPS h{Sf co),

(January

s < üf S > ßy

one proves that ha is the free space translation representation of the eigenfunction of Za(t) corresponding to the eigenvalue e^'". The above ideas can also be employed to obtain an explicit description of the incoming and outgoing spectral representations of U{t) from which we will in turn be able to obtain an explicit formula for the scattering operator S(z). We shall denote by /o, / - , and /+ the free space, incoming, and outgoing spectral representations respectively of a given initial data / . These spectrsJ representations are given by scalar products of / with certain improper eigenfunctions of C/^o(0» respectively U(t). We shall show that these improper eigenfunctions consist of exponentials plus certain incoming and outgoing solutions. We recall that the free space spectral representation for C/o(0 is simultaneously incoming and outgoing. Thus i>i. and D% map onto A-(N) and A+(N) respectively, while Z?_ and D+ map onto e~*''M_(iV) and e*p''A+{N) respectively. We shall limit our considerations to the case n = 3. THEOREM

12 (SPECTRAL REPRESENTATION FOR Uo{t)).

(3.10)

/o(z,«) = (/,M-,2,«))

where ( , ) denotes the Ho inner product and

The main tool employed in the derivation of (3.10) is the Fourier transform. THEOREM 13 (INCOMING AND OUTGOING SPECTRAL REPRESENTATIONS FOR U{t)). Let v+(f_) he the outgoing {incoming) solution of the

reduced wave equation Av + zH = 0 satisfying v+e-*"'*' = 0 on the obstacle. Set 4T''V±(^I Z> W)

= [v±(Xy z, w), izv±ix, z, «)]

and define 0± = <^o + l^±. Then (3.11)

/±(2,«) = ( / , < ^ T ( - , 2 , « ) ) ,

where the ( , ) denotes the inner product in H, Note the switch in signs, 25

1964I

SCATTERING THEORY

141

SKETCH OF PROOF FOR THE INCOMING REPRESENTATION FORMULA.

Step one. To verify (3.11) for data in D-. It is required that data in D- have the same representation as in the free space spectral representation. This in turn requires that (/, ^+) = 0 for all / in D-. Now f o r / = Uo{—T—p)w where w has support in \x\ < T , it is clear that (/, ^+) = (w, Uo(r + p)^+) = 0 since f/o(T+p)^+ vanishes for \x\ < r . It is proved in [9] that linear combinations of s u c h / are dense in £>_ 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\
29

DECAYING MODES FOR THE WAVE EQUATION

739

if the obstacle satisfies certain geometrical conditions described below and if the space dimension is odd^ then all such solutions decay exponentially for fixed x as t tends to infinity. I n fact, for large t such solutions behave asymptotically as follows: 00

(1.7)

u{x,t)^2c^e''''w^{x), fc=0

where the numbers Cj^ depend o n the initial data but the numbers ?^^ and the functions Wj^ are determined solely by the obstacle & and are in a generalized sense eigenpairs for the operator A in the exterior domain. Each Xj^ has a negative real part and they have been indexed so that (1.8)

0 > .^/^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 <^{z) is an operator on i^2('^2) ^ ^ ^ ^^^ be represented as

(2.1)

^ ( z ) = / + ^^^(z),

32

742

p . D. LAX AND R. S. PHILLIPS

where K^^{z) is an integral operator with kernel (2.2)

K^'^{oj,B;z)

= ! i ^-(co, -6;

z) ;

here k^^{cjo^ 0; z) is the transmission coefficient (see [4], Theorem 5.4 of Chapter 5). We shall be concerned with values of z in the lower half-plane, J'^n z ^ 0, and for such z the transmission coefficient is determined by the solution of the reduced wave equation: z^v + /\v = 0

in

^ ,

(2.3)

(Dirichlet problem) v{Xy CJO; z) = exp {izx • a>}

on

d^ .

It can be shown that the asymptotic behavior of the solution v for large |;c| is given by

k''{w, Ö; z) + 0 | -

v{rdy (o; z) = r

where Ö is a unit vector and x = rd. Thus, k^^{cx)^ 6; z) represents the asymptotic scattering amplitude created by a plane wave traveling in the co-direction and reflected by the scattering body in the direction Ö. It is known that the transmission coefficient is smooth in co, Ö and analytic in z in the lower half-plane. We begin by deriving a useful integral representation for the transmission coefficient. Since we shall be concerned with the purely imaginary zeros of c^(z) in the lower half-plane, it is convenient to work with a = zz. In what follows, a will denote a positive real number and, again for notational convenience, we set k{(x), 6; a) = k''{a). 6; -ia)

,

(2.4) X(a>, Ö, cr) = — k{ay, -d;a)

= K''{a), Ö, - m ) .

277

In this case, (2.3) becomes ah -

Av = 0

in

^,

(2.3) p

(Dirichlet problem)

v{x, (o; a) = ^^^'^ on and the asymptotic behavior of v is given by

v{rd, co; a) =

a^ .

k{(jo, 6;(y) + 0

0]

T h e transmission coefficient for the corresponding Neumann problem is defined

33

DECAYING MODES FOR THE WAVE EQUATION

743

in terms of the bounded solution v of aH -

[\v = Q

in

^ ,

(2.3)^ dv[x, (jo; a) de^^'"^ _^^ 5 = ~5— ^^ ^^ ^ on on where n denotes the outer normal to 9 ^ . Again, the asymptotic behavior of v near infinity is given by

k{a), d;a) + 0 ( -

r THEOREM 2.1.

If we denote by q^ the bilinear form q^{u^ w) = a'^uw + Vw • ^w y

then the transmission coefficient is given by k{w, Ö; c) = —

^^(^^^-^ ^^^•^) dx

477 JO

.

(2.5)

477 J ^

it;Ä^r^ a = 1 ybr the Dirichlet problem and — 1 for the Neumann problem^ v is the solution of (2.3) and (9 the interior^ ^ the exterior of the obstacle. Proof: Since the proof is essentially the same for both the Dirichlet and the Neumann problem, we shall limit ourselves to the former. Let ^j^ denote the domain ^ f^ {x: \x\ <, R}, Making use of the fact that exp {(Tx • o)} and v{xy co; a) both satisfy the differential equation L^u = a^u -— Aw = 0 5 we obtain from Green's formula the relation

0 =

[e'^'^-^L^vix, a};a) - v{x, co; a)L^e''^'^] dx

(2.6) =

[e^^'^ d^v{x, a>; (T) - v{x, co; a) d^e'^'^]

dS.

Now, d^ji consists of two parts: d^ and {x: \x\ = R}; denote the integrals on the right-hand side of (2.6) over these parts by I^ and I^ , respectively. O n d^

34

744

p . D. LAX AND R. S. PHILLIPS

we have v{x^ o)-^ a) =• exp [ax • oj] and hence I^ can be written as I^=

\

[v{x, d; a) d^x,

(o;a) -- e^^'^ d^^'']

dS ,

Next we apply the Green's formula

J

uL^w dx =

^^(w, w) dx — \

u d^w dS

to the functions v{xj 0; a) a n d v{x, o); a) in the exterior ^ and to the functions exp {ax ' co} and exp {ax - d) in the interior 0. Using the fact that the exterior normal to ^ is the interior normal for (9^ we see that I^ is equal to the righthand side of (2.5). Setting X = rjR on {x: \x\ = R] and making use of the asymptotic form of v given in (2.3) j) , we get

^-^k{co,

= -a Ju\=\

V; o) dn \-^)\R'dr^

+ 0

[^^

^~crii-n^e)R (1 + ^ . e)k{w, r];a) Rdrj + 0 (-] \R/

.

T h e kernel R exp {—-a{\ — rj - d)R] peaks a,t rj = 6 and it is easy to show that, as R becomes infinite, a

i ? e x p { ~ ( r ( l - Tj ' d)R]dr] = 27ra Jkl=i

i ? e x p { - ( r ( l - r)/?} ^ r - > 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;
T h e symmetry of A:(co, 6; a)^ that is, k{a)y 6; a) = k{d, co; a) ,

follows from (2.5) and the symmetry of the form q^ . Since v is real-valued, so is k{(o^ d; a) and, since k is smooth,

/J

\k{(D, d; a)\^dMdQ < oo ,

35

DECAYING MODES FOR THE WAVE EQ^UATION

745

SO that the operator is Hilbert-Schmidt. It remains to show that CL{ka^fl)= a

A:(co, Ö; G) a{a)) a{d) doj dd ^ 0

for all real-valued a{') in Z2(>5'2). Setting

(2.7)

I-

A{x)

^^^•^ a{a)) dco

for

x

in

v{x, co; a) a{a)) dco

for

x

in

]•

&,

it follows from (2.5) that (2.8)

0L{ka, a) =—

\

[aU^ + (V^)^] dx ^ 0 ,

which proves the corollary. We note that A satisfies the differential equation (2.9)

a^A - A ^ = 0

in (P KJ ^ and in the case of the Dirichlet problem that A is continuous across d^, whereas in the case of the N e u m a n n problem d^A is continuous across 9 ^ , but A itself is not. In both cases, A{x) tends to zero as \x\ —> 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^ ^ <^2 "" ^1 and ^ 2 5 further, d^C = 0 on both sides of d^i , and d^C is continuous across d^2 - Writing E{A, ^ ) =

[crM^ + (V^)2] dx

we have (2.11)

EiA,,

^ , ) = E{A, + C, ^ , ) = E{A,,

^ , ) + E{C,

'^,)+Q,

where Q = 2

[a^^C

+ V ^ i • VC] dx .

Integrating by parts over ^^ — ^ 2 ^ ^ ^ ^ 2 separately, and using the fact that d^C = 0 on 9 ^ 1 and is continuous across d^2 ? ^ ^ S^^ Q = 2(

[{a'C-AC)A,]dx-\-

(

A,d„CdS+(

A,[d,CU

- d^CUdS

Substituting this into (2.11) and recalling that E is positive, we see that E{A^ , ^ 0 = E{A,,

^ , ) + E{C, ^ i ) ^ E{A,,

37

^,) .

== 0 .

DECAYING MODES FOR THE WAVE EQUATION

747

It follows that A7r<^{k^a, a) = E{A, , R3) = E{A„ (9,) + E{A, , ^^) ^ E{A,,

&,) + E{A,,

^ 0 = E{A,,

[R3)

= 4:7roi{k2ay a) . This proves the inequaHty (2.10). According to Corollary 2.2^ oik{a) is the metric operator; in Section 5 we shall show defines a strictly positive operator. Since interior contains a sphere, we conclude from

kernel of a non-negative and symfor a spherical scatterer that (xk{a) every scatterer with a nonempty this and the monotonicity theorem:

COROLLARY 2.5. If the scatterer (9 has a nonempty interior^ a times the transmission coefficient is the kernel of a strictly positive operator, A few concluding remarks about the boundary value problem (2.3)j) : T h e operator cr^ — A satisfies the maximum principle; consequently, the solution v which has positive boundary values and vanishes at infinity is non-negative throughout ^ . This implies that the transmission coefficient A: is a non-negative function. Further, the function exp {ax • a>} is itself a solution of the reduced wave equation. O n the boundary it equals v and, from the asymptotic description of V for large |x|, we see that exp {ax • a>} 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)

<^^^vf

and

/c<^> ^/c^f^ .

We shall postpone the proof of this theorem in its full generality until the next section. However, when (9 is centrally symmetric the discussion is considerably simplified and it is both instructive and useful to have a special proof for this case: Step L (3.4)

If & is centrally symmetric, then k{a) and W c o m m u t e : k{a) W = Wk{a) ,

and hence K{a) is symmetric for each c > 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 <^ &^ ^ where (9^ is starshapedy and if c9^ has n purely imaginary zeros {—za^} with cf\^ a^ then ^ has at least npurely imaginary zeros {—ia^ with a^ ^ cr. It is clear that a comparison of (9 with contained and containing spheres will furnish us with good estimates of the purely imaginary zeros of ^ . This will be discussed in Section 5. Let (9^ be an obstacle for which one of the KJ^Ö^ OQ) has more than one intersection with the line K = —I. It is instructive to observe what happens when (PQ is deformed continuously into another obstacle &i ^ say a star-shaped one, for which K^{ay 6^ is monotonic in a. W e start with Rellich's observation that even when K{a) has multiple eigenvalues at some cr's, the eigenvalues can be continued as regular analytic functions of a across these points. Let a^(ö') denote the regular branches of these eigenvalues. It is easy to show that if one of the K^{a) is not monotone and crosses the line K = — 1 more t h a n once, then one of the analytic branches a^((T) will also cross the line a = — 1 more t h a n once. O n the other hand, if the K^{a) are monotonic, so are the a^(cT). Suppose now we deform (9^ continuously into (9-^ through an increasing family (9^ of obstacles, 0 ^ / ^ 1. Denote by a^(o', /) the analytic branch corresponding to €)^ . As t increases we shall eventually reach a t^ for which a^((T, t^ has a local maximum on the line a = — 1 . For i < ?Q , the curve a^(ö', t) has at least three intersections with a = — 1, two of which coalesce at i = fQ and disappear for ^ > ?o ? becoming in the process a pair of complex zeros of the scattering matrix. This process of annihilation and creation of purely imaginary zeros out of pairs of complex ones may be worth further study. 4. A c o m p a r i s o n T h e o r e m for K W e now return to the proof of Theorem 3 . 1 ; we shall in fact give two proofs. T h e first is fairly direct and is accomplished by means of a symmetric operator with the same spectrum as K. T h e second proof gives a minimax characterization of the eigenvalues of K in terms of K itself in the setting of a Hilbert space with an indefinite metric. Throughout this section, k will denote a compact strictly positive symmetric operator, W a symmetric involution (that is M^ = PF* and W^ = / ) and (4.1)

K =

kW.

Setting A equal to the positive square root of A:, (4.2)

A = p/2 ,

we now define (4.3)

C =

AWA.

42

752

p . D. LAX AND R. S. PHILLIPS

It is clear that C is compact and symmetric and further that (4.4)

KA^

AC.

Actually K and C are even more closely related. LEMMA 4 . 1 . K and C have the same eigenvalues with the same multiplicities. K-eigenspace of K is the A-transform of the K-eigenspace ofC. Proof:

The

If {v^ K) is an eigenpair for C, that is, if Cz; = KV^ then by (4.4) KAv = ACv = KAV ,

so that {Av, K} is an eigenpair for K. Conversely, if Kw = KW^ then since K = A^W v/c see that w lies in the range of ^ and hence that v = A'^^w exists. Again by (4.4) ACv = KAv = KAV , and hence {A^'^w^ K} is a n eigenpair for C. Since A is one-to-one, it follows that A maps the eigenspaces of C onto the corresponding eigenspaces of K. We have now reduced the study of the eigenvalues of K to that of the more tractable compact symmetric operator C. W e proceed to investigate C LEMMA 4.2.

Order the eigenvalues of C taking account of multiplicities :

Then (4.5)

v^ = max V

min wev

( Ww, w) {A-^w.A-^w)

VczRiA)

and (4.6)

K^ = m m V

( Ww, w)

max wev

{A-^w.A-^w)

VdRiA)

where the maximum and minimum with respect to V are taken over all subspaces V of dimension n. Proof: (4.7)

According to the familiar minimax theory, v^ = max V

min

{Cvy v) —,

veV

\Py^)

43

DECAYING MODES FOR THE WAVE EQUATION

753

where the m a x i m u m is taken over all rz-dimensional subspaces V. Writing {Cv^ v) = {WAvy Av) a n d setting w = Av^ we get {Cv, v)

(Ww^ w)

{v, v)

{A'-'^w, A-^w)

Now i; in F can be written as w in AV. T h e assertion (4.5) is therefore a simple consequence of (4.7); (4.6) is proved in an analogous way. W e wish to make use of (4.5) a n d (4.6) to establish a comparison theorem for the eigenvalues of two different operators K^ = k^W a n d K^^^ k^W when k^ ^ k^ . This requires a n additional lemma proving in effect that A:~^ ^ k~^. Again we set A, = kV^. LEMMA 4.3.

If 0 ^ k^ ^ k^, then R{A^)

lMi"'«'Il ^ lMr^"^ll Proof:

for

c R{A^

all

w

and

in

R{A^) .

Assume to begin with that Q <, dl "^k^^k^

a n d write

k{t) = ^1 + {k^ - k{)t,

0 ^ f ^ 1.

T h e n k{t)~'^ is uniformly bounded a n d dk(t)-'^ - ^ = -k{t)-^[k,

-

k,]k{t)~^

is obviously non-positive. I t follows from this that

(4.8)

V ^ k-^ .

If we merely assume that 0 ^ Ar^ ^ Arg , we can obtain the desired result by a limiting procedure. T o this end define k,{n) = ki + n-U

and

A^in) = k,{nf^^ .

Since A^{n) is invertible, given u in H there will exist a u^ such t h a t (4.9)

A,{n)v„ = A,{n)u .

T h e inequality (4.8) takes the form \\A,{n)-^wP and setting w — A^{n)u = A2{n)v^\^e (4.10)

^

\\A,{n)-^w\\\

get

\\vj ^ \\u\\ .

44

754

p . D. LAX AND R. S. PHILLIPS

Since the v^ are bounded in norm, a subsequence will converge weakly to some limit v and ||r;|| ^ lim \\uj\. O n the other hand, A^{n) = [k^ + n-^y^'^ tends in norm to A^. Hence letting n tend to infinity in (4.9), we obtain, as a weak limit on the left and a strong limit on the right, A2V — A^u .

This proves that R{A^) c: ^ ( ^ g ) ^^d also by (4.10) that \\A~^w\\ ^ \\A-^w\\ for ?i\\winR{A^). W e now have all of the ingredients at h a n d for the proof of our comparison theorem. THEOREM 4.4. Suppose k^ and k^ are compact strictly positive symmetric operators such that 0 < A:^ ^ ÄJg and set K^ = k^W. Order the eigenvalues of K^ taking into account multiplicities: yU) ^ ^(i) ^ . . . > 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


-2\yJ Jul

where / „ = 0.66274 • • • . If in addition & is star-shaped, then lim sup —

_

'


W e surmise that the limit lim exists. Formula (5.15) shows t h a t X^^ defined by (5.14)5 is positive for all n. This proves that the operator {a\27T)(xk^{^d) is strictly positive for all o" > 0 ; but then according to Theorem 5.1 so is {aj27r)oLk{a). This fact was anticipated in the proof of Corollary 2.5. 6, T h e M o n o t o n i c i t y T h e o r e m f o r S c a t t e r i n g b y Positive Potentials In this section we consider scattering by a positive potential and show t h a t the operator whose kernel is the transmission coefficient depends monotonically on the potential. T h e dynamic equation governing the motion is u^^ — iS.u — qu . As before the transmission coefficient is defined by means of the reduced equation (6.1)

a^u -

^u + qu^O

;

we consider solutions of the form (6.2)

u{x, a};a) = ^^^'^ — v[x, co; a) .

T h e function v satisfies the inhomogeneous equation (6.3)

-Av

+ aH ^^ qv =

P
It is hard not to deduce from this that if q{x) exp {(yx • co} tends to zero fast enough as |^| -> oo (see condition (iii) below), then (6.3) has exactly one solution V which vanishes at infinity; for large \x\ this solution has the following

53

DECAYING MODES FOR THE WAVE EQUATION

763

asymptotic behavior:

(6.4)

v{r, d;o)

=

k{co, Ö; (T) + 0

©:

We shall assume that q satisfies the following requirements: (i) q{x) is non-negative and Holder continuous except at a finite n u m b e r of points, (ii) q is locally square integrable, (iii) q decays exponentially, i.e., there exist positive constants a, C, R such that \q{x)\ ^ a ~ « l ^ '

for

1^1 > - 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—
= —,

:•

47r 1^ — JV| T h e asymptotic behavior of this kernel for fixed y and large \x\ is readily seen to be

(6.6)

G{rd,y; cr) = 1 ^-^ 477 r

^üQ,v ^

Q

(;y

Writing the right side of (6.5) as an integral and substituting into it the above asymptotic expression for G{x^y\ cr), we obtain an asymptotic description for v{x^ (jd\ a) for large |;c|. Finally, recalling the definition (6.4) of A:, we have A:(ca, 0; (T) = lim re^^v{r^ oj'^ a)

(6.7) = —

e^^-yq{y){e^<^'y - v{y, (o; a) dy .

Next we solve for v in (6.5); the following lemma shows t h a t this is possible. LEMMA 6.1.

The operator I + Gq is invertible zn 1.2(^3)•

Proof: I t is easy to establish that Gq is compact, in fact Hilbert-Schmidt. It follows by the Fredholm alternative that I + Gq is invertible if the only

54

764

p . D. LAX AND R. S. PHILLIPS

solution of (6.8)

w + Gqw = 0

which Hes in L2{P^s) is w = 0, T o verify this condition note that (6.8) implies that w = {GqYw and it can be shown from this by familiar estimates that w{x) is bounded. Thus^ qw belongs to L<^(JR>^ and we can take the L2{^^ scalar product of (6.8) with qw^ denoting this L^ scalar product by brackets we get {qWj ui] + {qw^ Gqw] = 0 . We have remarked earlier that G is a strictly positive operator and, by assumption, the potential q is non-negative. It therefore follows from the above relation that qw = 0; but it then also follows by (6.8) that w = 0. This completes the proof of the lemma. We can now express v from (6.5) as v=

{1+ Gqy^qe{co) ,

where e{co) stands for the function e{x, co) = ^^^-"^ ; notice that qe{co) belongs to ^ 2 ( ^ 3 ) . Substituting this expression for v on the right side of (6.7) and using the above abbreviation, we get the following representation for k: k{co, d;a)

= T - J ^{y, Ö)?(j){^(j>', M) — v{y, G>; 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---<- .

Finally, let v = v{x, ca; a) denote the corresponding signal scattered by the

57

DECAYING MODES FOR THE W A V E EQUATION

767

obstacle (P u n d e r Dirichlet boundary conditions; that is^ v is the solution of (7.2)

-Av

+ a^v = 0

V = ^^^'^

on

in the exterior ^ of & with (7.3)

(9

and vanishing at infinity. THEOREM 7.1. For each a ^ 0 the limit lim v^{x^ a>;

(T)

= v{x^ co; cr)

71-•00

exists uniformly for all x on compact subsets of ^ and all co in S^ . Before presenting the proof of Theorem 7.1 we shall point out the most important implications of this result. COROLLARY 7.2. Denote by k^{a}^ 6; a) the transmission coefficient associated with the potential q^ and by k{cjo^ 6; a) that associated with the obstacle & under Dirichlet boundary conditions ; then lim k^{(D^ d; a) = k{co^ 6; cr) n-^-oo

uniformly in co and 0. Proof of Corollary 7.2: T h e transmission coefficients k^ and k are determined from the asymptotic behavior at infinity of the scattered waves v^ and v; and this asymptotic behavior can in turn be related to values of v^ and J; in a compact portion of the exterior domain. T o find such a relation we apply Green's formula to the function v^ , respectively v^ and the fundamental singularity J

G{x,y;a)

^-(T\x-y\

= — 47r \x — y\

in the exterior of the sphere {x : \x\ = R} so chosen that the obstacle lies entirely inside the sphere. For |^| > 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

<^.

768

p . D. LAX AND R. S. PHILLIPS

and similarly

Substituting these estimates into (7.4) and recalling the definition (6.4) of k^ , we get k^{a), 6; a) = lim re''^v^{rd, a>; 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\
dx = \ U, drU, dS J\x\=R

+ (gn + 0^)<]

Since the sphere {\x\ = R} lies in a compact subset of ^^ we can assert that u^ and d^u^ are uniformly bounded in n and co on this sphere. It therefore follows that the integrals on the left in (7.7) are also uniformly bounded in n and co. Now the potentials q^ are defined as ng where g{x) is positive throughout (P^ vanishing in ^ . W e can therefore conclude from the boundedness of (7.7) that, for every compact subset K of the interior of 0^

lim lim r

(7.8)

uldx = 0

n-*oo JK JK

uniformly in coNext we prove LEMMA 7.3.

lim

ul dS = 0

uniformly in (o. Proof: After a partition of unity it suffices to show for every small boundary patch S on the boundary of & that (7.9)

lim luldS

==0

n-*co JS

uniformly in co. It is convenient to introduce new coordinates in Ö? near S^ i.e., a smooth mapping / o f the cylinder «^ x [0, 1] into (P^ so that ^ X Ois S. T h e image of ^ X [d^ 2d] is a compact subset K^ of & and hence, according to the assertion (7.8), (7.10)

e{n,d)=\

uldx

= \ \

wlJdyds

tends to zero uniformly in co as n —> oo; here w^ = u^of-^ and J denotes the Jacobian of f. If we now apply the mean value theorem to (7.10), we see that there is a value s{n^ d) (here we omit the dependence on a>), (7.11)

d <s{n,d)

61

<

26,

DECAYING MODES FOR THE WAVE EQUATION

771

such that (7.12)

\ wl{y,s{n,d))dy

=

'^"'^^ JÖ

'

where j is the value of J at some point. Next we make use of the identity (7.13)

p ' ' ' f d,wl dyds = ( wliy, Jo

J9

s{n, d)).dy - f wKy, 0) dy .

J9

J9

T h e integrand of the left side can be written as 2w^ ^s'^n ? ^sing the fact established earlier that the functions u^ , and thereby the functions w^ , are uniformly bounded in n and w^ we can estimate the left side of (7.13) by the Schwarz inequality and see that it is bounded by / p^ r \i/2 const. ( (5 {dsW^^ dy ds\ According to a calculus inequality,

J

i^s^nV dy ds -^ const. 0 J9

(AwJ^ dx ; J\x\
and according to (7.7), the integrals on the right are bounded uniformly in n and CO. Putting these estimates together, we see that the double integral on the left in (7.13) is bounded by const. (5^/^, the value of the constant being independent of w or co. T h e first term on the right in (7.13) is expressed by (7.12); consequently the remaining term in (7.13) can be bounded as follows:

i 9

^liyy

0) ^ ^ const. ^1/2 _|_ const. ^^"' ' O

We can choose d first and then n so that the right side of this expression is less than any preassigned number uniformly m n ^ n a n d a>; this shows that lim

wl{y, 0)dy = 0

n->oo JSi

uniformly in co; it is clear that (7.9) is equivalent with this and thus the proof of Lemma 7.3 is complete. Now by (6.2), o
62

772

p . D. LAX AND R. S. PHILLIPS

and by (7.3), V = ^^^*^

on

d(P .

Hence, Lemma 7.3 can be expressed as (7.14)

lim f 1^; - vJ^dS

= 0

uniformly in co. According to (7.1) and (7.2) both v^ and v are annihilated outside of & by the operator — A + cr^ and hence so is their difference v^ — v. We can therefore represent v^ — v 3.1 any point A: in ^ in terms of the value ofv^ — v on d(P with the aid of the Green's function F for the exterior domain ^ : (7.15)

v^{x) - v{x) =\

{v^-

v)F{x,y;

a) dyS ,

J 50

Expressing F{x^y; a) as the sum of the fundamental singularity and a regular part, we see, with the aid of the m a x i m u m principle, that \F{x,y;a)

^ l-^e-"^

,

where d is the distance of ;^ to d(9. Estimating the right side of (7.15) by the Schwarz inequality and using the above estimate for F as well as (7.14), we conclude that v^{x^ o); a) -> v{Xj co; a)y uniformly for all x bounded away from d(P and for all co. This completes the proof of Theorem 7.1. 8. An Integral E q u a t i o n f o r t h e E x p o n e n t i a l l y D e c a y i n g M o d e s o f t h e Wave E q u a t i o n There is an interesting relation between the purely decaying modes of the wave equation and those of the transport equation. I n order to bring out this similarity we shall give a different and more direct proof of the fact that there are infinitely many purely decaying modes for the wave equation with the following simple type of potential: r^ in (P, [0

outside

&,

where c is a positive constant. We shall use a direct method to prove that there are infinitely many purely decaying modes. Let u = e~^^w{x)y cr > 0, be a purely decaying m o d e ; w satisfies the relation (8.2)

a^w - I^w + qw = 0 .

63

DECAYING MODES FOR THE WAVE EQ^UATION

773

We introduce the following abbreviations: L^ is the operator

e^ denotes the outgoing fundamental solution of L^ :

(8-3)

^.^ = T T '

it satisfies

(8.4)

Ke„ = ^-

We denote the translates of ^^ by e^y :

they satisfy (8.5)

L„e,_,=^ö{x-y).

Equation (8.2) can be abbreviated as (8.6)

L^w

+ qw = 0 .

In addition to satisfying the eigenfunction equation (8.6), the purely decaying mode w has to be outgoing at oo. Among the several equivalent formulations of this property the following is the most convenient to use here: A function w which satisfies L,w = 0 outside of a fundamental We shall We shall use

bounded domain (^ is outgoing if it is a superposition of outgoing solutions e^y ^y inside (9. derive now an integral equation for outgoing solutions w of (8.5) • Green's formula

f

r / ^^

^A

{vL^ z - zL^v) dx = \ \z— - v—\ J9 Jd^ \ an an/ LEMMA 8.L

dS.

Let QJ he any domain^ p and y any pair of points contained in 2); then

Jd^ \ ^'^ dn

^'^ dn /

Proof: Using Green's formula we can express the above boundary integral as the volume integral

J

{^„.y L„ «^.j, - ^^,p L„ e„_y) dx .

64

774

p . D. LAX AND R. S. PHILLIPS

Using (8.5) we see that the value of this integral is

which b y definition equals ^a{p -y)

- e^y

- P) •

Since e^{x) is a n even function of x^ the above quantity is zero. This proves L e m m a 8.1 when/? 7*7 J^; the case/? = y can be handled by letting p approach y. LEMMA 8.2. Suppose that w is an outgoing solution of L^ w = 0 outside (9; then^ for any domain Ql such that d^ lies outside 0^

hsf \ ^*^ dn

dn J

Proof: Outgoing means that, outside of 0^ w is ?i superposition of functions e^ y ^y in (9. Forming a superposition of the integrals asserted to be zero in L e m m a 8.1 we obtain L e m m a 8.2. W e apply now Green's formula to w a n d e^,^^ over any domain containing 0. According to L e m m a 8.2 the boundary integral is zero; therefore so is the volume integral. Since L^ e^^ = d{x — p) and;, according to (8.6), L^w= —qw^ that volume integral is (8.7)

w{p) +

\ e,^^qwdx

= 0.

Recalling the definitions of e^^^ and of q we can rewrite this as 1

(8.8)

r ^<^i^-^i

---w{p)=\ C

w{x) dx . J(!)\X — p\

Conversely, let jx; be a function defined in 0 which satisfies equation (8.8) for p in (P; let us define w outside of & by (8.8). Clearly, the function w thus extended satisfies L^ w + q w = 0 andj being a superposition of function e^^^ , x in (P, is outgoing. W h a t we have shown so far can be summarized as THEOREM 8.3. The wave equation with the potential (8.1) has an exponential mode e~^^w if and only if —Xjc is an eigenvalue of K^{0)^ an integral operator over 0 with kernel ßo\x-p\

65

DECAYING MODES FOR THE WAVE EQUATION

775

W e turn now to investigating the eigenvalues of K^ . Since the kernel of K^ is only mildly singular^, K^ is a compact operator and has a standard discrete spectrum. Also^ the kernel of iT^ is real and symmetric so that the eigenvalues of K^ are real. Denote by K^{a^ (9) the rz-th negative eigenvalue of iC^ when K^ has n negative eigenvalues. Otherwise set K^{a) = 0. LEMMA 8.4. Proof:

K^{(y) is a continuous function of a for each n.

It follows from the minimax characterization of eigenvalues that \'
It can be shown easily that K^ depends continuously on a in the norm topology, from which L e m m a 8.4 follows. LEMMA 8.5.

K^{a)

= 0 for

a ^

0 for all n.

Proof: W e shall prove that K^'^0 for o" ^ 0; clearly this implies the lemma. T o prove the positivity of K^ we note that K^ is the restriction to & of convolution with

'V\' For (T < 0 this function has the Fourier transform 1 l^ + (r2 which is positive. This proves the positive definiteness of iT^ , cr < 0. LEMMA 8.6.

For all n^ l i m K^{a)

=

— 00 .

Combining the last three lemmas we see that each K^{a) is a continuous function whose value changes from 0 to — oo as cr goes from 0 to oo. Therefore, according to the intermediate value theorem, each equation 1 C

has at least one solution a^ > 0. This proves THEOREM 8.7.

There are infinitely many decaying modes.

66

776

p . D. LAX AND R. S. PHILLIPS

Proof of L e m m a 8.6: W e choose a ball inside 6; denote its diameter by D. We select n distinct points ATJ , • • • , :v^ on the surface of the ball, so chosen that no two are antipodal. Denote by d the maximum distance of any point x^ to any point x^ or x^ ^j y6 i^ where x^ denotes the antipode oi x^ . By construction, d
.

Let B^ be the ball of radius r and center x^, B^ its antipode. enough these balls belong to (9 and are disjoint. Define n functions u^ ^j = 1, • • • , n^, as follows ( a

in

Bj,

[—a

in

Bj 5

For r small

uj{x) = a so chosen that the Lg norm of Uj is 1. Since the M^ have disjoint supports, they are orthogonal. We claim that for any u belonging to the span of the {u^}, for a large enough and for r small enough,

(8.9)

{u,K„u) ^ -,(^^)<^ 11^1,1 .

T h e assertion of L e m m a 8.6 follows from this via the minimax principle. W e have u = ^a,' u^ ; since the Uj are orthonormal,

ll"f = !«?• Now

(8.10)

''' = 2 ("t' ^o "i) «i + 2 ("t' ^(T "i) «i « i ' i^o

i

and by definition, [Ui, K^ Ui) =\\

Ui{x) Ui{y) dx dy .

Since the support of u^ consists of two antipodal balls whose minimum distance is Z) — 2r, and since u^ was chosen as odd^ we see that for a large enough (8.11)

{u^ , K^ u,) ^ ~ c o n s t . ^^<^-2r) ^

O n the other hand, the maximum distance of any point x in the support of u^ to any point y in the support of u^ is, for i j ^ j \ less than d + 2r. Therefore, (8.12)

\{Ui, K^ z/,.)| ^ const. ^^<^+2r) ^

67

DECAYING MODES FOR THE WAVE EQUATION

777

Substituting (8.11) and (8.12) into (8.10), we see that for r < ^{D — d) inequality (8.9) holds. It follows from the minimax principle that the eigenvalues oi K^{&) depend monotonically on 0] this implies that for star-shaped 0 the n u m b e r of negative eigenvalues o^K^[0) increases with a. Using the folk theorem of Section 7 this yields yet another proof of Theorem 3.5.

9. E x p o n e n t i a l M o d e s o f t h e L i n e a r B o l t z m a n n E q u a t i o n W e shall derive now an integral equation for the exponential modes of the linear Boltzmann equation for the transport of mono-energetic neutrons in homogeneous domains. This turns out to be very similar to the integral equation obtained for the wave equation in the preceding section, het f =^f{ty x^ v) be the distribution function of the neutron density, where x is position in space, V velocity. Since the particles are assumed mono-energetic, the velocities all have the same absolute value, say 1. T h e transport equation asserts that

(9.1)

/^ + ,./^ + A / - J

fdv^O.

Here X is the removal cross section and c the combined scattering a n d fission cross section; X a n d c are constants. T h e equation asserts that the rate of chafige of f along a particle path is the sum of two terms: one, proportional to f^ is due to the removal of scattered particles from the beam, the other, proportional to § fdvy is due to particles scattered into the beam or created by fission; both processes are assumed to be isotropic. T h e boundary conditions are that no particles enter 0 from the outside; for convex 0 this means that (9.2)

f{t,

x,v)=0

at all boundary points for those velocities v which point into (9, Let us look at exponential solutions f = e~^^g{x^ v); we get (9.3)

^_fj,^

X)g +

v^g^-cw^O,

where w = w{x) stands for (9.4)

w{x) =

g{Xy v) dv.

68

778

p . D. LAX AND R. S. PHILLIPS

Abbreviating // — A by or we can write (9.3) as (y g + — g{^ — ^ ^y ^) = as

—(^^'

Multiplying by e"^^ and integrating with respect to s from 0 to d^ where d is the distance of ^ to the boundary in the direction v^ we get, using the boundary conditions, ra gi^Xj v) = c\ e^^w{x — s v) ds . Substituting this into the definition (9.4) of w we obtain (9.5)

w{x) = ^

I e^^w{x — s v) ds dv .

j\v\=\Jo

W e introduce instead of ^ and v a new variable of integration y: y =

X — s V .

Since dy = s^ ds dv and s = \x — y\^ (9.5) becomes 1 r e^\x-v\ - ^{^) = ] r. ^(y) c J \x — y\r P

dy

We can summarize this result as follows: The Boltzmann equation (9.1), (9.2) has an exponential mode e~^^g if and only if Ijc is an eigenvalue of H^{6)^ a = fi — X^ where H^{0) is the integral operator over 0 with kernel p(T\x—y\

\x

-y\^

This kernel is very similar to the kernel of the integral operator K^{(9) which appeared in the description of the exponential modes of the wave equation with potential c over 0. T h e same argument as was employed in Section 8, except that now we choose even instead of odd test functions Uj, shows that the w-th positive eigenvalue of H^ tends to oo as cr —> + 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
f

u^ dx c:^ \ (t^u^ dr dd .

\t\-c<\x\<]t\+c

Motivated by these considerations we study the following limits^ for ^ in R and 6 in S2 '. (A.3)

k^{s, 6) =

lim tu^{{t + s)d, t) . t-^±oo

T h e functions k_^ and k_ assigned in this way to a given solution u are called the outgoing and incoming translation representations^ respectively. ^ Limits of this kind were first considered by F. G. Friedländer, [2].

70

780

p. D. LAX AND R. S. PHILLIPS

THEOREM A.l (i) The limits (A.3) exist for a dense set of u. (ii) The representations defined by (A.3) are unitary; that is, the i>2(^^ ^^2) ^orm of k^ is equal to the energy norm of u:

H-,t)\\l

= ^ J[l^-«l' + l"*l'] ^^ = JJ l^±(^' ^)l' ^^ '^^'

and every kj^ in /.2('R^ »^2) represents some u of finite energy. (iii) Translation of u in t corresponds to translation of k in s: U{X, t +

T)

^^^ k^{s — Ty 6) .

T h e proof of this theorem can be m a d e to depend on a related result for the wave equation in free space. T o this end we associate to a given k in L^i^j Sc^ the function v{Xy t) defined as (A.4)

Xyt) t) =— = v{x,

\ k{x'd)

- t, d) dd .

LEMMA A.2. (a) The function v satisfies the wave equation in R 3 (b) For smooth k with compact support, the limit lim tVi{{t + s)d, t) = k{s, d) t->±co

exists.

(c) M-,t)\\% = ^j\k{s,d)\^ddds= u r (d) If k{s, d) = 0 for s K p, then v{x, ^) = 0 for |;v| < p + t; similarly, if k{s, 6) = 0 for s ^ —p, then v{x, t) = 0 for \x\ K p — t. (e) Conversely, if v is a solution of the wave equation in R 3 which has finite energy and which vanishes for |x| < p + ^5 then v can be represented in the form (A.4) with k[s, d) = 0 for s
71

DECAYING MODES FOR THE WAVE EQUATION

781

the truncated cone |.^| < p + t. Then, for ^ > 0, w does not interact with the scatterer and hence can be thought of as a solution of the wave equation in free space. According to p a r t (e) of L e m m a A,2y such a u can be represented for / > 0 in the form (A.4) with a k which is zero for i" < p. If this k turns out to be smooth and has compact support, then according to p a r t (b) of L e m m a A.2 the Hmit (A.3)^ exists. T h e set of all such nice k is dense in the Zg sense among all k with support in (/>, 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<^'Ydihxcd) functions and that ^{z) is an operatorvalued {L2,{S^ -> L^iS)) function.

72

782

p . D. LAX AND R. S. PHILLIPS

We wish to derive an explicit expression for 6^{z)] (A. 7) In order to compute ^{z) (A.8)

^(z)

by definition^

={e'''S{r)dr,

it is convenient to determine how S acts on k_{s) = e-'^'n,

n

in

Z2(6'2);

for purposes of exposition we overlook the fact that such k_ are not square integr a t e . I t follows from (A.5) and (A.7) t h a t (A.9)

k^{s) = e-^''y{z)n

.

T o find an explicit relation between k_ and k in (A.8) and (A.9), we examine the solution u which they both represent. First of all, since both k_ and k, are exponential functions of s^ it follows from part (iii) of Theorem A.l that u also depends exponentially on t: (A.IO)

u{x,t)

=e''^w{x)

.

Clearly, w is a solution of the reduced wave equation (A.ll)

z^W'\'Aw

= 0

in

^,

which satisfies the boundary condition (A. 12)

M; = 0

on

the

obstacle .

O n e way of constructing solutions of this type is to solve ( A . l l ) for solutions w_^ and w_ which are outgoing and incoming, respectively, in the sense of Sommerfeld and which are both equal to some given smooth function on the obstacle (see T h e o r e m 4.2 of Chapter V in [4]). T h e n , (A. 13)

w{x) = w^{x) — w_{x)

satisfies both ( A . l l ) and (A. 12); moreover, for large \x\,

and e^^\^\

[ x\

73

/ 1 \

DECAYING MODES FOR THE W A V E EQUATION

783

Next we evaluate the limits (A.3); it follows from (A.13) and (A.14) that these limits exist in some mean, say Cesaro, sense ^ : k^{s, d) =C

- lim tu^{{t + s)d, t)

= C - lim izte''^w{{t + s)e)

(A.15X i{t-\-s)z

^i{t-}-s)z

C — lim izte^^^

L ^ +^

"^

t +s

Similarly we obtain, noting that for t large negative \{t -\- s)d\ = —s — t, (A.15)_

k_{s, 6) = ize-'''
.

Comparing this with (A.8) a n d (A.9) we see that

(A.16)

3^{z) : cp_{-d)->
In what follows we need an expression for (p_ and 97^ directly in terms of w; as above this can be obtained from (A. 13) and (A. 14): (p^{d) =C

-lim

re'^'w{rd) ,

r—>00

(A. 17)
= e''"^'"^ .

Clearly, e{x) is a solution of the reduced wave equation; equally clearly it does not vanish on the obstacle nor at infinity. Ignoring these defects for the moment, we evaluate the limits (A. 17) and obtain after a brief calculation 27T

C ~ lim re'''e{rd, co; z) = — ^_«,(ö) , ••—•00

^^

(A.18) 277

C - lim re-''''e[rd, co; z) = — d^{d) ; here d^ denotes the (3-function at the point co. T h e limits (A.18) exist in the topology of distributions in 6; it is reassuring to note that we can recover ^ The limits do not exist in the ordinary sense because the solution u in question has infinite energy.

74

784

p . D. LAX AND R. S. PHILLIPS

e'^^^e{x^ co; z) from (A.4) if we set k{s^ 6) = 27Te~^^^ö_^ , as required by (A. 15), (A. 17) and (A. 18). Next let v{x, co; z) denote that solution of the reduced wave equation (A.l 1) in the exterior domain which satisfies the boundary condition (A. 19)

v{x) = ^^'^^-^

on

the

obstacle,

and which is outgoing at infinity, i.e., has the asymptotic form (A.20)

v{rd, co;z) =-—

k'^{oy, Ö; z) + 0 ( ^ - j .

As above, this boundary value problem has a unique solution. T h e function (A.21)

w{x) = e{x) -

v{x)

is then a solution of the reduced wave equation in the exterior domain which is zero on the boundary of the obstacle. From (A. 18) and (A.20) we get the following values for (p__ and 99^ associated with the function w defined in (A.21): V-{Q) = - ^ J ö ) , zz

^^(0) = -

iz

ö _ J 0 ) + /:-(co, 0; z) .

Hence, using the description (A. 16) of ^ ( z ) , we have ^{^)

• ö-Jd)

-> 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]{
dd = a{(o) + ^

J^*''(a>, - 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)

<

f
dn

at all points interior to B. Let V be the fundamental eigenfunction of — A over some domain D contained in a ball B of radius p, subject to the boundary condition (1.3); denote by A the lowest eigenvalue. Since Z;Q > 0 in B^ we can write v as V

Vc^W

=

.

Substituting this into the eigenvalue equation for v^ and using the eigenvalue equation which v^ satisfies^ we get the following equation for w:

(1.10)

A«; + 2 — " • Vu; + (A - yl«)«; = 0 .

O n the boundary of D we have

(1.11)

dwV

// II dv, öy„

\

än

\vQOn

/

T o prove that A ^ A^ we assume the contrary; then solutions w of equation (1.10) satisfy the m a x i m u m principle. Suppose w takes on positive values; it

82

ON THE SCATTERING FREQUENCIES OF THE LAPLACE OPERATOR

89

assumes its maximum at some point y on the boundary of D, At y^ relation (1.11) holds; using (1.9) and the fact that w{y) > 0 we deduce from (1.11) that dw —

on

+ (-^0+

K)W<0.

By (1.4), KQ :^ K; therefore we see that, at y, dw/dn
Frequencies

An operator-valued inner factor 0 ( { ) is an analytic function of ^ defined for J^<^M ^ ^ 0^ whose values are operators mapping some Hilbert space into itself and which has the following properties: (i) | 0 ( O I ^ 1

for

J^^ ^ < 0,

(ii) O(^) is unitary for ^ real . LEMMA 2.1. (2.2)

Let

^{Q

be an operator-valued inner factor

and suppose that

\^-\U)\^c-^

for some t,^ = ^^ -\- irj^ , T^Q < 0. Then 0 ( Q can have no vectors for any ^ in the circle of radius 2^ |?/ol (^ "~* c'^)~^ ao,>?o(I + ^ ^ ) ( 1 -c')-')Proof: Suppose that 0 ( ^ i ) n = 0^ J W ^ < 0^ and follows from (2.1i) that, for arbitrary m of norm one,

nontrivial null about the point

choose [^z] = 1.

It

fa) ^ {^{Qn, m) is holomorphic and of absolute value at most equal to 1 in the lower half-plane; in addition, / ( ^ i ) = 0. Hence, / can be factored as

/a) = ^-=4Vi(o, where f^ is again holomorphic in the lower half-plane. Moreover, since (^ — ^i)(^ — C^i)""^ is of absolute value one for real ^, the m a x i m u m principle

83

90

PETER D. LAX AND RALPH S. PHILLIPS

implies that i / i ( O I ^ 1 for all ^ in the lower half-plane.

for

Consequently,

Jm ^ < 0 .

Since this holds for all m^ we see that

for I b

J^m i < 0 .

4l

According to the hypothesis (2.2), ^0-^1

c^\mo)n\^

^0-^1

and it follows from.the extreme members of this inequality that ^^ lies outside the circle of radius 2^ |?yol (1 ~ <^^)~^ about the point (%o?^o(^ + ^^)(1 — c'^)"^)^ as asserted. COROLLARY 2.2. Suppose that the inner factor O(^) is holomorphic for J ^ ^ ^ and that for some real y^^ (2.3)

|(D'(;^„)| =

0

a.

Then 0 ( Q is one-to-one for all t, in the circle of radius a~^ about the point (Xo y —OL~-^). Proof:

Expanding O(^) about the point [^ =: ^o ^ ^^ ^^^ ^^^^

^(%o + ^"^o) = ^iXo) + iVo^'(Xo)

+ 0{r]l)

for

T?« < 0 .

Since 0(xo) is unitary by (2.1), we have ^*(Zo)^(Zo + iVo) = / + ^•^o^*(%o)^'(%o) + 0{rjl) , and making use of (2.3) we obtain \^-\Xo

+ iVo)\ = l[<E>*(;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 ^<^n I,
^v + t,^v = 0

in

G,

subject to the boundary condition that v -f exp { — it,x • co) has zero b o u n d a r y data on 5G; the boundary data can be of Dirichlet, N e u m a n n or R o b i n type. T h e solution v is required to vanish at 00; this implies that v tends to zero exponentially as |;^| —> 00^ and has there the asymptotic behavior

(2.8) T h e function coefficient.

v{rd;oy,Q =

'%(ö,co;0 + o(-

k appearing in this formula is defined to be the transmission

85

92

PETER D. LAX AND RALPH S. PHILLIPS

It follows from (2.4) that ^ is unitary on the real axis; therefore, it can be continued into the upper half-plane by the Schwarz reflection principle. This shows that the poles of ^ ( ^ ) in the upper half-plane are reflections across the real axis of those points ^ in the lower half-plane where ^{Q is not invertible. Since 6^ is the sum of the identity and a (compact) integral operator, ^{Q is noninvertible if and only if it has a nontrivial nullvector. W e shall apply Corollary 2.2 to the inner factor appearing in (2.4). Differentiating (2.4) with respect to ^ and setting ^ = 0, we get (2.9)

a)'(O) = ^ ' ( 0 ) - 2 z > ^ ( 0 ) .

Setting ^ = 0 in (2.5) we see that (2.10) LEMMA 2.3.

6^{0) = I. For the scattering matrix

(2.11)

^

^ ^ ( 0 ) = icP ,

where c is a number satisfying (2.12)

0 ^c^2p

,

and P is the orthogonal projection of L2{S2) onto the constant functions. Proof: Differentiating (2.5) with respect to ^^ using (2.6) a n d setting ^ = 0^ we see that 6^^{0) is an integral operator with kernel —{i/27T)k{ — d, w^ 0), defined by (2.8). T h e boundary condition for v when ^ = 0 is independent of the parameter CJO; therefore so is k{d^ co; 0). T h e function v defined by (2.7) is harmonic a n d hence asymptotic for \x\ large,

(2.13)

k v{x) c:::^ — ^ \x\

^o = const, ;

this shows that k{dy (o; 0) is independent of 6 as well as a>. T h u s the kernel of the integral operator ^' [f] is a constant so that ^' [v) is a constant times the orthogonal projection P onto the constants, as asserted in (2.11). T h e value of the constant c in (2.11) is (2.14)

c=-2k,,

where k^ is the constant appearing in (2.13).

86

ON THE SCATTERING FREQ^UENCIES OF THE LAPLACE OPERATOR

93

Let US consider the Robin boundary condition: (2.15)

— V + KV + K = 0 dn

on

dG,

where K is some positive function defined on the boundary. (2.16)

- 1 ^ v{x) ^ 0

in

We claim that

G.

Proof: Since v tends to zero at oo if it has negative values^ it will, by the m i n i m u m principle, assume its m i n i m u m value at some boundary point y. At this point the outward normal derivative of v is at most 0, so it follows from (2.15) that v{y) ^ — 1. Using the maximum principle we deduce similarly that V cannot assume a positive value; this completes the proof of (2.16). Letting AC -> 00, respectively —> 0, we conclude that (2.16) holds also in the limiting case of the Dirichlet, respectively N e u m a n n , boundary conditions. It follows from (2.16) that the harmonic function

v{x) + -^

\x\

is non-negative on the sphere \x\ = p; therefore by the m i n i m u m principle the same is true for \x\ > p. I n particular, it follows that ^o + P = 0? where k^ is the constant in (2.13). It follows likewise from (2.16) that k^ ^ 0. This shows that c = —2A:Q lies between 0 and 2p, as asserted in (2.17) and completes the proof of L e m m a 2.3. Substituting (2.10) and (2.11) into (2.9), we get -iQ)'{0)

= cP - 2pI = {c - 2p)P - 2p[I - P] .

Since c lies between 0 a n d 2/9, we obtain (2.17)

l^'(0)| ^ 2 p .

THEOREM 2.4. The acoustic scattering matrix^ associated with an obstacle contained in a ball of radius p and subject to Dirichlet, Neumann or Robin boundary conditions, has no pole in the disk of radius 1/2/> about the point (0, i[2p) shown in Figure 1. Proof: W e apply Corollary 2.2 to the inner factor O appearing in the factorization (2.4) of 6^, Inequality (2.17) shows that (2.3) is satisfied for ^^j == 0 with a = 2 p ; hence, according to the corollary, O does not vanish in the indicated circle. It follows that ^{Q is invertible in the same circle, a n d

87

94

PETER D. LAX AND RALPH S. PHILLIPS

FIGURE 1

therefore ^ ( 0 is pole-free in the reflected image o f t h a t circle; this completes the proof of T h e o r e m 2.4. I n view of the relation between poles of ^ and scattering frequencies^ we have COROLLARY 2.5. In the exterior of an obstacle contained in a hall of radius p, all scattering frequencies t, of — IS. lie outside the disk described in Theorem 2.4. 3. A n A d d i t i o n a l F r e q u e n c y - F r e e

Region

In this section we show that there is an open set of the complex plane, containing the open interval {—n/p^ Wp)? which is free of scattering frequencies of — A for any domain whose complement is contained in |;c| ^ p, and on whose boundary the Dirichlet condition i; = 0 is imposed. A similar result holds for the Robin boundary condition

(3.1)

dv + Kv = 0 dn

on

except that now the excluded interval is

( - - • - ) >

\

P p/

dG ,

K > /Co > 0 ,

ON THE SCATTERING FREQUENCIES OF THE LAPLACE OPERATOR

95

where v is the smallest positive solution of (3.2)

tan v

1

P^o

W e recall that the scattering frequency functions v are defined as: (i) Az; + ^2 y = 0 (3.3)

(ii) V = 0

on

^, y<^jt ^ > 0, and scattering eigen-

in

G,

dG

(Dirichlet condition) ,

(iii) V satisfies the radiation condition (1.2) • This last property is equivalent with the following: for any surface S containing 0 but not containing x^

(3.4)

.r;W=£{.( ^ = 1 lv{y)?^i^^-e{x-y)^^dS,

where e is the fundamental solution

e{x, Q =

47T\X\

O u r proof relies on the energy identity, derived by multiplying (3.3i) by v, a n d integrating over the truncated domain Gjj defined as Gj, = Gn {\x\ < R} . After one integration by parts we get

(3.5)

f {\vJ'-C'\vndx

=(

JGR

v^dS.

J\X\^R

on

T h e imaginary part of the above relation is

(3.6)

(r - D f \v? dx = f

( 4 ' - 4 1 dS.

J \x\=R \ on

JGB

Next we introduce the n o r m

(3.7)

ll.f = f \v{x)\'Ux JGio

89

on)

96

PETER D. LAX AND RALPH S. PHILLIPS

and normalize all scattering eigenfunctions by ||z;|| = 1. We denote by V^ the set [v] of all scattering eigenfunctions thus normalized, corresponding to all smooth obstacles contained in the ball {\x\ < p) a n d to any ^ in some compact set Z of the complex plane. LEMMA 3.1.

The set F ^ is contained in a compact set in the sense of the norm

- [

\vrR=

J

\V{X)\UX,

p<\x\
R arbitrary. Proof:

Every

v in

V^

satisfies an elliptic equation (3.3) for

Because of the imposed normalization,

I

\v{x)\^ dx '^ \\

\x\ > p.

elliptic estimates

p<|a;|<2p

applied in the shell p < |A:| < 2p show that all v{x) and v^{x) are uniformly bounded on a sphere {|^| = d]^ p d}: (3.8)

\d:v{x)\^c{K,n)

for all V in F ^ , all x in K and all ^ = 0, 1, • • • . In order to estimate the right side of (3.5), we apply (3.8) for n = 0, 1 and take K as {\x\ = R}^ i? > 2 p ; we see that for all v in F ^ , (3.9)

f

\vJ'dx^c{R),

JGB

Since Gj^ includes the set p < |;^| < i?, L e m m a 3.1 follows from Rellich's compactness theorem. Let {Gj} be a sequence of domains whose complements are contained inside the p-sphere, ^y a sequence of scattering frequencies in Z for A over Gj which tend to a real value x- Let v^ denote the corresponding scattering eigenfunctions, normalized by (3.7). According to L e m m a 3.1 w^e can select a subsequence which converges in L^ locally outside the sphere \x\ = p. LEMMA 3.2. Denote by u the local L^ limit in \x\ > 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 <. \x\y

(3.11)

v{x) =

\v {y)

j\v\=d

de{x — jv)

dn

L

, X dv e[x - y) — dSy ; on

(iv) for every i? > 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<\x\
the theorem of Arzela-Ascoli we can select a subsequence such that Vj a n d /\v^ converge uniformly inside any ball. T h e limit v satisfies the limiting equation (3.10). Relation (3.11), where e is the fundamental solution, follows by taking the limit of the analogous integral relation (3.4) satisfied by each v^. Likewise, we see that (3.12) is the limit of the analogous relation (3.6) for each Vj if we observe that the leftside of (3.6) tends to zero when ^^ approaches a real value X; here we make use of the uniform boundedness of J |z^>,|^ dx over G^ . THEOREM 3.3. There is an open set of the complex t^-plane containing the interval {—TTfpyTTJp) which is free of scattering frequencies of —A vuith Dirichlet boundary conditions for all domains exterior to obstacles contained inside the ball {\x\ < />). 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<\x\
Letting J? —> oo, we get

1. 2p<\x\

so that V is constant; because of (3.15), that constant is zero. Again we see that 27 = 0 for |^| > p. Having shown that v ^ 0 for \x\ > 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\
Since we saw earlier that, as j —> 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<5 < |;c| < 2p, imply that

(3.19)

j

\vj\^ dx —>- 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

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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.

103

622

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 <» in the direction ß. Similarly we define a horosphere through a point w in non-Euclidean space as a limit of spheres through w whose centers tend to the point ß at 00. Since geodesies from the (non-Euclidean) center of a sphere are perpendicular to the sphere, it follows that all geodesies from ß are perpendicular to all horospheres through ß. From this we deduce that horospheres through ß are spheres tangent to the boundary of TT at ß; see Figure 1.1 for a geometric proof, keeping in mind that Euclidean and non-Euclidean angles are equal, since the two metrics are conformaL In model (i) the point at infinity on the ß-plane is an exceptional and an exceptionally simple point at infinity for II; we denote it by ß«- The horospheres through ß« are the planes z = const., and the geodesic lines through ßoo are parallel to the 2-axis.

105

624

p. D. LAX AND R. S. PHILLIPS

geodesic horosphere

ß Figure 1. Geometric proof.

Since horospheres are defined in terms of the metric, it is clear that non-Euclidean motion carries one horosphere into another. It is often convenient to specialize to horospheres through ßoo. In the Euclidean case a convenient parametrization of planes is their normal ß and their distance s from the origin. Similarly we parametrize horospheres by the point ß at oo through which they pass, and their distance s from a reference point which in model (ii) is taken as the origin and in model (i) as (0,0,1). We denote this horosphere as 2(s, ß). Note that, in contrast to the Euclidean case, a horosphere divides II into an inside and outside. For a point w in Euclidean space and unit vector ß the scalar product w • ß can be interpreted geometrically, aside from sign, as the distance of the origin from the plane through w with normal ß. The analogous concept in non-Euclidean space is: (w, ß>= ±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)


This is obvious for r which keeps / fixed. In general, we factor r = /uur, where a keeps ß fixed and /m keeps j fixed, and verify (1.13) for cr in place of r; this can be done by inspection in model (i) if ß is chosen as ß«,It is easy to see in model (i) that (1.14)

<w,ßoo) = logz.

106

TRANSLATION REPRESENTATION FOR THE WAVE EQUATION

625

since non-Euclidean distance along the geodesies through ß«, are the logarithms of the Euclidean distance. For arbitrary ß we deduce, using (1.13), that in model (i) 1 + lßP (1.15)

<W,ß) = l 0 gIw-ßP^' :'"" -

where 11 denotes Euclidean distance. The analogous formula in model (ii) is (1.16)

<w,ß) = log

1-lwP l - 2 w ß + |w|

where 11 and • denote the Euclidean distance and scalar product. In model (ii) the points at infinity are represented as the unit sphere; we endow B with the standard surface measure of the unit sphere. The mapping relating model (ii) to model (i) carries this into the area element

The element dß is invariant under all motions r that keep / fixed; under arbitrary r, (1.18)

dirß) = exp {2
This may be verified with the aid of (1.13). 2. The Radon Transform in Euclidean Space We start with the Fourier transform of a function /(w) in R^:

(2.1)

^ ^ ^ ^ ^ ^ ( 2 ^ |e"^"''/(w) dw.

It is convenient to express | as Aß, where A = |^| and ß is a unit vector; then we break up the integration in (2.1) by first integrating over the planes w • ß = s and then integrating with respect to 5. We get

(2.2)

^^'^^^ ^ ( 2 ^ j^'^^'fo^ ß) ds,

107

626

V^.D, LAX AND R. S. PHILLIPS

where (2.3)

&,P)-f

fiw)ds;

f is called the Radon transform of /. Thus / is the Fourier transform of / with respect to s. From this we deduce that (2.4)

I f(\ß)e^\^

d\ = j ^ ^ d^fis, ß).

Fourier inversion of (2.1) gives (2.1)'

fM^-^^lme'-'d^-

We set ^ = Aß, d^ = k^ dXdß; extending the range of A integration from IR+ to IR and using (2.4) with 5 = w • ß, we get

(2.5)

^^"^^" 8 ^ 1 ^^^f^"" ^ß^ß)dß^

the inversion formula for the Radon transform. Note that the integration in (2.5) takes place over planes that pass through w. It follows from (2.2) that A/(Aß) is the Fourier transform in s of djis, ß); thus, by PlanchereFs formula, 477^1^ A^|/(Aß)PdA = | | a J ( s , ß ) P d s . Substituting this into the Plancherel relation between / and /: 1^ j°^l/(Aß)PA^dAdß = ||/(w)Pdw, we get (2.6)

g ^ I \dM ßt äsdß^j

108

\f(w)\^ dw,

TRANSLATION REPRESENTATION FOR THE WAVE EQUATION

627

the Plancherel relation for the Radon transform. From (2.6) we deduce

(2.6y

^

j djdjds dß = I fgdw.

We state now the main properties of the Radon transform. THEOREM

2.1. (i) For every /, / is even:

(2.7)

/ ( - s , - ß ) = /(s,ß).

(ii) The Plancherel relation (2.6) holds. (iii) Every even function k(5, ß) for which d^k is in L2 is the Radon transform of some f in L2. (iv) For any Euclidean motion r define (2.8)

r ( w ) = /(T-^w).

Then (2.9)

r(5,ß) = /(s-T(0)-ß,r-^ß).

(v) For fin Cj, (2.10)

dj=ß

dJ,

d^ = gradient.

A/= d^f,

A = Laplace operator.

(vi) For every f in C^, (2.11)

(vii) The inverse of the Radon transform is described by (2.5). Proof: Property (i) follows directly from (2.3). We have already proved (ii) and (vii). Property (iv) is obvious for translations and rotations; therefore by composition it holds for all r. To prove (iii) we observe from relation (2.2) that every even function in Co is the Radon transform of some square integrable function /; from this and (ii) we deduce (iii). Property (v) follows from (2.9) applied to translations, and (vi) follows from (v). We turn now to the wave equation (2.12)

dfu-Au = 0.

109

628

p. D. LAX AND R. S. PHILLIPS

It is easy to prove that the energy, denoted by |M|| and defined by |w||=kf~(i^,Au) (2.13)

= |kP+KPdw,

is independent of time. We denote the Cauchy data of u at time t by Fit): F(t) =

{u(t\u,(t)}.

Conservation of energy shows that F{t) is determined by F(0); we denote the operator relating F(0) to F(t) by U(t). Conservation of energy also shows that U(t) is unitary in the energy norm. We denote by ^ the completion in the energy norm of C^ data. It is known (cf. [6], page 95) that the first component of every F in ^ can be identified with an element of V^. The wave equation can be solved with the aid of the Radon transform. Taking the transform of (2.12) and using (2.11) we get

which can be factored as (2.14)

id, + dj(d,ü-d,ü)

= 0.

This shows that dtü—dsü is a function of s —t alone, and suggests the DEFINITION.

Let F = {/i,/2} be data in M\ the operator R^ on M is defined

by (2.15)^

R^F= cidth-dj^),

c = ->(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|0. (iv) u(w, 0 = 0 for | w | < a ~ | t | if and only if THEOREM

(2.21)

k(s,ß) = ZPm(s)Y^(ß)

for

|s|
where Y^ is a spherical harmonic of degree m, and Pmis) a polynomial of degree less than m. Proof: The "if" parts of Theorem 2.4 are obviously true. Let <^(0 be a C^ test function; multiply (2.20) by {t) and integrate; interchanging the order of integration and introducing w - ß — f = s as a new variable, we get (2.22)

[ it)u{w, 0 dt = \c\ \\iw^ ß - s ) k ( s , ß) ds dß.

In case (i) the left side is 0 for all w. Differentiating the right side of (2.22) m

112

TRANSLATION REPRESENTATION FOR THE WAVE EQUATION

631

times with respect to w and setting w = 0 gives

(2.23)

0 = 1 1 ß^(^<^(-s))k(s, ß) ds dß,

where N is any multi-index of order |N| = n. Fundtions of the form d^y 4> G C^, are dense in 1^(1^), since their orthogonal complement, which would consist of L2 polynomials, is 0. So it follows from (2.23) that k is orthogonal to all functions of the form 4^is)Yn, i^ in L2. Since these functions span L2, k = 0 as asserted. In case (ii) the left side is equal to 0 for jw|<S if the support of is confined to t>S>0. Then (2.23) holds for all such <^. Since functions of the form d^(s), eC^, <^(t) = 0 for t>5, are dense in L2(IR_), it follows as before that k(5, ß) = 0 for s < 0 . This proves (ii); the proof of (iii) is analogous. In case (iv), (2.23) holds for all <(> supported in [s| do not span I^i—a, a); their orthogonal complement consists precisely of all polynomials of order less than m. It foUows then from (2.23) that |ß^fc(s,ß)dß are polynomials of degree less than n = |iV|. This completes the proof of Theorem 2.4. A solution M(W, 0 of the wave equation is called outgoing if it vanishes in the forward cone |w|
Finally we turn to the question of characterizing the Radon transform of functions that vanish outside a ball. The answer is contained in THEOREM 2.6. Let f be an L2 function which is equal to 0 for |w|>r. Denote f by h. Then (i) h(s,ß) = 0/or |5|>r, (ii) for every non-negative integer n.

(2.24)

I s^'his, ß) ds

is a polynomial in ß of degree at most n.

113

632

p. D. LAX AND R. S. PHILLIPS

Converselyy if h is an even function such that d^h is in L2 and if h satisfies (i) and (ii), then h—f where f is in L2 and equal to 0 for |w|>r. Proof: We have already shown in the course of proving Theorem 2.3 that if /(w) = 0 for |w|>r, then /(s, ß) = 0 for |s|>r. To derive property (ü) consider (2.25)

I (w • ßTf{ w) dw,

clearly a polynomial in ß of degree at most n. Performing the integration first on the plane w • ß = s and then integrating with respect to s, we see that (2.25) = (2.24). To prove the converse, we assume that / satisfies (i) and (ii) and deduce from this that /(w) = 0 for w > r by showing that (/,g) = 0 for all g that vanish for |w|
(/,g) = c 2 | d i a j d s d ß .

Define M(W, 0 = c^ jdsg(w •ß-t,ß)dß

;

clearly u is a solution of the wave equation, and since dsgis,ß) is odd, M(W,0) = 0; by (2.5), ii,(>v,0) = -g(w). Since g(w) = 0 for |w| < r + 8 , it foUows by the classical uniqueness theorem for solutions of the wave equation that M(W, 0 = 0 for | w | < r + e - | t | . Then part (iv) of Theorem 2.4 is applicable, with k = c^d.g. In view of (2.21), c'a.g = Sp.(s)Y„.(ß), p^ being a polynomial in s of degree less than m, Y^ a spherical harmonic of order m. Substituting this into (2.26) and using (2.24) we get (f,g) = =

l\dJp^{s)Y^{ß)dsdß I.\qm(ß)Y„(ß)dß.

where q^(ß) is a polynomial of degree less than m; therefore, Jq^Y"* dß = 0 and Theorem 2.6 is proved.

114

TRANSLATION REPRESENTATION FOR THE WAVE EQUATION

633

Theorem 2.6 is the Paley-Wiener theorem for the Radon transform; the above proof is from [7]. A slightly different version of the result is contained in [9]; another version is given in [2]. An entirely analogous result holds for data; see page 113 of [6]: 2-7. Let F={fi, /2} be data withfiniteenergy, and suppose that (i) L/i = 0,/2 = 0/or |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 A<0. For verification we refer to [3]. Denote by T the operator defined in (3.1) linking / to /, and by S the inverse operator defimed in (3.7). Since the kernel of S is the conjugate transpose of that of T, S and T are adjoint to each other: (3-8)

(T/,g) = (/,Sg)

for all C^ f and g. Setting g = / we see using (3.7) that Sg'^Sf^f. Substituting this into (3.8) we get

(3.9)

l/T = l/P,

116

TRANSLATION REPRESENTATION FOR THE WAVE EQUATION

635

where the norm on the right is L2(n), on the left £2(1^+ x B) with respect to the measure k^dkdß; (3.9) is the Plancherel relation. If in the definition (3,1) of / we first integrate over <w, ß) = s and then integrate with respect to s, we get (3.10)

/(A, ß) = ^ ^ ; ^ I exp {(1 - iA)s}/(s, ß) ds,

where (3.11)

fis,ß)=\

f{w)dS,

dS being the non-Euclidean surface element on the horosphere {Wyß) = s. The function / is called the Radon transform of /. Formula (3.10) shows that / is the Fourier transform of e^f with respect to 5; from (3.10) we deduce that (3.12)

f

/(A, ß)e^U^ dX = --yflhrdteiis,

ß) .

Substituting (3.12) into (3.7), after extending the A integration to all of R, we get fM = T^\ LIT

(3.13)

=-T^l

exp «w, ß» dUiis, ß)L=<w.ß> dß JB

exp{2(w,ß)}

x[/«w, ß), ß)+2a,/«w, ß), ß)+a^/«w, ß), ß)] dß. This is the inversion formula for the Radon transform. Note that integration is over all horospheres that pass through w. Just as in the Euclidean case, we deduce from (3.10) that irA/(A, ß) is the Fourier transform in s of d^e'fis, ß); thus, by Plancherel's formula,

TT^ I

A^1/(A, ß)P dA = j \d,ei(s, ß)P ds.

Integrating this with respect to dß and using Plancherel's relation between / and / given by (3.9), we get (3.14)

^\^|a,67Tdsdß

=\

117

l/(w)Pdw,

636

p. D. LAX AND R. S. PHILLIPS

the Plancherel relation for the Radon transform^ By polarization we obtain, from (3.14), (3.14)'

:^\\d^eid,e^ldsdß=^^fgdw.

We now state the main properties of the Radon transform. 3.1. All functions f in C^ijl) have the following properties: (i) The Plancherel relation (3.14) holds. (ii) For any motion r define

THEOREM

(3.15)

r(>v) = /(T~^w);

then (3.15)'

r U ß) = fis-H

ß), r-' ß) .

(iii) For any a > 0, (3.16)

e^ I f(a, ß) dß = e~^ j /(-a, ß) dß.

(iv) (3.17)

Lf^e-^dUi.

where L is defined in (3.5). (v) The inverse of the Radon transform is given by (3.13). Proof: We have already established (i) and (v). To prove (ii) we perform a change of variable w — rw' in (3.11): r(5, ß) = f

r(w) dS = f

=f

fiw')d^S.

fir-' w) d^S

Using (1.13), with T~^ß in place of ß, we have
So the domain of integration above is <w',T-^ß> = s-<77,ß>.

118

TRANSLATION REPRESENTATION FOR THE WAVE EQUATION

637

Substituting this in the above integral yields (3^15)', We note in particular that if we specialize r to be a rotation around /, that is to a T for which rj = /, then
r(^,ß) = /(5,T--^ß).

If f is rotationally symmetric, that is if f^=/, then we conclude that f{s, ß) = /(s, T~^ß) for all such T. Since any point of B can be moved to any other by a rotation about /, it follows that / is independent of ß. We begin the proof of (iii) by setting (3.19)

fa.(w)=Jr(w)dT, fa.(w)=jf

where the integration is over the rotation group keeping ; fixed. Qearly, /^^ is rotationally symmetric so that its Radon transform is independent of ß. Hence if we take the Radon transform of (3.19), making use of (3.18), we obtain (3.19)'

Lis)=\f(s,r-'ß)dT.

Obviously, (3.19r

j /(s, T-^ß) dr = I /(s, ß) dß.

Thus in order to prove (3.16) it suffices to show for a rotationally symmetric function / that (3.20)

ei{s)^e-i(--s).

To this end we consider two horospheres X+ = S(s, ß) and 2 - = S(-s, ß). Denote by A^{r) and A^{r) the siuiace area of that part of X^ and %^ whose distance to / does not exceed r. We claim that, for all r, (3.21)

A^{r)^e^AM'

Since / is rotationally symmetric around /, fM = g(r),

119

r = distance of / to w,

638

p. D. LAX AND R. S. PHILLIPS

and thus f(s)=\

/(w)dS = j g ( r ) d A 4 r ) ,

/(-s)=J

/(w)dS = | g ( r ) d A _ ( r ) .

From this and (3.21) we deduce (3.20). The proof of (3.21) is a calculation best performed in model (ii). It sufläces to verify that d 1 ^ ~ A^ = C — A^ I

dr

dr

we leave the calculation to the reader. To prove (iv) we apply (3.12) with Lf in place of /, and make use of (3.6): dU'Lf = - V W Lfe'^^A^ d\

It follows from this that^^Lf — d^e^f+a + bs. Since, for / with compact support, /, and likewise Lf^ have compact support in s (see the proof of Theorem 3.7), a = 6 = 0 and (3.17) follows. In the course of proving Theorem 3.1 we have also proved: COROLLARY 3.2. Let f be a C^ function in U with rotational symmetry^ that is /(TW) = /(w) when rj = j . Then /(s, ß) is independent of ß and (3.20) holds.

A more complete treatment of the Radon transform in the setting of model (iü) can be found in [1], Chapter V; see also [5]. It should be pointed out that condition (3.16) is the non-Euclidean analogue of the evenness of the Radon transform in the Euclidean case (see Theorem 2.1, (i)). Helgason [4], page 473, has shown that this property characterizes the range of the Radon transform. We turn now to the non-Euclidean wave equation: (3.22)

M«-LM = 0 ,

where L is defined in (3.5).

120

TRANSLATION REPRESENTATION FOR THE WAVE EQUATION

639

The following facts are furnished by the classical theory of hyperbolic equations: (i) The initial value problem for (3.22), M(W, 0 ) = /I(W) ,

w,(w, 0) = /2(w),

has, for arbitrary C^ functions / i and /2 defined in 11, a unique C* solution M(W, 0 in H X R . (ii) Signals propagate with speed at most 1, i.e., the value of M(WO, t) does not depend on values of /i and /2 at point w whose distance from WQ exceeds (iii) The non-Euclidean energy (3.23)

|u|| = |M|i+kP,

where (3.24)

|u|i = ~ ( u , L u ) ,

is independent of t Yet another property of solutions of (3.22), Huygens' principle, will be stated later as Theorem 3.7. We claim that \u\l is non-negative^ for u in Co (11). In fact, it follows from the definition of L and an integration by parts, with dw — dxdydzlz^, or du-ectly from (1.10) that

-{u,Lu)=

-

\uz^d-du

+ ü^\dw

Another integration by parts shows that for M in C^

so that we can write (3.24)'

\u\l=^z\vl+vl+vl)dy^,

^We shall show in a subsequent publication that, in contrast, energy over the fundamental domain with respect to some discrete subgroups can be negative.

121

640

p. D. LAX AND R. S. PHILLIPS

where (3.24)"

v=-u/z.

Even more is true. THEOREM

3.3. For any compact subset K of H there is a constant c^ such

that (3.25)

\u\j,^\

uUw^cMl

for all u in Cl (11). Proof: We define a new norm

|U|L,JK

by

I"IIK-|U1!.+|U||

and work in the function space obtained by completion of Co with respect to this norm. It follows easily from the Rellich compactness theorem that |u||: is compact with respect to |U|L,K- Therefore, |U|K can be written as (3.25)'

| U ! | = (U,TKU)^K,

where TK is a compact selfadjoint operator. We claim that all eigenvalues A of TK are less than 1. For if TKU

then by (3.25)', get

|M|K = A|U||,IC-

= Xu,

Substituting this into the definition of

|U|L,K

we

(l-A)|u|tK=l"ll. Since, by (3.24)', |u|i^O, it follows that A ^ l . K A were equal to 1, |U|L would be 0. To show that this cannot happen, we recall that |U|L is invariant under all motions. Therefore, we can apply (3.24)', (3.24)" to u'^ and obtain l"li=lw1i=J2:^0wt^^)'dw, where

122

TRANSLATION REPRESENTATION FOR THE WAVE EQUATION

641

Thus |M|L = 0 implies that t;^= const, and so w^ = const, z; but this can hold for all T only if M = 0. Since T^ is compact, its eigenvalues accumulate only at 0. Denote by A the largest of them; as shown above, A < 1 . Since TK is selfadjoint. \u\l =

iu,T^u)j:^^^\\ul JL,K=^\^\UK

Substituting |M|i,K=|w|i+klK gives (3.25), with CK = A / ( 1 - A ) . We shall denote by 5if the completion in the energy norm (3.23) of all eg(11) data. It follows from Theorem 3.3 that for all data F = {fi,/2} in 9if I/IIK^CK|F|H.

Let u be a Cl solution of (3.17). We denote by Fit) its data {M(0, Utit)} at time t. It follows from conservation of energy and positivity of energy that if F(0) = 0, F(0 = 0 for all t. We denote by 17(0 the operator relating F(0) to F(f); since the wave equation (3.22) is hyperbolic, there exist solutions with arbitrary initial data. Thus the I7(t) are unitary operators with respect to the energy norm, and they form a one-parameter group. We denote the generator of this group by A, i.e..

In view of (3.22),

-di) As in the Euclidean case, the non-Euclidean wave equation (3.22) can be solved by the Radon transform; taking the transform of (3.22) and using (3.17) we get

Defining m as e^w we can rewrite the above in the form

which can be factored as (3.26)

(d,±dJid,mTd,m) = 0.

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642

p. D. LAX AND R. S. PHILLIPS

It follows that

is a function of s^t alone. Denote the initial data of u by M(W,Ö) = / I ,

^(w,0) = /2.

We define the incoming and outgoing translation representersfe_and k+ of F = {/i,/2}by

where c = — (V^ir)"^. We denote by 1?± the operators (3.27)' THEOREM

.328)

R+F=fc+,

K_F=fc_.

3.4. (i) For euery C^ data F,

R+t/(OF=Ms-f,ß), R_t7(0F=fc_(s + r,ß),

where k^ are defined in (3.27), (3.27)'. (ii) For every F in C^,

(3.29)

1F1| = |1?,FP = |J?_FP,

where the norm on the right is the L2 norm with respect to the measure ds dß, as defined in (1.17). (iii) JR^. and R^ each map C^ onto a dense subset of L2(ß x B). Note. It follows that R^ and jR_ can be extended by completion to all of ^y and give the unitary translation representations of ^ for the group U(t). The relation of these representations is furnished by the scattering operator which was described in the introduction. Proof: Let u be the solution of (3.22) with initial data F; then (3.26) is satisfied. Applying d^ to (3.26) we obtain, in the notation (3.27), id,±h)R^F{t)^0

124

TRANSLATION REPRESENTATION FOR THE WAVE EQUATION

which shows that J R + F ( 0 is a function of s-f, F(0=l/(OF, (3.28) follows. We turn to (ii); by definition,

R-F(t)

643

of s + t Since

To prove (3.29) we have to show two things: (3.30)

1F|| = c^l j |d^e»AP+1 ke^/ap}

and (3.30)'

\dUiidsei2

= 0.

By definition (3.23), \F\l = -(fr,Lf,)

+ \fJi\

We shall verify (3.30) by showing separately that c ^ j ke72p = l/2p and c^ j |a^e7iP = -(/i,L/i) • The first equality is just the Plancherel relation (3.14) with /a in place of /. The second follows from the bilinear Plancherel relation (3,14)' upon setting f = fi', g = Lfil for, using (3.17), we get (A ,Wi) = c^\ a,(e7i) dsie'Qx) ds dß =

c^jdMi,)d^Aeii)dsdß

=

-c^^mei,rdsdß.

This completes the proof of (3.30). Relation (3.30)' is easily proved when both /i and /a have rotational symmetry. For then, by Corollary 3.2, djC'/i is an even, dje^/z is an odd function of s, and both are independent of ß. Next we consider the case that fi is rotationally symmetric but /2 is not. Then only /2 depends on ß. Recall the definition of /„„ in (3.19):

fa.(y^)=lr2( w) dr.

125

644

p. D. LAX AND R. S. PHIIXIPS

where the integration is over the rotation group keeping j fixed; it follows from (3.19)' and (3.19)" that

Lis) =

\f2(s,ß)dß.

We can therefore write (3.30)' as I I d^sie'h) a,(6^2) dßds = ^ a^(e7i) dAeia.)

ds

which is zero since both /^ and f^^ are rotationally symmetric. Next we take the case when /2 is arbitrary and fi has rotational symmetry around some point other than j ; i.e., we take /i = h \ where h is symmetric around / and r is arbitrary. By (3.15), P = h(s-
ß » d^ei^is, ß) ds dß .

Introducing t = s — {TJ,ß) as a new variable of integration gives I df exp{f+
dtdß.

We introduce 7 = T~^ß as a new variable of integration. Relation (1.23), with T""^ in place of r, yields d(T-^ß) = exp{2
+

{TUTy),Ty)dydt.

We use now (1.13), with W = T~^/, ß = 70 = 0; T7) =
From this relation and (3.15)' we get f2it +
126

TRANSLATION REPRESENTATION FOR THE WAVE EQUATION

645

where g = fl'"^. Therefore, the above integral can be written as

jdWhit)d,e'iit,y)dydt, an integral previously shown to be zero. To complete the proof of part (ii) of Theorem 3.4 we have to show that every /i can be approximated by linear combinations of functions symmetric about some point. A function h(z, w) which is rotationally symmetric around z is simply a function of the non-Euclidean distance from z: (3.31)

h(z,w) = g(dist.(w,z)).

Obviously h(z, w) is symmetric in z, w. Given any e>0 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=<^.3>
Now it follows from (3.16) that

is an even function of s; hence p'{s) is odd and therefore vanishes at s = 0: (3.38)

p\0) = ^d,e%Uodß

= 0.

Consequently, (3.37) holds for w = /. One could argue that all of our concepts are invariant under the non-Euclidean motions and that, under a different parametrization, (3.38) implies (3.37) for arbitrary w. We shall make this argument explicit. For given w let T denote the motion interchanging / and w: (3.39)

T = T~*,

rw^jy

Tj = w.

By (3.15)', (3.40)

flis, ß) = Us - (rj, ß), r-' ß);

by (1.13), (3.40)'

<w,ß)=-<w,Tß),

and by (1.18) (3.40)"

d(Tß) = exp {2
Replacing /z in (3.38) by fl and making use of the relations (3.39)-(3.40)", we get 0 = f d,ei2{s + <w, Tß), Tß)L=o exp {2(w, rß» dirß) .

130

TRANSLATION REPRESENTATION FOR THE WAVE EQUATION

649

Finally, making the substitutions rß -» 7

and

s -> 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,/) be any Co function; we form (3.44)

ü(w)= J M(W, 0<^(0 dt.

Substituting for u as defined in (3.41) we get, after the change of the variable s =<w, ß ) - r in the integration, that t;(w) = j I exp «w, ß)}k«w, ß ) - f, ß)it) dß dt (3.45) = I exp{<w, ß)}fc(s, ß)c^«w, ß)~s) ds dß.

132

TRANSLATION REPRESENTATION FOR THE WAVE EQUATION

651

This formula shows that v(w) is a C°° function. If w(w, 0 = 0, then t)(w) = 0, and in particular all its partial derivatives with respect to w at j are zero. We compute now these derivatives by differentiating (3.45) and setting w = /. Let N = (iVi,-• • ,N„) be a multi-index. Using the abbreviation dj^ = l l a 5 , ^e form (3.46)

a^t)L=,- = j a^[exp «w, ßMiiw, ß>- s)L=ife ds dß.

Clearly, (3.47)

a^[exp{<w, ßMiiw, ß)-s)L=,- = Q^{d,, ß),

where Q^ is a differential operator of order \N\ whose coefficients depend on ßThe leading coefficient of Qj^ is easily computed; it is 2'^'ß^, where we have used the fact that a<w, ß)/3wi|«=, =2ßj, a fact easily deduced from formula (1.16) for <w, ß). So we set (3.48)

QN(s,ß) = ß^PN.

We claim that all the coefficients of F^ are independent of ß. To see this we set (3.48) into (3.47) and (3.47) into (3.46); we get (3.49)

a^t)(w)L=, = j kis, ß)ß''Pr,(ds, ß)4> dsdß.

We take now any rotation around /, i.e., a motion for which pj = j. According to (1.13), with p for T and p~*ß for ß, (3.50)

(pw,ß) =

(w,p-'ß),

since ((w, y)-s)k{s, py) ds dy. Analogously to (3.49) we get after differentiation (3.51)

3Xpw)L=,. = j k{s, py)7^PN(a„ y) ds dy.

133

652

p. D. LAX AND R S. PHILLIPS

Next we compute d^v{pw) by using the chain rule. We note that in model (ii), with / = 0, non-Euclidean rotations p are Euclidean rotations: (P^)k = Z Pki'^i • Consequently, r)

I

and

Using formula (3.49) for derivatives of order \N\ of v{w) at w = j we get a^i;(pw)L=.,. = I fcU ß)(p^ßrP^,(d,, ß)cf> 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 <^ in Co . Denoting the adjoint of P by P*, it follows that Pnids)K = 0All solutions of this ordinary differential equation are of the form n

(3.52)

fe„(s)=

S fc^.„ eXp {XrnS} , m= l

134

TRANSLATION REPRESENTATION FOR THE WAVE EQUATION

653

where P^(A„) = 0; since P„ is of order n, there are n roots Aj, • • •, A„. We claim that A^ = —m; to see this we substitute (3.53)

M 5 , ß ) = e-"*^y„(ß),

m = l,--.,n,

into (3.41)+: uiw, 0 = J exp «w, ß » exp {(-<w, ß) + Om}y„(ß) dß = 6"-^ exp{<w, ß)(l-m)}y„(ß) dß

where in the last step we have used formula (1.16) for <w, ß). Qearly for m = 1, • • •, n the first factor is a polynomial in ß of degree less than n, and thus is orthogonal to Y^. This proves that, for k of the form (3.53), u(w, t) = 0. Since we have shown that all fc for which this is true are of the form (3.52), it follows that the values — 1, • • •, - n belong to the set {A^}. Since this set consists of at most n numbers, it follows that A^ = — m, m = 1, • • •, n, as asserted in part (i) of Theorem 3.8. The other parts of Theorem 3.8 are proved in the same way. To show the sufficiency of condition (3.43) we note that the value of u(w, t) as given by (3.41)+ for (w, t) in the forward cone only depends on values of k^{s, ß) on IR_, while the values of u in the backward cone only depend on the values of k^ on U+; the value of u for dist. (w, j ) < a —|t| only depends on values of k^ in (-a, a). To show the necessity of condition (3.43) we argue as before, except that in case (ii) we restrict the support of to [R+, in case (iii) to IR_, and in case (iv) to (-a, a). This completes the proof. We denote as before by U{t) the solution operator of the wave equation (3.22). We define the outgoing and incoming subspace 2l+ and 2)^ of ^ as follows: 2)^ consists of all data F in ^ for which Uit)F vanishes in the forward cone: dist. (w,y)
135

654

p. D. LAX AND R. S. PHILLIPS

^a consists of all data in ^ that are equal to 0 inside the ball: dist. ( w , ] ) < a . 3.9. (i) R^3)^ = L2(ß+), the space of L^ functions k+{s,ß)^Q s<0. (ii) R+3}^ consists of all functions k+ in L2 that satisfy (3.43) for s>0. (iii) R+^a consists of all functions k^ in JL2([R) which satisfy condition (3.43) for -a<s
for

jR_@_ = L 2 ( R J .

(ii)_ R^Q}^ consists of all functions k+ in L2(H) which satisfy (3.43) for s>0. (iii)^ R-^a consists of all functions fc+ in L2(IR) which satisfy condition (3.43) for -a<s
3)1 = U(a)3}^,

21 = U{-a)2.

.

Since JR+ and jR_ are translation representations, see part (i) of Theorem 3.4, part (t)±, (ii)^ of Theorem 3.9 yield

136

TRANSLATION REPRESENTATION FOR THE WAVE EQUATION

655

COROLLARY 3.10. (i)^ R+3)^ consists of all l^ functions k+is, ß) that are zero for s—a. Similarly, {i)l JR_2lf consists of all L^ functions k^ that are zero for s - a .

LEMMA 3.11.

The closure of 21 +3)1 is ^«.

Proof: It follows from the definition of 3)^ that every F in U(a)3}+ is zero in the ball dist. (w, j)
k(s,ß)=

X

Kn^ne-'^Y^iß)

for

-a<s
We define fc+ and fc_ as follows: k_(5, ß) = < ^ 1 Z

fem,ne~^Yn(ß)

'^''•'Hli^ß)-Sfe.,ne~^yn(ß)

for a < 5 , for

s
for a < 5 .

It follows from (3.55) and part (ii)^ of Corollary 3.10 that k_<^R+21, and from (i)+ that fc+ciR+SJ^. Since, clearly k_ +fe+= fc and k = R+f^^, we have accomplished the decomposition of p^^ into a sum of an element of S»" and S)+. Since \f-p^\-*0 as N-^<», the lemma is proved. The following assertions are equivalent for data G in 3^: Gl^„, G121 and G±2-, Lgi = 0 and g2 = 0 outside the ball Bij,a) of non-Euclidean radius a

THEOREM3.12.

(i) (ii) (iii) about j , (iv)

8f c^gi = 0 = a,e^g2 for s > a.

Proof: Lemma 3.11 implies that the orthogonal complement of '^a is the intersection of the orthogonal complements of ^" and 3)1. Both are easily

137

656

p. D. LAX AND R. S. PHILLIPS

determined. Suppose Gl^a^ that is, (G,F)E = 0

for

aU

F

in

^„.

Then, by definition (3.23), (3.24) of the energy inner product, we have (3.56)

(G, F)E = -(Lgi, /i) + (g2, h)

for all / i , /2 in CQ. Gl'^a implies that this is zero for all F in Co which vanish in the ball B(/, a). Since such F are dense in ^a, we conclude that G l ^ a if and only if (iii) holds. On the other hand, according to parts (i)+ and (i)" of Corollary 3.10, G is orthogonal to Si+ (respectively, 2-) if and only if R+G = 0 (R_G = 0) for s>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 s<—a, nor is g required to satisfy (3.57); rather, these properties are consequences of the fact that d^e^g vanishes for s>a. 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<^, ß2+^sinösin<^, r(l-cosö)) Jo

JO

(3.60) sm d , , a-cos Of Making use of (3.59), condition (3.58) becomes

(3.58)'

a/l/(r,ß)^ = 0 for

r
Our strategy is to reformulate this condition as a Volterra integral equation from which the conclusion of the theorem will follow. Let us suppose to begin with that / belongs to Cä(II). We denote the partial Fourier transform in the x, y direction by /(|, z): (3,61)

/ ( e z) = ^ I

exp {-i(x|i + y|2)}/(x, y, z) dx dy.

We now derive an expression for the ß-Fourier transform of / in terms of /:

i- f exp{-iß-|}/(r,ß)dß

= f^'f"^ exp\iv{r, e,<(,) • $}f{lr(l-cos Jo Jo

fl)),. ^"'^,,,dd(Uf>, (l-cosö)

where v{r, 0, ) = (r sin Ö cos , r sin 6 sin <^). Interchanging the order of integration and performing the (^-integration we get ij:|exp{-iß-C}/(r,ß)dß (3.62) =2.7

Jo(r\^\sme)Mril--cose))—^^^—^de, Jo

( 1 —COSü)

140

TRANSLATION REPRESENTATION FOR THE WAVE EQUATION

659

where 1 C^'^ Jo(u) = T— I exp {iu cos 4>} 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 rv) = 0 for almost all w with 2<2a. This completes the proof of Theorem 3.14. We note that Theorem 3.13 follows from Theorem 3.14. The analogous generalization of Theorem 3.12 is THEOREM

(3.65)

3.15. Suppose F={/i,/2} belongs to 5if. Then l>/i = 0 and f^^Q

outside the horosphere X if and only if both JR+F and R^F vanish in all horospheres lying outside of X. Proof: We see from (3.27) that both R^F and R^F vanish in all horospheres lying outside of X if and only if (3.66)

3^^eir = 0^d,ei2

in all such horospheres. Hence the assertion of the theorem for the second component of F, that is for /2, follows directly from the previous theorem. Remark. Set G={/i,0}. K G belongs to D{A\ then AG={0,L/}e9if. According to (3.17), Lft = e-^dth and hence as above the assertion of the theorem for the first component would follow from the previous theorem. K /i is not smooth, we can achieve the same end by mollifying. Choose <^ in Co(-l, 1) to be even with <^^0, I <^ = 1 and set sit)=^^(t/s), G,=^^ At)U{t)Gdt. Then G^ belongs to D(A), converges to G in 9if as e -> 0 and (3.67)

R^G^ = I <^,(t)fc(s=Ff, ß) dt = 14>As-t\)kit,ß) dt.

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TRANSLATION REPRESENTATION FOR THE WAVE EQUATION

661

where by (3.27) k =-R^G = cdUii ^ Working with model (i) and specializing X to the horosphere z = 2a, we can be quite explicit. Assuming (3.66) and making use of (3.59), we see that fc(5, ß) vanishes for all s>log[(l + |ß|^)/2a]. It therefore follows from (3.67) that jR±Ge vanishes for all s>log[l + Ißp)/2a]+8, in other words on all horospheres lying in the set z < 2ac""^. By the above remark we conclude that I^SBA

~ 0 ^^^ 2: < 2ae""^.

Taking the limit as s -> 0^ we see from Theorem 3.3 that g^ 1 -> /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 z<2ae~^. Multiplying (3.68) by ^(t) and integrating with respect to r, we see that 0 = (G3,H)H=-(gs4>i''tl) + (ge.2,M

for all such H. It follows that Lg^i = 0 for all z <2ae~^^ and, since G^ belongs to D{A)y we conclude from the above remark that ds^*ge,i = 0 on all horospheres lying in the set z<2ae""^. Since cd,V&,i = l?Jg,,i,0) and since {ge,i,0} converges to G in the energy norm, we see that ^±{ge,i>ö}~^^±^='^^^^/i i^ the L2 norm and hence that d^e^fi vanishes on all horospheres lying outside of X. This concludes the proof of Theorem 3.15. 4. The Scattering Operator A scattering operator S for the group U is defined by means of two translations representations Ti and T2 as the mapping (4.1)

S: TiF->T2F.

143

662

p. D. LAX AND R. S. PHILLIPS

In the case of the non-Euclidean wave equation on 11 the obvious candidates for Ti and T2 are R+ and JR^; we shall in fact use jR^. and a slightly modified We shall require of a translation representation T that it maps ^ unitarily onto L2(IR,N), where N = L2(B), and that (4.2)

T17(0F=L7,TF,

where U^ denotes translation to the right by t units; that is, [l/tfc](s) = k{s-t). According to Theorem 3.4, R^ is a translation representation of 17 in the above sense whereas 1?_ fails only because it translates to the left instead of the right. This defect is easily remedied by reflection; we set (4.3)

T, = R^,

T2 = R--.

where fe^(s) = fc(-s). For FeSif, we write (4.4)

fc,(s,ß)=T,F.

The corresponding spectral representations 3'^ and ^2 ^ ^ obtained by Fourier transformation with respect to s from Ti and T2: (4.5)

^i=^^'Tr.

that is, (4.5)'

k,(cr,ß) = ^,F = - ^ j e ^ k , ( s , ß ) d s .

In the spectral representation the action of U{t) is multiplication by e^. The spectral representation of S is (4.6)

^ = -'S,

called the scattering matrix. It is easy to show (see [6], Chapter 2) that S has the properties (i) S is unitary on L2(!R,N), (ii) S commutes with translation. The corresponding properties of S^ are: (i)' &" is unitary in L2(IR, N), (ii)' 5^ commutes with scalar multipliers.

144

TRANSLATION REPRESENTATION FOR THE WAVE EQUATION

663

It follows from this that 5^ is a multiplicative operator, that is (4.7)

k^i(r)^Se(a)Ua),

and that 5^(o-) is unitary on N for each a. If in addition S has the causal property: (iii) SL2(!R_)cL2(IR_), then obviously (iii)' 5^3if2^^2, where ^2 denotes the Hardy class ~^L2(!R_). In this case it can be shown that 5^(0-) is the boundary value of an operator-valued function Sfiz) which is holmorphic in the lower half-plane, ^m z < 0, where 9^(z) is a contraction on N. LEMMA

4.1. Denote by 3)'^ the orthogonal complement of 3)^. Then

(4.8)

2>L<=2i^

and (4.9) Proof:

S i J. 2»;. According to Theorem 3.9, K+2»+ = L2([R+) and

l?+2i_3L2(IR_).

It follows from the left relation that (4.10)

R+2)1 = L2(IR_)

and this together with the right relation shows that SJ^cr 2)_. The "plus" part of (4.8) is proved similarly. Relation (4.9) is now immediate since 2>Lc2>_ COROLLARY

4.2.

and

2i_±2):^ = S ; .

S has property (iii).

Proof: In view of (4.10), L2(IR_) = Ti(SiL). Likewise, K_(®;) = L2(IR-) so that (4.10)'

LM^) = L2m-r

= K - ( S ; ) " = T2(2>^) .

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, s<0.

Note that (4.16)'

-1

..


-i

1 a—I im

Since Sf is an inner factor in the sense of Beurling, the obvious candidate for y is (4.17)

5^o(
where

B„(
k|=l,

is the Blaschke product with the same poles as the 0~^e^, m = 1,2, • • •, n, and 0„ is the orthogonal projection of N onto the spherical harmonics of order n. We can express Q„ as an integral operator with kernel -J

P„(ß-Ö),

where P„ is the Legendre polynomial of order n and ß, Ö are unit vectors. THEOREM

4.4. ITic scattering matrix 5^ mapping &'xF-^6'2F is given by

(4.17).

147

666

p. D. LAX AND R. S. PfflLLIPS

Proof: Since |B„(cr)| = l for real cr and since the 0„ span N, it is clear that Sfo(a) is unitary on N for each real a. Sfoicr) obviously commutes with scalar functions of a; Bnicr)^ being holomorphic and bounded in the lower half-plane, maps the Hardy class into itself and the same is true of 5^o- Thus 5^0 satisfies properties (i)'-(iii)/» Moreover, SfoM commutes with rotations and has the maximum amount of indeterminacy permitted by (4.14). It remains only to show that the range of 6^o and ^2 is ^2(®-)- To this end we prove: (I) 5^0(^2) ^^2(2>i). 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 <^ + iP

obviously belongs to the Hardy class, the usual contour integration argument shows that the integral in (4.18) vanishes, as desired. (II) 5^o^2=^ ^2(®-)- Denotingthe conjugate Hardy class by 34^2, it follows from properties (iy--(iiiy that 5^o^2=^^2- Given ke^2 we now compute the ^2 component of 9^ok; i.e., the projection of ^&^ok into L2(R-)-

^^ok = Z-i=Y;.(ß) ^ -JITT

\e-^ll^^^K(a)d
1^ cr
Sincefc„belongs to the conjugate Hardy class (and hence is holomorphic in the upper half-plane), the usual contour integration argument shows that, for s<0, ^5^0fc is a sum of the form (4.15). Thus

and since 5^o is unitary and 35^2= ^2 > we conclude that 6^0(^2)^ (34^2 0 ^2(30)-' = ^2(®i). Together (I) and (11) prove that ^0(^2) = ^2(S>1). Consequently, for a suitable choice of constants {e„}, &'Q = ^. This completes the proof of Theorem 4.4.

148

TRANSLATION REPRESENTATION FOR THE WAVE EQUATION

667

Bibliography [1] GeFfand, I. M., Graev, M. I., and Vilenkin Ya. N., Generalized Functions, Vol. 5, Academic Press, New York, 1966. [2] Helgason, S., The Radon transform on Euclidean spaces, compact two-point homogeneous spaces, and Grassman manifolds. Acta Math. 113, 1965, pp. 153-180. [3] Helgason, S., l i e Groups and Symmetric Spaces, Battelle Rencontres, Benjamin Pub. Co., New York, 1967. [4] Helgason, S., The surjectivity of invariant differential operators in symmetric spaces, I, Ann. Math. 98, 1973, pp. 451-479. [5] Helgason, S., Harmonic analysis on homogeneous spaces, Proc Symposia in Pure Math., 26, Amer. Math. Soc. 1973, pp. 101-146. [6] Lax, P. D., and Phillips, R. S., Scattering Theory, Academic Press, New York, 1967. [7] Lax, P. D., and Phillips, R. S., The Paley-Wiener theorem for the Radon transform, Comm. Pure Appl. Math. 23, 1970, pp. 409-^24. [8] Lax, P. D., and Phillips, R. S., Scattering Theory for Automorphic Functions, Annals of Math. Studies, No. 87, Princeton Univ. Press, 1976. [9] Ludwig, D., The Radon transform on Euclidean space, Conun. Pure Appl. Math, 19, 1966, pp. 49-81. [10] Pavlov, B. S., and Faddeev, L, D., Scattering theory and automorphic functions. Seminar of the Steklov Mathematical Institute of Leningrad, 27, 1972, pp. 161-193. [11] Semenov-Tian-Shansky, M. A., Harmonic Analysis on Riemannian symmetric spaces of negative curvature and scattering theory. Math. U.S.S.R. Izv. 10, 1976, pp. 535-563.

Received October, 1978.

149

BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 2, Number 2, March 1980

SCATTERING THEORY FOR AUTOMORPHIC FUNCTIONS BY PETER D. LAX AND RALPH S. PfflLLIPS^ ABSTRACT. This paper is an expository account of our 1976 monograph [6] on Scattering theory for automorphic functions. Several improvements have been incorporated: a more direct proof of the meromorphic character of the Eisenstein series, an expUcit formula for the translation representations and a simpler derivation of the spectral representations. Our hyperbolic approach to the Selberg trace formula is also included.

1. Introduction. In 1972 Faddeev and Pavlov [2] discovered a revealing connection betv^een the harmonic analysis of functions automorphic with respect to a discrete subgroup of SL(2, K) and the Lax-Phillips scattering theory as applied to the non-Euclidean wave equation. Their work is based on the spectral theory for the Laplace-Beltrami operator previously developed by Faddeev [1] using elliptic arguments. In our 1976 monograph [6] we redid the Faddeev-Pavlov paper entirely within the framework of our theory, basing our development on the non-Euclidean wave equation. We obtained new treatments for (i) the spectral theory of the Laplace-Beltrami operator over noncompact domains of finite area; (ii) the meromorphic character of the Eisenstein series over the whole complex plane; and (iii) a new form of the Selberg trace formula. In this paper we sketch a revised version of our monograph including a more direct proof of the meromorphic character of the Eisenstein series, an expHcit formula for the translation representations and a simpler derivation of the spectral representations. The harmonic analysis of automorphic functions has been extensively studied; references to the pertinent parts of this theory are contained in our monograph. We recall that the Poincafe plane II, that is the upper half plane w = X + i>,

y>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)<^^.

U (/(0^~=^=

for r < 0 and

U U(t)^^.

U{t)<^^ c ^ ^

for / > 0,

(1.11)

An example of an outgoing subspace is given by the following: % =« L2(R, ^ ) , 91 some auxihary Hilbert space, U(t) translation to the right by an amount /, and ^ ^ = L2(R^y 91). It tums out that this class of examples exhausts all possibihties since every Hilbert space % with a unitary group for which there is an outgoing subspace can be represented as above, essentially uniquely; this is called the outgoing translation representation of %. Corresponding to an incoming subspace there is an incoming translation representation in which ^ _ maps onto L2(R-., 91). In the appUcations of the theory we are about to describe, / represents time and the group U(t) describes the propagation of waves. The significance of incoming and outgoing subspaces for wave propagation is that they furnish an asymptotic description of signals in the remote past and distant future respectively. The scattering operator S relates the incoming translation representer of a given data to its outgoing representer. Thus S gives a description of the scattering process, i.e. it relates directly the asymptotic behavior of waves for large negative and positive times. S maps L2(Ry 91) onto itself; since it relates two translation representations it is unitary and commutes with translation. Therefore S is convolution with an operator valued distribution (91 -^ 91) called the scattering function and denoted as S(t) (cf. [7]). This approach is particularly fruitful when ^ _ and ^ ^ are orthogonal to each other; in this case 5 is a so-called catdsal operator^ in the sense that the scattering function 5(0 = 0 for / > 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)^ © <^^). The resolvent of the infinitesimal generator of Z turns out to be intimately related to the scattering matrix. Before returning to the automorphic case we would like to indicate how these ideas are used to study wave propagation in Euclidean space in the presence of an obstacle 0 . In the exterior & of the obstacle the signal satisfies the Euclidean wave equation: u,, = At/.

(LI3)

The interaction of the wave with the obstacle is described as a boundary condition on w; if the interaction conserves the energy contained in the wave, the boundary condition makes A a selfadjoint operator on ^2(8). Typical of such boundary conditions are w = 0 (or 9^M = 0) on 90. The underlying Hilbert space is the space of initial data {M, WJ defined on 6 and normed by the energy norm: ^ s ( " ) == - (^> A")s + ("P t/,)g,

(1.14)

where ( , )g denotes the Lj inner product over S . Since A is selfadjoint, E^u) is independent of / for solutions u of (1.14) satisfying the boundary conditions. This shows that the operator U{t) relating initial data at time 0 to data at time f is an isometry; since the initial-boundary value problem is reversible in time and can be solved for a dense set of data, the operators U{t) are actually unitary and form a group. We call a solution u outgoing if w(x, 0 = 0 at all points x whose distance from the boundary is less than t; an incoming solution is described similarly with / replaced by - / . ^ _ and ^ ^ are defined as the initial data of incoming and outgoing solutions, respectively. Properties (i) and (ii) can be immediately verified; property (iii) Ues considerably deeper and says essentially that all incoming signals eventually become outgoing signals after a comphcated process of reflection from the boundary. Roughly speaking, the same picture can be painted of automorphic waves over noncompact fundamental domains. Signals coining in from of going out to infinity can be defined analogously. Again one of the main results to be proved is that all incoming signals are eventually tumed into outgoing signals. The analysis is comphcated by the possible presence of standing waves due to a rather rich point spectrum, and of exponentially increasing solutions due to the indefiniteness of the energy form. A treatment of the corresponding free space problem can be found in [8]. 2. Hie non-Euclidean wave equation» The first step in our analysis consists in finding a suitable Hilbert space setting for the initial value problem 153

SCATTERING THEORY FOR AUTOMORPHIC FUNCTIONS

265

associated with the non-Euclidean wave eijuation: «// = Lu, u{x. 0) = f,{x\

t/,(x, 0) = h{x\

(2.1)

For a more complete treatment we refer the reader to [6, § 5]. In terms of the L2{F) inner product ( , )^, the energy form for the wave equation is ^F(«)

= - (w>

^«)F

+ (wp u,)p.

(2.2)

Since L is selfadjoint in L2(F), Ep{u) is independent of / for automorphic solutions of (2.1). In general Ep is indefinite for automorphic data; this can be seen if we bring (2.2) into a more symmetric form by an integration by parts:

£,(«) = / / j l",P + Kl' - | f + ^ }
(2-2)'

The indefiniteness introduces complications analogous to those produced by bound states in the theory of scattering for quantum mechanics (see [5, Chapter 6]). It is useful at this point to write the energy form in a way which depends on the shape of the fundamental domain F. In the case of the modular group with fundamental domain described in (1.10) we split F up into two parts: ^o = ^ n {y
(2.3)o

which is compact, and a neighborhood of infinity Fj = F n { y > a } ;

(2.3)|

here a is restricted to be > \. 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
(2.5)

where

154

266

P. D. LAX AND R. S. PHILLIPS

Notice that for data with support in F, the E and G forms are the same. It can be shown by simple estimates that G is positive definite and equivalent with G'iu)^fJ

j k P + KP + J^jt/xJK

+ / / {K?'^yk^f]dxdy Fl { I v>^ 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
l/(0^±C^:.

(ii)

n ^ ( 0 ^ - = (0) = n U{t)<^^.

(2.13)

for^^O,

(2.14)

The change of variables (2.9) maps the energy form E defined by (2.2)' into the classical energy form /{|t>5p + \vg\^} ds for (2.10). In these variables a trivial calculation shows that ^ . and ^+ are E-orthogonal

(2.14)"

Since data in ^ _ and ^ ^ are zero in FQ, it follows that ^ « and ^^ are also G-orthogonal. We denote the zero Fourier coefficient of/with respect to x hyf-^^i f'^Ky) = r^'fix^y) dx,

y>a.

(2.15)

''-X/l

Suppose the data/ = (/ij/j) in %G i^ ^'-orthogonal to ^ _ ; it is easy to show by using the definition (2.13)_ of ^ _ and the JE^-scalar product derived from (2.2)" that the components of/°> satisfy

Similarly if it is jE'-orthogonal to ^^ then

^ 0 ) ^ ^ 3 / 2 9 / ^ 1 lory >a,

(2.16)^

If/in %Q is orthogonal to both ^_ and ^+ then (2.16) implies that the zero Fourier coefficient of/is of the form /(O) = ( e V y , 0) for>^ > a.

(2.17)

We denote the set of all such data by %i

gc = x^e(^^e^j.

(2.18)

The next theorem is the basic technical tool in this appHcation of our scattering theory.

156

268

p. D. LAX AND R. S. PHILLIPS

THEOREM 2.1. For every \ in the resolvent set of A, the resolvent operator (\I — A)"^ maps the unit ball in % into a compact subset of %Q.

We give a new proof of this theorem in the appendix. COROLLARY 2.2. A has at most a denumerable set of eigenfunctions in % and the corresponding eigenvalues are discrete.

3. The associated semigroup of operators. In this section the group of operators U and the incoming and outgoing subspaces are combined to form a semigroup of operators which plays an important role in our theory. To this end we introduce the orthogonal projections P_ and P^ which remove the ^ _ and ^^ components, respectively. Again since the G and £-forms are the same for>^ > 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 \

(4.8)

and$inC°^(R)sothat

We now define / equal to e except for the zero Fourier coefficient in the region >^ > b where we set /(0)(^) = {>.V2^/z^^3/2(^/zyj + ^(^)j^i/2~/.^ izy'/^-^]. (4.9) It follows from the above discussion t h a t / belongs to %Q and from (2.16)_ that/is orthogonal to ^ _ . Comparing (4.7) and (4.9) shows that

/ = e + {^•/2(^ _ \)/^y/\i^ - \)/^yy^

(4.9)'

clearly/belongs to D{A) when Im z < - 1 / 2 . It follows then from (4.5) that Af^izf+k,

(4.10)

k = ^(0) = {j>;^/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
4.3. The poles of s(z) occur at most at the points -ia^B).

5. The translation representations. So far we have managed to avoid the compHcations which result from the energy form being indefinite; in fact E has played only a minor role in our discussion. We must now confront this problem. By definition G = E + 2K; since K is compact with respect to G, E is positive on a subspace of finite codimension. It is clear from the expression (2.2) that E is nonnegative on the ^-orthogonal complement of the positive eigenspaces of L. Denote the positive eigenfunctions and eigenvalues of L by ( i . V),y = 1,. • . , m with \ > 0: H = %-

(5.1)

^.- = {qjy ±}^qj].

(5.2)

We set

It is easy to verify thatj^~ belongs to %Q and that Af^- = ±A^^; Eiff^Jk") = 0 = E{f,-jn 160

(5.1)' for ally, k;

(5.3)

272

P. D. LAX AND R. S. PHILLIPS

and

We now set ^^ = span of {j^.-^},

6>. = span of{j(;-),

6^ = ^^ + ^ ^

(54)

and denote the ^'-orthogonal complement of ^ in %^ by %Q. E is nonnegative on %Q^ positive if ^ has no null vectors. We denote by Ö' the jF-orthogonal projection of %Q onto SC^; using formulas (5.3) and (5.3)' we can write

Qj-'f+^

a^f; + 2 apr

(5.5)

where a/ = E{fjr)/>}, a- = E{f,J,-)/)^\

(5.5)'

If A has null vectors, they will span a finite dimensional subspace f and on the quotient space E is positive definite and equivalent to G. Clearly %^ is an invariant space for the operator U(t). It is easy to show that ^ . ± ^ + , ^^±^^ and ^ n ^ ^ = {0} = ^ n ^ + , (5.6) but it is not true in general that ^ « ± 9 « or that ^^±^^; thus the subspaces ^_ and ^ ^ do not in general he in %ß. The next best thing is to replace ^ ^ in our considerations by their E-orthogonal projections ^ ^ in 9C^; that is by ^ ; = ö'^±-

(5.6)'

Strictly speaking one has to mod out ^ in the right member of (5.6)'; however since ^± n ^ == {0} this causes no confusion. The projection operator Q' conmiutes with U{t). It follows from this and properties (2.14) (i) and (ii) that (i)

f/(/)^4 c ^ ;

forr^O,

(ii) n f/(o^: = (0} = n ^(0^;-

^^'^^

It can be shown that ^^ ^^^o satisfy

(in)

U u(t)<^i = 3c; = U f / ( 0 ^ ; .

where %^ is the jE-orthogonal complement of the eigenfunctions of ^ in 9C^. For the proof of this property we refer to Corollary 6.14 of our monograph [6]; the main ingredient of the proof is Theorem 2.1. According to the translation representation theorem (see Chapter 2 of [6]), any subspace satisfying (i)_-(iii)_ (or (i)^.~(üi)+) of (5.7) can be used to 161

SCATTERING THEORY FOR AUTOMORPHIC FUNCTIONS

273

construct a translation representation of DC^', i.e. a unitary map T_ (or T^) of DC^' onto a vector valued space of functions of the form L2(R, 91), 91 being an auxiliary Hubert space, with the properties: r ^ : / e OC; -> Ä:^ G L^CR, 91),

(5.8)

(a)

E{f)^j \\k^\\%,ds,

(y)

T^ maps ^ ; onto L2(R^, 91).

We need not rely on this general existence theorem for we can, as we shall show below, obtain explicit formulas for these primed translation representations. We treat only the ^^-representation but it will be clear that the ^1-representation can be treated analogously. We begin by constructing a translation representation for the elements of U{t)^^. Given automorphic d a t a / = (/i,/2) defined on all 11 with locally finite G-norm, denote, as in (2.15), hyfP the zero Fourier coefficients of ^, i = 1, 2: fP{y)

= C^^f,{x,y) dx,

0
00.

(5.9)

•'-1/2

We define the translation representer o f / t o be 7-+ / = : ^ [9.(^"'/^'^(^0) - e-^/'fi\e%

-oo < ^ < oo. (5.10)^

It is easy to see that T^ is a Hnear mapping, continuous from %Q into

4^(R). LEMMA

5.1. r^. commutes with translations: T^U(t)f^T{t)T^f;

(5.11)

here T(t) denotes translation to the right by t units. PROOF. Let t/(x, y, t) be the solution to the non-Euclidean wave equation with automorphic initial data/:

u,^ = Lw; w(0) = / ,

and w^O) = / j -

Then for each time /, u will be automorphic and in particular it will be periodic in x of period 1 for all ^ > 0. The zero Fourier coefficient u^\yy t) will therefore satisfy the equation t/,f = A i ; > + t/<^>/4 forall>^>0.

(5.12)

As we have seen in §2, the change of variables s := log>^,

V = w
(5.13)

transforms (5.12) into the classical wave equation: t.« = Vss;

(5.14)

the initial data goes over into ü(0) = e-*/^/«>(eO

and Ü,(0) = e-*/^2<«>(e'). 162

(5.15)

274

P. D. LAX AND R. S. PHILLIPS

Setting k{sy t) = v,- ü„

(5.16)

it follows from (5.10) and (5.15) that k{s,0)=V2T^f.

(5.17)o

Since the solution data at time / is U{t)f, we have similarly k(s,t)=V2T^Uit)f.

(5.17),

Combining (5.14) and (5.16) we see that d,k + a,Ä: = 0

and hence that k{s, t) = k{s - t, 0).

(5.18)

Finally on combining (5.18) with (5.17) we obtain the assertion of the lemma. Note that in the proof of Lemma 5.1 we used only the fact that / was periodic in jc with period 1. We therefore have COROLLARY 5.2 Let f be data periodic in x with period 1 and with locally finite G-norm. Define T+ as before by (5.9) and (5.10)+. Then Lemma 5.1 holds forf.

Next we consider data t/g defined as in (2.13):

äo^{y'^My)>-y'^'¥iy)}

(5.19)

where cp lies in C^(R) and vanishes for y
2 d,(yw). (5.20) Tooxr Since

^ < a, d^iw) = 0 for y
(5.21)

Qearly UQ(W, t) is periodic in x. The automorphic solution of the nonEuclidean wave equation with initial data d given by (5.20) is « ( w , 0 = S UQ{yw,t). (5.20), r«\r The reasoning used above shows that u(w, t) = UQ{W, t) for w in F and / > 0. LEMMA 5.3. For dQ and d given by (5.19) and (5.20), respectively, (a) T^do = T^d;

0>)EAd) = f'^ iT^dfds.

163

SCATTERING THEORY FOR AUTOMORPHIC FUNCTIONS PROOF.

Since u - UginFfort

275

> 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[^^)^^ - ^-^]

<^ *•

Performing that x integration, we can replace /j and /j by their zero Fourier coefficients ff^^ and f^P; changing the other variable of integration y into s = log>^, 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^ log a. Comparing this with (5.27) we see that T^ ff = 0. LEMMA 5.6. (a) Let p be an eigenfunction of L with negative eigenvalue -v^. Denote by g^ = {/?, ± ivp] the corresponding eigenfunctions of A. Then

(b) Let f be an arbitrary automorphic data in %^ and denote by f^ the projection off into %^. Then T^f^T^f^.

(5.28)

PROOF. The proof of (a) is entirely analogous to that of (5.26) since g^ belongs to the point spectrum of A in SC^ and because of this can easily be shown to be orthogonal to ^ 4 ^^^ hence to be orthogonal to ^ ^ . Part (b) follows from part (a) since / — X' ^^^ ^^ expanded as a convergent series in eigenfunctions of the kind considered in part (a). Recalling that T^ is continuous from %Q to L^ and that the G and E norms are equivalent on %ß^ assertion (b) now follows. THEOREM

5.7. T^ is the ^'^-translation representation of 5Q.

PROOF. We shall verify properties (5.8) where auxiUary Hilbert space 91 is C: By Lemma 5.1, T^ is a translation representation and hence satisfies property {ß). By part (b) of Lemma 5.3, T^ is isometric on ^^.. Formula (5.22) shows that T^ maps ^ ^ onto L2{R + log a, Q. Since by Lemma 5.5, T^d'^ = T^d^ and by (5.25) Ef{d'^) = E^id^X it follows that T^ maps ^+

165

SCATTERING THEORY FOR AUTOMORPHIC FUNCTIONS

277

isometrically onto L2(R + log a, Q . This proves property (y). Since by property (iii) of (5.7) the translates of ^ ^ by U(t) span OQ and since the isometry persists under such translations, it follows that T^ is unitary and hence is an outgoing translation representation of 3Q with respect to ^:^. TTie analogue of Theorem 5.7 holds for the ^1-translation representation of 9C; with T^ replaced by r ^ / = - ^ [ 9 . ( e ^ / ' / r ( e - ) ) - e^^^Pie-^

(5.10).

Formulas (5.10)^ are the exphcit translation representation formulas alluded to in the introduction. We recall that for the Euclidean wave equation, the translation representation is given in terms of the Radon transform of the data defined by integrals along all straight lines in the plane. The integrals in (5.8) are over a one-parameter set of curves in F that are the images of the horocycles through infinity under the projection of the Poincare plane 11 onto F. The inversion of T^ can be accompHshed in three steps. Let k(s) "^^ T^ f and define (p by


(5.29)

•'-oo

(i) Set

SoM = [y'^Myl -y'^Wy)Y

(5.30)

(ii) Define g(H')= 2

^O(Y>^)-

(5.31)

(iii) Project g into %Q : /=ß'g-

(5.32)

PROOF. Suppose / belongs to ^ ^ . Then by definition there exists a ^^. in ^ ^ such that Q'd^ ^ d'^^f. It is clear from (5.19), (5.20) and (5.22) that g, defined as above, is this d^\ this proves (5.32) when/ G ^ ^ . Next suppose t h a t / = UiOd^; then, since Q' commutes with U{t%f^ Q'U{t)d^. Comparing the relations (5.19) through (5.20^ with (5.29) to (5.31) we conclude that g = U(t)d^; the relation (5.32) therefore holds in this case as well. Finally since U f/(0^+ is dense in 3Q, it follows by continuity that (5.32) holds for all/in 9C;. Using Lemma 5.6, we see that formula (5.32) can be used to project a n y / in %^ into 3C;. We conclude this section by defining the scattering operator S' as the mapping of the ^1-representation onto the ^^-representation:

S'lT^f^T^f

for/in DC;.

(5.33)

6. The spectral representation. A representation in which the action of U{t) is multipHcation by exp (iat) is caDed a spectral representation. It is clear that the Fourier transform of a translation representation satisfies this criterion. In this section we shall study the spectral representation of t/, restricted to 9C;, 166

278

P. D. LAX AND R. S. PHILLIPS

SO obtained from the ^+ and ^ 1 translation representation treated in §5: (^^ f){o) = - ^ V2^

r

e'-(r^ f){s) ds.

(6.1)

•'-00

We now prove THEOREM

6.1. TTie ^i-spectral representation can be expressed as ^^ f{o) = -^E,{f,

M^),

(6.2)

where e^(z), z = a + /r, are analytic continuations of the Eisenstein series generated by ıK^) = -^{>^'/'^^~/^'/'^'^)V2 iz

(6.3)

REMARK. It should be noted that h^ and h_ are closely related to but somewhat different than h in (4.1)- In particular

h_{z) = - / i ( - z ) / V 2 iz; since the Eisenstein series e corresponding to h converges for Imz < - 1 / 2 , the Eisenstein series e_(z) converges for Imz > 1/2, and the zero Fourier coefficient of e_ is related to that of e by e(z) = - ^ e<^>(-z). (6.4). y/iiz An expHcit formula for e^^^ is given in (4.7). On the other hand h^{z) is essentially the same as (V2 izy^h{z) except that its second component has the opposite sign. This sign change has no effect on the convergence of the corresponding Eisenstein series e^(z) nor on its analytic extension. The analogue of (4.7) is now V2 iz e f (z) = [7^/2-^^ -/zy^/2+^} + 5(z){>^^/2^^ -/zy^/^-^}. (6.4), PROOF. We shaU treat only the ^^-representation. Since ^ , is the composite of the two unitary maps T^ and the Fourier transform, it is itself unitary. Thus ^ , is continuous; so it suffices to prove (6.2) for a dense subset of 3Q; for our purpose this will be the set of data/for which T^ fis smooth with compact support, and for which

j

T^f{s)ds^O.

We shall derive bounds for such/by inverting T^; see the end of §5. Set

and Then ggiy) has compact support in^'; as a consequence only a finite number 167

SCATTERING THEORY FOR AUTOMORPHIC FUNCTIONS

279

of images y'^F of F lie in that portion of the support of g^ which is contained in the strip 5 = ( - 1 / 2 < x < 1/2,7 > 0 } . Thus the sum giy^)-

S goiw) (6-5) Too^r has only a finite number of nonvanishing terms on F and is of compact support in F. g is automorphic and, as shown at the end of §5, f^Q'g,

(6.6)

T^f^T^g.

(6.6)'

and by (5.26) By definition (5.5) of Q\ f-gE:^. (6.7) N.b. It is easy to show that f — g actually belongs to ^ _ , but the weaker result (6.7) suffices, here. Substituting (6.6) into (6.1) we get V2^ 5.10)+ defining T+ and making the change of variable: Recalling formula (5.10)+ s -> log>', the relation (6.8) becomes

^^^-i^r[/-:;>Wir)-^}'^

y (6.8)'

The remainder of the proof of Theorem 6.1 splits into three parts: Step L Continue (6-8y analytically into the half plane I m z < - 1 / 2 and then rewrite it as 1 %f-^-J=rEs{g. h^(z)); (6.9) here E^ denotes the invariant form of the energy integral given in (2.2)' but integrated over the strip S instead of F. Step 2. Replace the integral over the strip by a sum of integrals over all the y'^F's contained in S. Using the fact that g is automorphic this gives V2ff

r«\r

1 2 Ep{g, V2ir r„\r

--^EJg,

2

h+{yw,z))

h^{rw,z)) = Ep{g,7:jI)).

(6.10)

Note that the sum defining the Eisenstein series converges for Im 2 < - 1 / 2 168

280

P. D- LAX AND R. S. PHILLIPS

and that e^(z) can be continued analytically back to real z. Step 3. Show for the eigenfunctions {j^~} that Ej.(f,'^.e^{o))^0.

(6.11)

Since the j^~ span ^ and since by (6.7)/ — g G ^ , the relation (6.2) is an inunediate consequence of (6.10) and (6.11). PROOF OF STEP 1. Since T^ g(s) has compact support, it is clear that ^^ / as defined by (6.8), and hence (6.8)', is entire in a and can be continued analytically into the half-plane Imz < - 1 / 2 , z = 0 + ir. Notice that the support for the individual terms in (6.8)' are bounded away, from y ^ 00 but not from 7 = 0. It follows that each of these terms is integrable for large y; for small y the factor y^ decays fast enough if Im z < - 1 / 2 , and we now make use of this fact to show that (6.8)' exists as a double integral. As noted before, all but a finite number of the terms in (6.5) vanish on F. Thus g{w) is bounded on F and since g is automorphic it is also bounded on 5^. It follows that for Im z < - 1 / 2 the second term on the right in (6.8)' is integrable near>^ = 0. The first term can be written as

the same reasoning as above shows that the contribution of the second term on the right in this expression to (6.8)' is also integrable for small y. Thus it only remains to consider the integrabihty of y'^-'^gv

(6.12)

Recall that by (6.5)' Tooxr where g^ is a smooth function of y alone with compact support, say in the interval (m, M). Hence ^0(1^) = ^o(f)

where aw + b M + It? = yw =cw + d 7

. y and v =|cw + d\^

Now

9£ dy

dv

dy

dy

<

1

\cw -\-df

y

It follows that 9^O(YH')

9V

M < const —

for all y.

y

Since the number of terms in (6.5) which do not vanish on any transform yF of F is finite and independent of y, we conclude that

169

SCATTERING THEORY FOR AUTOMORPHIC FUNCTIONS

281

9^1

const — , y This estimate is sufficient to assure the integrability of (6.12) over the strip S when Im z < - 1 / 2 . We may now proceed formally to bring (6-8)' into the desired invariant form. We write dy

/ / y%[-^)

d^^ = ff {y''-'%gi-lr'-'^'gi} dx ^; (6.13)

Splitting the first term, this can be rewritten as

and integrating the middle term by parts gives

The integrated term vanishes; substituting the remaining term back into (6.13) we get

Note that this integration by parts is the reverse of the one that brought (2.2)' into the form (2.2)". The right member of the above equation is in invariant form as can be seen from (1.5) and (1.6). Substituting this back in (6.8)' and making use of the definition of Ä^ as given in (6.3) we obtain (6.9). PROOF OF STEP 2. In order to justify the interchange in the order of summation and integration occurring in (6.10), it suffices to recall the fact, noted in §4, that the Eisenstein series converges in %Q when Im z < - 1 / 2 , except for the first term A^.(w). However h^(w) is locally in %g and since g is of compact support in F, even this term causes no difficulty. According to Theorem 4.2, the Eisenstein function e^{z) is meromorphic in the local %Q topolo©^. Hence if we again make use of the fact that g has compact support in F, we see that Ej,(gy e^{z)) is meromorphic and can be continued analytically back to the real axis. It follows that % f{o) - -^Efig.

"M^).

(6.9)'

PROOF OF STEP 3. This proof is based on the observation that bothj^" and e^(a) are eigenfunctions of the generator A and that the standard proof for the orthogonaHty of eigenfimctions of a skew-Hermitian partial differential operator can be applied in this case even though one of the eigenfunctions is a generalized one. As defined in (5.1)

f/

- {qj, ±X,qj],

where qj is an eigenfunction of LiLqj = A?^, belonging to L^F). The zero

170

282

P* D. LAX AND R. S. PHILLIPS

Fourier coefficient of qj satisfies the ordinary differential equation

—-ir*U'"=Hir' and hence is of the form qf) = cyV^-^^ + 4^^/^-^ fory > a. Since qj is square integrable on i s the coefficient c must be zero. On the other hand e^{o) is in %Q except for its zero Fourier coefficient which by (4.7) is of the form

V2I iz It follows that the Ep inner product of j^~ and e^{o) is integrable and that integration by parts is permissible. This gives ioEp{jj^. e^{o)) = Ep{ff,

Ae^{o))

= '-Ep(Aff, J ^ ) = T\^Ep{ff,

7>0)(6.14)

from which (6.11) follows. This completes the proof of Theorem 6.1. In analogy with Lenmia 5.6 we have LEMMA 6.2. Let f be an arbitrary automorphic data in X^ and denote by f^ the projection off into %^. Then

ErU^ e^{o)) = ^^(/;, e^{o)).

(6.15)

Since / differs from^' by a sum of genuine eigenvectors of Ay this follows by essentially the same argument as that used in (6.14). REMARK 6.3. It is easy to see that A has only a discrete set of purely imaginary point eigenvalues. In fact since the zero Fourier coefficient of an associated eigenfunction is of the form and since a point eigenfunction is by definition of finite G-norm, both c and d must be zero. It then follows by (2.17) that any such eigenfunction belongs to the subspace 5C; the assertion is therefore an immediate consequence of Corollary 2.2. We conclude this section by obtaining the spectral representer for the scattering operator 5" defined in (5.33). To this end we prove LEMMA 6.4.

e^{o) - -s{o)e^{o). PROOF.

(6.16)

It is readily verified that h^ satisfy locally Ah^ = -izh^.

It follows, along the lines indicated at the beginning of §4, that e^ also satisfy locally Ae^ = -ize^. 171

SCATTERING THEORY FOR AUTOMORPHIC FUNCTIONS

283

It follows from (6.4)^ that while according to (6.4). and (4.7) ~V2 ize^^\z) - {7^/^-^ -/zy^/^-'^} + s{-z){y^/^'''\

^/zy^/^^^}.

Therefore

/ = e^{z) + s{z)e^{z) satisfies /(0)(^) = const {>^'/^-'^ -/zy^/^-^'*^). Thus for Im z > 0,/^^\z) belongs to OC^. According to Lemma 4.1, e^(z) — e^P(z), belong to 9C^ except when z lies in "-ia(5); it follows that for all such z with Im z > 0,/(z) belongs to %Q. Since/is a linear combination of e^. and e_, it satisfies

Af^-izf. Now ^ has only a finite set of nonimaginary eigenvalues. These are the values { ± A^.} appearing in (5.1)' which correspond to the positive eigenpairs of L. In fact, as explained in §5, in the quotient space %ß = ^ G / ^ > where ^ is the null space of ^, £ is positive definite and equivalent to G. Since A restricted to 3Q is skew-Hermitian, it has no nonimaginary eigenvalues on 9Q. Thus for Im z > 0 and z not in {/A^}, it follows from the above that/(z) = 0. By analytic continuation this holds for real z as well, as asserted in Lemma 6.4, THEOREM

6.5. The spectral representer of the scattering operator: S':^^f^%f

(6.17)

is the multiplicative operator: SXo)--s(o). PROOF.

(6.18)

It follows from Theorem 6.1 and Lemma 6.4 that

as desired. In the case of the modular group, the scattering operator S' can be expressed in terms of the Riemann ^-function and the gamma function as in (4.7y. The poles of S' are the nontrivial zeroes of ^(1 + 2/z). Thus if the Riemann hypothesis is true these poles will He on the line Im z « 1/4. A further connection between our theory and the ^-function can be expressed in terms of the semigroup of operators Z" defined in analogy with Z of §3 using ^Z ^ ^^ n%E instead of ^^. Pavlov and Faddeev [2] have formulated the Riemann hypothesis in terms of properties of Z": The Riemann hypothesis is true if and only if Z"{t) decays exponentially. Specifically, in [6] we have shown: The Riemann hypothesis is true if lim sup r^logllZ^XO/b < - 1 / 4 (6.19) for a set of/dense in SQ'. The relation (6.19) is an assertion about the rate at which energy is propagated to infinity for those solutions of the automorphic 172

284

P. D. LAX AND R. S. PHILLIPS

wave equation which contain no standing waves, i.e. are orthogonal to point eigenvectors. This latter condition prevents the appHcation of hyperbolic techniques for proving local energy decay via nonstandard energy identities. Another possible approach is to use in (6.19) vectors / obtained by appUcation of Z " ( 0 ^ ^1; this is discussed in an appendix to §7 in [6] entitled "How not to prove the Riemann hypothesis''. Hubert is reputed to have envisaged a proof of the Riemann hypothesis by constructing a skew-hennitian operator on a Hilbert space whose eigenvalues are of the form K — 1/2, ic being the nontrivial zeroes of ^. By an analogue of Corollary 4.3 the infinitesimal generator JB" of Z " has as its spectrum the values {(ic - l)/2}. It follows that 2 5 " + 1/2 (6.20) is the kind of operator on 3C^' that was envisaged by Hilbert; all that is missing is an appropriate inner product on 9Q which makes (6.20) skewhennitian. 7. The point spectnim. In addition to the continuous spectrum analysed in §§4, 5 and 6, the operator A has a substantial point spectrum. As noted in Remark 6.3, the point spectrum is discrete. In the case of the modular group, analysis of the point spectrum is made particularly easy by exploiting the bilateral synmietry of the fundamental domain F described in (1.10), that is the mapping j c - » - x which takes F into itself. Since the operator L is invariant under the reflection: x -» -jc, the domain of L can be reduced to the direct sum of even and odd functions. The boundary conditions induced by a function being automorphic become Dirichlet boundary conditions on half of F for odd functions and Neumann boundary conditions on half of F for even functions. It is not hard to show that under Dirichlet boundary conditions on half of F, the proper eigenfunctions form a complete orthonormal system. Thus the point eigenvalues corresponding to odd eigenfunctions of L over F accumulate at infinity. It can also be shown that the number of proper eigenvalues corresponding to even eigenfunctions over F is also infinite. The asymptotic distribution of the proper eigenvalues of L can be related to the asymptotic behavior of the scattering operator, see Theorem 8.6, p. 205 of [6]. Finally we note that if -v^ is an eigenvalue of L with eigenfunction/?: i4> = -y^P.

(7.1)

then

g± = ( A ±iyp) are eigenvectors of A with eigenvalues ± />: Ag^ = ±ivg^.

(7.2)

8. The Selberg trace formula. The hyperboHc theory developed here lends itself to two distinct versions of the formula which Selberg derived xising the concept of trace for operators of trace class. On the one hand the trace of such an operator is equal to the sum of its eigenvalues; on the other hand when the operator is an integral operator on L2 with a smooth kernel, it is 173

SCATTERING THEORY FOR AUTOMORPHIC FUNCTIONS

285

equal to the integral of the kernel along the diagonal. For Hennitian trace class operators this is standard; the general theorem is due to Lidskii (see [3]). In our first version of the trace formula we need only the Hermitian case, but in our second version we require the general theorem» We now give a simple illustration of how the trace formula is used. Let q>(t) be a real valued smooth even function which decays reasonably fast as |/| -» 00. Given a positive real number /?, let H be the space of /^-periodic functions endowed with the 1^2(0,/;) topology. Finally define K ^ K^ as the convolution operator:

( V ) W = r 'P(')>"(^ -t)dt = q> *f.

(8.1)

•'-00

Clearly K^ maps L2(0, p) into itself and is Hermitian. When written as K^(x) = 2

r^'^'


+ rq,)f{t) du

we see that its kernel: 2 ^>{x - t + np) is smooth; therefore K^ is of trace class and its trace is equal to

The eigenfunctions of K^ are exp {2mmt/p% m G Z; the eigenvalues are

where

P^


= S ^{^}

(8-2)

this is the Poisson summation formula. Returning to the problem at hand, let U(t) be the solution operator for the automorphic wave equation, defined as in §2, and let A denote the infinitesimal generator of the group U. As we saw earher, the domain of A can be reduced to two ^'-orthogonal subspaces, one spanned by the proper eigenvectors of A and the other spanned in some sense by the generalized eigenvectors corresponding to the continuous spectrum of ^. We denote the restrictions of U to these subspaces by U^, and f/^, respectively; clearly U{t) = U^{t) + U^it).

(8.3)

Next we denote by %, the space of initial data of the form {0,/}. Note that E is positive definite on Xj, in fact X, == L2{F). Suppose that u{t) is a solution of the automorphic wave equation with initial data in 0C|, that is u(0) = 0. Then -u(-t) will have the same initial data as w(0, which shows that u(t) is an odd function of t. It follows that U{t) + U(-t) maps % into %. Let cp be a real valued function which is even and decreases exponentially as |/| -» 00. We define Ä^ = Ä^ by K^^r

U^{tMt)dt

174

(8.4)

286

P. D. LAX AND R. S- PHILLIPS

restricted to SC,. Cleariy K^ maps OC, into OC, and is Hermitian. If -vf is a negative eigenvalue of L with eigenfunction/?y, then {O^pj} is an eigenvector of K^ with eigenvalue /•OO

e'Vcp(0^r = (p(i.^.);

(8.5)

as before 9 denotes the Fourier transform of cp. As we noted eariier, L also has a finite number of positive eigenvalues A/, 7 = 1, • . . , m (see (5.1) and (5.2)); these give rise to eigenvalues 00. To this end we further impose some mild assumptions on

tr A^ = E Hvj) + S
(8.6)

here rj is the multiplicity of the zero eigenvalue of L. Next in order to evaluate the trace of K^ in another way, we express K^ as an integral operator. Using (8.3) and (8.4) we write K^'-J[U{t)-U,{t)]q>{t)dt.

(8.7)

We can express U^ in terms of the operational calculus as 1/^(0 = exp^^r, where A^ is the absolutely continuous part of A. Now in § 6 we have expUcity determined the spectral representation of A^. The generalized eigenfunctions of A^ are given by the extended Eisenstein series e(w, a). Since A^ acts as multipUcation by ia in the spectral representation, any function $(-L4^) of -lA^ is an integral operator with kernel J ^{a)e{wy o)e{w\ o) do. In particular / ^c(0v, o)e{w\ o) do. (8.8) / Next we show how to write U{t) as an integral operator. Denote by U^it) the solution operator for the non-EucHdean wave operator over all of 11. Since L is invariant under the non-Euclidean motions, so is U^i)\ in particular if /Q is automorphic in 11, so is UQ(t)fQ. Thus if we denote the restriction of/Q to F by / ~ XFJ(P

175

SCATTERING THEORY FOR AUTOMORPHIC FUNCTIONS

287

then Since

/o=Sr V> where we conclude that

U{t)f^XF2

UJJ.

(8.9)

r The non-Euclidean wave equation in 11 can be solved explicitly: For initial data w(w,0) = 0, t/,(w,0)-/(>v), we have V8 TT ^r<\t\ Vcosh t — cosh r where r is the non-Euclidean distance from w to w' and ^ ' is the nonEuchdean area element. Using this formula we can represent the restriction of UQ to Xj as an integral operator with a distribution kernel Ä, called the Riemann function: R{w, w\ t) = ^ ^ d. ^ —. V8 TT Vcosh t — cosh r Substituting this into (8.9) we see that the kernel of / U{t)(p(t) dt is

- i -

/
V 8 TT -^

r

,

'

^^

(8,10)

(8.11)

ycosh r - cosh r^

Following Selberg we separate the sum in this integral into four parts corresponding to the identity, the hyperbolic, the eUiptic and the parabolic elements of V. Within each part we group the terms into conjugacy classes of r . Accordingly, (8.11) is broken into the sum of four kernels, three of which are of trace class. The trace of the part coming from the identity is

area(F) r

^\t)

f^fn'"-

(«•'^)-

47r J sinh t/2 For the modular group area (F) = 7r/3. The trace of the part coming from the hyperbolic elements is the weighted sum

? J o 2|smh(W2)| "^"^

^^'^^

where the first sum is over all inequivalent primitive hyperbolic elements T of r , and / = /(T) is defined by 2 cosh / = tr T. 176

(8.13)

288

P- D. LAX AND R. S- PHILLIPS

The trace of the part coming from the elliptic elements is a sum of weighted integrals: T

1 J

smh H/L + sm kn/m

where the first sum is over all inequivalent primitive elliptic elements r and m = W(T) is the order of r, i.e. the smallest m such that T'^ = /. Alternatively 2 cos ir/m = tr T. The part corresponding to the parabolic elements is not of trace class, but its difference from cp(-i^e) i^- Using (8.8) the trace of the difference can be expressed as a weighted integral of 9; for the modular group this difference is the sum of two terms: 1 cp(O) - log 2
/
(8.12),

and \
/ ^^^^\ -£ log sio) do;

(8.12),

here s(a) is the scattering function appearing in formulas (4.7)' and (6.18). For arbitrary discrete subgroups (with noncompact fundamental domain of finite area) s(o) has to be replaced by the determinant of the so-called scattering matrix, and the three terms in (SA2)^^j, have to be multipUed by the number of cusps, i.e. points at infinity, of the fundamental domain (which is the same as the number of inequivalent primitive parabolic elements in T). The trace formula says that (8.6) = (8.12)ia + (8.12)hyp + (8.12)en + (8.12)par + (8.12),oBf For further details we refer to §9 of [6] and to Kubota [4]. It should be remarked that a similar trace formula has been derived by Selberg for discrete subgroups whose fundamental domain F is compact. The formula and its derivation are simpler in this case since there is no continuous spectrum and since T contains no parabolic elements; see [9] for an illuminating exposition and a geometrical interpretation of the trace formula. We remark finally that when the procedure described above is applied to the trivial case of the classical wave equation for functions of x and / which are periodic in x, the operator (8.4) reduces to the operator (8.1) and, as remarked earlier, the trace formula becomes the Poisson summation formula^ We turn now to a description, even sketchier than the foregoing, of our other approach to the Selberg trace formula; this is described more fully in an appendix to §9 of [6]. It is based on the notion of trace for non-Hermitian operators. Let Z(t) denote the semigroup of operators Z ( 0 = /^H.t/(0^-,

(8.15)

where, as in §3, P^ are the orthogonal projections onto ^ i and U(t) is as before; we take as the domain of Z{t) all of the original Hilbert space %Q. As so defined Z(t) does not converge strongly to the identity as / -> 0; however 177

SCATTERING THEORY FOR AUTOMORPHIC FUNCTIONS

289

the restriction to ^ = 9C^ 0 (^^ © ^ ^ ) does have this property. We take a function
Z{t)q>{t)dt.

(8.16)

We claim that C^ is of trace class. For proof we refer to [6], where two demonstrations are given. In the second proof, contained in Appendix 2 to §9, it is shown that C^ is an integral operator whose kernel is smooth except for a jump dicontinuity along the line>' = a (caused by the action of P^ and that the kernel decays faster than any power of >^ as >^ -^ oo. The argument showing this decay is based, appropriately enough, on the Poisson summation formula. We now determine the eigenvectors of C^. Aside from the null vectors of C^, which do not contribute to the trace, all of these are eigenvectors of Z{i) restricted to 5C. It is shown in the appendix to Chapter 9 of [6] that these eigenvectors are obtained, by P^ projection, from the eigenvectors of U{t) which are orthogonal to ^ Q . We treat first the proper eigenvectors of U{t). We saw in §7 that L has infinitely many eigenfunctions/? corresponding to negative eigenvalues -v^ Lp = -v% Integrating this relation with respect to jc, we get for the zero Fourier coefficient/7^^^ the ordinary differential equation:

all of whose solutions are of the form Since -v^ is in the point spectrum,/? belongs to L2(F) and this requires both a and b to be zero, i.o.p^^^ = 0. We saw in §7 that ifp is an eigenfunction of L, then g± = {Py ±i^P) are eigenvectors of U(t)y with eigenvalues exp(±i>/). The condition/?^^ = 0 impHes that g^ are orthogonal to both ^ _ and ^ + , i.e. that P^ g:t = ^+ ^± = g±' It follows that g^ are eigenvectors of Z{t) as well, with eigenvalues exp(± ivt); hence g^ are also eigenvectors of C^, with eigenvalues 4>{±pj).

(8.17)

We have noted in §5 that L has a finite number of positive eigenvales X"^; denote the corresponding eigenfunction by q. Then {q, Xq] is an eigenvector of C/(0, with eigenvalue exp(X/). As remarked in (5.6), for X > 0, {^, Xq) is orthogonal to ^ . ; from this one can deduce, using the relation (3.7), tiiat ^+ i^y M] is an eigenvector of Z(t) with eigenvalue e x p ^ ) , and therefore of Cy with eigenvalue
(8.18)

290

P. D. LAX AND R. S. PHILLIPS

The vector (^, -\q} is also an eigenvector of (/(/), ^ this case orthogonal to ^ ^ , but not in general orthogonal to ^ _ . However if it happens also to be orthogonal to ^ _ , then as above {^, -\q} is an eigenvector of Z(t) with eigenvalue exp(-AO, and of C^ with eigenvalue 4>{i^).

(8.18)'

We turn now to the generalized eigenvectors of U(t) which are orthogonal to ^ _ • The corresponding eigenvalues appeared in the proof of Theorem 4.2 as poles of the meromorphic continuation of the scattering function s{z) into the upper half plane. According to this theorem, these poles can occur only at points of -ia(B); here B denotes the infinitesimal generator of Z(/) restricted to %. Taking the residue of the Eisenstein series about such a point, it is clear from (4.7) that the resulting generalized eigenvector of A will be orthogonal to ^ _ and hence, after projection by F^., will be a proper eigenvector of B. The corresponding eigenvalue of J5 is a complex number ft^. Re jt^ < 0, that of Z(t) is cxp(fijt) and that of C^ is
(8.19)

Finally we denote by TJQ the multiplicity of 0 as an eigenvalue of B; then (p(0) is the corresponding eigenvalue of C^ and it has multiplicity %• Thus the sum of the eigenvalues of C^ is obtained by adding (8.17) to (8.18), (8.18)', (8.19) and to the above. Altogether we get

2 4>{±^) + 2 4>{-ih) + S '
2 H-%) ~ E''
/ [4>{o) +
(8.21)

Since the poles of s(z) in the upper half plane coincide with the set -iiif and the so-called relevant iX^% this formula can be verified by deforming the integral on the right from the real axis off to infinity. This can be justified on the basis of appropriate estimates for s(z) away from the poles; for the modular group such estimates are available. In order to verify (8.21) in general we use a "mini-trace formula" for the operator

c;^jz"{t)
SCATTERING THEORY FOR AUTOMORPHIC FUNCTIONS

291

on %'\ where and Z"(/)=

PlU{t)Pl;

here ^4' denote that subspace of ^.^ which is orthogonal to the subspace ^ defined in (5.4) and P^ are orthogonal projections onto OQ © ^Z- As a by-product of this analysis we find that T/Q = Irf. For details see the appendix to §9 of [6\. Appendix: A new proof of Theorem 2J. Theorem 2.1 is a restatement of Theorem 6.6 in [6]. Since the initial steps of the two proofs are the same we shall refer the reader to [6, pp. 126-130] for the proofs of some of the initial lemmas. THEOREM 2.1. For every X in the resolvent set of A, QJ — Ay^ maps the unit ball in % into a compact subset of %Q. REMARK. Since it suffices to prove the theorem for a single value of X, we shall for convenience take A to be real > 4. Theorem 2.1 is a consequence of a simpler result: THEOREM

A.l. Let cp be any C^ function on R^. Define the operator

M^M^by M^^r

q>{t)U{t)dt.

(A.1)

Then M^ maps the unit ball in % into a compact subset of %Q. The proof of this theorem is based on the following two propositions; LEMMA A . 2 . (a) The set

u = Mf G{f) < 1, isprecompact with respect to the norm Gy(u) = / /

{|«,..|^ + \u,J

(A.2)

+ \u,f + \u,\^} dx dy

(A.3)

for any Y; here F( Y) denotes the domain F{Y) = F r\

{y
(b) Given any e > 0, there is a Y > a such that

G^iu)^fl {wa + y%^

V7

"•

>u.<'

+ ^]Jxdy<

eG{f)

(A.4)

for all u of the form (A.2) with fin %. Part (a) is a simple consequence of Rellich's compactness criterion. We note that the imposition of a finite number of linear continuous constraints on the domain of M does not affect the precompactness argument. Now fl^\a) == 0 is just such a constraint since by Lemma 4.2 PROOF OF PART (b).

180

292

P. D. LAX AND R. S. PHILLIPS

in [6] l/r(«)P < f

l/i(^. «)P dx < const GUy

•^-1/2

Since for / in %, fPiy) = 0 for all y>a,ii follows by (2.17) that we can assume without loss of generahty that/^^\>') = 0 for all 7 > a. Fixing ' = be^) will not get above >^ = 2be^^ (or below 7 = b) during time t e (0, T). According to (A.l) and (A.2), w is a superposition of values of f/(/)/, 0 < / < T. It follows that values of u at points where7 > 2be^^ (ory < b) do not depend on values off below>^ = 2be^ {or above>? = be^). This suggests that we spht/as f-g

+ h

so that g = 0 lory < be^y /i = 0

for>^>2^e^

and so that G ( g ) < const G(/). We can do this in such a way that the constant is independent of our choice olb > a, which will be fixed later on. Since part (b) of the proof is concerned only with values of u for larger, h plays no role in what follows. We therefore need only consider g and to keep our notation simple we shall hereafter denote g by /. We may therefore assume that/satisfies /=0

forall^
(A.5)

In this case w = 0 for all y
'1/2

e-'''''u{x,y)dx

(A.6)

'-1/2

are obtained directly from M acting only on the corresponding Fourier coefficients of/; that is, the Fourier coefficients of f/(0/remain uncoupled as long as the signal stays above >^ = a as it does for 0 < / < 7\ In particular t/<^> = 0. We note that G and E are identical for data vanishing for y ^, we have

(?(/)=/ / /

laj.P + j

-1/2

v7

I/2P dx dy

\A''\ k¥^0 ^a

I

J

181

dy.

(A.7)

SCATTERING THEORY FOR AUTOMORPHIC FUNCTIONS

293

We now set p^{\I^

A)u=^j (X(p +

M)U{t)fdL

(A.8)

Clearly G{p) < const G{f).

(A.9)

Further writing (A,8) in component form, we get Atij — W2 = Pi,

A1/2 — Lui = P2J

(A.8)'

so that X^w, — y^^Ui — Wj/4 = Api + /^2 = ^•

(A.S)"

Applying (A.7) top, we get (A.10) ifc#0 -'b

and combining this with (A.9) we see that 2

/

k^o Jb

-^^-T" ^ < const G(/») < const G{f). yr

(A.11)

Finally it is clear from (A.8)" that the A:th Fourier coefficient of w, satisfies the equation

~ - 4 "1'^ + (^^'^' + -)\ "1'^ = ^ '

(A-12)

where ic = A^ — 1/4. LEMMA A . 3 , Let t? G C ^(R) am/ / ^ |t)p dy < oo, then there exists a sequence yj -> 00 for which Re ü'tJ|^ -^ 0. PROOF.

Obviously lim

(^^^ \v\^(fy^{).

By the law of the mean there is y/ in {j\j + 1) for which Moreover

v(y/)-^0.

HyUi)? - Hy;)\' = P^^ ^ |t^P 4^ = 2 p - Re (v'v) cfy. Since this expression converges to zero and since^^^2 "~ yj > h the law of the mean again furnishes us with a>^ in (yj, yj^2) satisfying the statement of the lemma. We return now to the proof of part (b) of Lemma A.2. Multiply the relation (A. 12) by MJ^^ and integrate by parts. Take the real part of the resulting relation and use Lemma A.3 at the upper limit of integration and the fact u\^\b) = 0 at the lower limit. This gives us

JJ' Iwn^ + (4^'^' + -^)l«!*f I dy^lRclJ" ^1^ dy. 182

294

P. D. LAX AND R. S. PHILLIPS

Applying the Schwarz inequality to the right member gives us ^b

^b

r

y^

^b

and transposing the last term on the right we get

•fb

y

Jb

y^

From the first relation in (A.8)' we get

^b

y^

^b


y^

^b

y

A'/"l«<*f^+/"b(*)p^l. -'b

Jb

(A.14)

J

It now follows from •2 Jh

I -^ V 1 7

I

Jb

y^

^b

'^b

together with (A.7) as applied to M„ (A.IO), (A.11), (A,13) and (A.14) that ^ . V const ^, .. G{u) < - — - G ( / ) , where the constant is independent of our choice of h. Hence if we choose h so that const/Z?^ < c and Y > Ibe^^^ then we get the desired inequahty for part (b) of Theorem A.l.

REFERENCES 1. L. D. Faddeev, Expansion in eigenfunctions of the Laplace operator in the fundamental domain of a discrete group on the Lobacevskii plane^ Trudy Moscov. Mat Obsc. 17 (1967), 323-350; see also English transL, Trans. Moscow Math. Soc. 17 (1967), 357-386. 2. L. D. Faddccv and B.S. Pavlov, Scattering theory and automorphic functions, Proc. Steklov Inst. Math. 27 (1972), 161-193. 3.1. C. Gohbcrg and M. G. Krcin, Introduction to the theory of linear non-selfadjoint operators, Transl. Math. Monographs, vol. 18, Amcr. Math. Soc., Providence, R. I., 1969. 4. T. Kubota, Elementary theory of Eisenstein series, Wiley, New York, 1973. 5. P. D. Lax and R. S. PhiUips, Scattering theory. Academic Press, New York, 1%7. 6. , Scattering theory for automorphic functions, Ann. of Math. Studies, no. 87, Princeton Univ. Press, Princeton, N. J., 1976. 7. , The scattering of sound waves by an obstacle, Comm. Pure AppL Math. 3© (1977), 195-233. 8. , Translation representations for the solution of the nan-Euclidean wave equation, Comm. Pure Appl. Math. 32 (1979), 617-667.

183

SCATTERING THEORY FOR AUTOMORPHIC FUNCTIONS

295

9. H. P. McKcan, Selberg's trace formula as applied to a contact Riemann surface^ Comm. Pure Appl. Math. 25 (1972), 225-246. 10. A. Sclbcrg, Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc. 20 (1956), 47-87. 11. M. A. Scmcnov-Tian-Shansky, Harmonic analysis on Riemannian symmetric spaces of negative curvature and scattering theory. Math. USSR Izvcstija, vol. 10 (1976), 535-563. COURANT INSTTTUTE OF MATHEMATICAL SCIENCES, N E W Y O R K UhOVERSITY, N E W Y O R K , N E W YORK

10012

DEPARTMENT OF MATHEMATICS, STANFORD UNIVERSITY, STANFORD, CALIFORNIA 94305

184

Translation Representations for the Solution of the Non-Euclidean Wave Equation. II PETER D. LAX Courant Institute

AND RALPH S. PHILLIPS Stanford University

Introduction This is a continuation of our paper [3]. The present paper begins with Section 5; references to Sections 1 through 4 pertain to the original paper. In Section 5 we derive Friedlander's formula for the outgoing translation representation of solutions of the non-Euclidean wave equation and prove the eventual equipartition of energy for such solutions. In Section 6 we study the Radon transform of Kf, where K is an invariant integral operator, and show that K can be expressed as a superposition of the operators U{t) with the Selberg transform of K as weight factor. In Section 7 we correct a small lapse in the proof of Theorem 3.8. In a subsequent paper we shall extend many of the results in these two papers to automorphic solutions of the wave equation.

5. Friedlander's Formula L e t / = (/,,/2) be initial data of finite energy for the non-Euclidean wave equation in hyperbolic three-space H and let k— r_^ / be its outgoing translation representer defined by formula (3.27). Denote byy the origin chosen in H and let (/,y) be non-Euclidean polar coordinates around y, i.e., w{t,y) is that point on the geodesic issuing from j in the direction y whose distance from j is /. The direction y can be thought of as a point on the infinity set B bounding H. THEOREM 5.1. Let u{wj) denote the solution of the non-Euclidean wave equation (3.22) with initial data f. Then

(5.1)

^lim/c(5,y;/) =

k{s,y),

where (5.2)

k{s,y;t)-^^smh{t

+ s)u,{w{t + s,y)J)

for

s>-t

and equals 0 otherwise; here the limit is in the L2{%B) sense. Communications on Pure and Applied Mathematics, Vol. XXXIV, 347-358 (1981) © 1981 John Wiley & Sons, Inc. CCC 0010.3640/81/030347-12$01.20

185

348

P. D. LAX AND R. S. PHILLIPS

Proof: We shall use formula (3.41)^. expressing u in terms of its outgoing translation representer k defined by equation (3.27): (5.3)

u{w,t) = \c\ rexp<w, ß}k{{w,

ß} ~ /,

ß)dß.

JB

Here it is convenient to use the unit ball model of 11 with j = 0; in this model (5.4)

<w,)S> = log

1 - kP l - 2 w i ß + |wp '

where [>v| denotes the Eudidean absolute value and w • ß the Euclidean scalar product (see (1.16)). In terms of EucHdean polar coordinates this becomes (5.4)'

<w,)ß> = log

^~^' 1 — 2rcos^ + r^

where r = |>v| and 9 is the angle between w and ß. The relation between the Euclidean distance r from j to w and the nonEuclidean distance R is given by

.« = i ± i .

(5.5)

1— r For w = >t^(^T) in (5.4)', we have R = t so that (5.5)'

^' = T ^

'

and 9 is the angle between ß and y. As ß varies over B, (w, ß} varies between logi^ I+r

and

log|±^, 1 —r

i.e., in view of (5.5)', between — / and /. We introduce in B spherical polar coordinates around y as pole; then (5.6)

dß^sm9d9dq)=

-dpdtp,

where (5.7)

p = cos9

and

cp is the angle of rotation around y.

Furthermore we introduce the new variable (5.8)

r = (w,ßy=^\og

186

\-'' ^. 1 — 2rp + r"^

TRANSLATION REPRESENTATION FOR THE WAVE EQUATION. II

349

Clearly, dr dp

2r 1 ~ 2rp + r^ '

so that (5.9)

e^'^'^Uß^

^"^^'

^dTd(p^

^LzAcirdq).

Differentiating (5.3) with respect to /, we obtain: (5.10)

u,{w,t) = -|c|Je<-^>Ä:X<w, iß> - /, ß)dß

which by (5.9) is equal to

We shall show first, when k is C j , that the Hmit (5.1) holds pointwise uniformly if s is restricted to a compact set. To this end we may replace, in (5.1), t\yj t — s and sinh(/ + 5) by \e\ and using (5.5)' and (5.10) write

(5.11)

= |C|J_"£ ^-j^k\T-t

+ S,ß)d7dtp.

Suppose that k{s, ß) is supported on — l^ s ^ I; then the r-integration in (5.11) is restricted to (/ — / — 5,/). It follows easily from (5.5)' and (5.8) that if r lies in this interval, then p tends to 1 as / tends to 00. Since p is the cosine of the angle between ß and y it follows that, for k G CQ, ß in (5.11) is restricted to a neighborhood of y which shrinks to y as / -> 00. Hence for such k we can replace ß by y in (5.10) at the cost of an error that approaches 0 as t-^00. Also, as /-> 00, r tends to 1; thus we can replace (1 + rf /Ir by 2, again at the cost of a small error. Consequently, (5.11) can be written as e'^(iv(r,y),/ - 5) = 477|c| V

k\r - t + s,y)dr+

Jt-l-s

= HC|/C(5,r)+£(/).

187

e{t)

350

P. D. LAX AND R. S. PHILLIPS

Since' c = -{2^/2 7Ty\ this shows that the relation (5.1) holds when k is C j , uniformly for s restricted to a bounded set. To complete the proof of Theorem 5.1 we need LEMMA 5.2. Denote by k(s,y,t) L2(R,B) norm by \'\. Then (i)

the function defined in (5.2) and denote the

\k{tf^2\k\\

(ii)

\k{t)\^{\

+ e{t))\kl

where the function €(/) tends to 0 as t-^ oo; €{t) depends on k. Before giving the proof of Lemma 5.2 we show how it can be used to complete the proof of Theorem 5.1. So far we have seen that (5.iy

\\m^k{s,y;t)=-k{s,y)

uniformly on compact subsets of R X 5 . According to part (ii) of Lemma 5.2,

where €(0->0 as r-^ oo. This combined with (5.iy implies that (5.1) holds in the L2 sense for initial data / in C j . To prove (5.1) in the L2 sense for all k in Lj we merely observe that Cj is a dense subset of L2 and that by part (i) of Lemma 5.2 the L2 norm of k{t) is uniformly bounded by twice that of k. The result then follows by the principle of dense convergence. We turn now to the proof of Lemma 5.2. Using the definition of k{t) we have \k{t)\^ = r r'^2sinh^(r + s)\u,{w{t +

s,y),t)\^dsdy

(5.12) = 2| J

smh^s\u^{\v[s,y),ty(^dsdy.

'According to (3.27), c = —(v2 77)"^ this is because m{B) was chosen to be TT. If a factor 4 is introduced in (1.17) so that the measure of B if 47r, then the factor in (3.1), (3.7) and (3.10) becomes (27r)~^/^ the factor in (3.12) becomes -(27r)'/^ the factor in (3.13) and (3.14) becomes (STT^)" ' and c becomes

188

TRANSLATION REPRESENTATION FOR THE WAVE EQUATION. II

351

Now the formula for the volume element in non-Euclidean polar coordinates is sm\i^s ds dy. Hence (5.12) can be rewritten as (5.12)'

\k{tf = l\u.{t)\\

i.e., as twice the integral of u^ squared over n . According to the definition of energy \u\^ in (3.23), \uV. PE = U/1 H I + \u. In view of the non-negativity of energy, see (3.24)', we conclude that |w^P = |t/|£. Finally, according to Theorem 3.4, formula (3.29), the translation representation is an isometry, i.e., (5.13)

\u\\^\k\^;

this completes the proof of part (i). To prove part (ii) we need the following result on the equipartition of energy: Let u{t) be a solution of the non-Euclidean wave equation which has finite energy. Then the total energy, defined by (3.23),

\\-\u\\ |2 is an invariant of the; motion; however the individual constituents |w(0!i ^^^ |t/^(/)P are not. Nevertheless the following asymptotic relation holds:

THEOREM 5.3.

(5.14)

As t-^oo,

\u{t)\l/\u,{tf-^\.

Note that (5.14) implies that 2\u,(t)\^-> \u\% and this combined with (5.12)' and (5.13) proves part (ii) of Lemma 5.2. The Euchdean version of Theorem 5.3 is known, see Corollary 2.3, Chapter IV of [1]; the key point of the proof is the orthogonality of ^ _ and ^ _^. Since this no longer holds in the non-Euclidean case, an additional argument is needed here. It suffices to prove Theorem 5.3 for a set of solutions dense in the sense of the energy norm. We choose for this dense subset the set of all data/whose outgoing translation representer k ^ k{s, ß) vanishes for 5 < —b,b some number. We decompose such a k into (5.15)

k^k^ + k^''^.

189

352

P. D. LAX AND R. S. PHILLIPS

where k;^ contains all spherical harmonic components of order at most N and /c^'^^ contains all of the rest. N is chosen so large that (5.16)

|A:^'^>|<€.

We further decompose kf^{s — t) into incoming and outgoing parts (see Theorem 3.9) as follows: (5.17)

k^{s - t) = 4'>(^,0 +

k)}\s,t),

where k^j^^ is of the form (5.18)

4'^(^,iß;0= [

o<mf„SN " ' " ^ ^ 0,

"^ ^ 5 < 0,

the coefficients a^„ being chosen so that k^fp is orthogonal to 4 ^ *

(5.19)

(4)(0,4'^(0) = 0.

This can be accomplished by applying the inverse of the Gram matrix of the functions e~'^^Y^{ß), 0 < m^ n^ N over R^ to the array of numbers (5.20)

/ / " ^ ( ^ "^ '^

ß)e-^'Y„{ß)dsdß.

Changing variables top = s — t, and using the fact that k{p) = 0 for/; < — ^, we can rewrite (5.20) for / > Z? as (5.20)'

e-'"'jj^k{p)e-^PY„{ß)dpdß.

Obviously the quantities (5.20)' tend to 0 as ?->oo; consequently a^„(t)-^0 t-^oo. It follows then from (5.18) that (5.21)

as

\im\kj,'\t)\==0.

We recall now the definitions of outgoing and incoming solutions of the non-Euclidean wave equations as ones vanishing in the respective forward and backward light cones: dist(>v,y) < / and dist(>v,y) < — t, respectively. Clearly, if u{t) is outgoing then u{— t) is incoming. The subspaces ^^ and ^_ are defined as the Cauchy data of outgoing and incoming data, respectively. Theorem 3.9 characterizes the outgoing translation representers of ^^, as consisting of those k{s, ß) which vanish on ^ < 0; and the outgoing representers of ^ _ are characterized as those k which are of the form (5.18) for 5 > 0.

190

TRANSLATION REPRESENTATION FOR THE WAVE EQUATION. II

353

Since k{s, ß) is assumed to vanish for ^ < —b, k{s — t^, ß\ which is the representer of {u{t^, «/(^o)X vanishes on 5 ^ 0 when t^^b. The same is true then of k^{s - /o, ß) and of k^^\s, ß; g . Denote by wj^^\>v,r;/Q) the solution whose initial data is represented by kj;J\s, ß; t^). Then it follows from Theorem 3.9 that (i) ^NXQ is outgoing, (ii) the initial data of ujJ\tQ) is orthogonal to ^ _ . As noted earlier, it follows from (i) that u^^\w, — t; /o) is incoming; therefore by (ii) its initial data and those of wj^^^w, t; tg) are orthogonal in the energy norm:

Thus the energy scalar product of these two is zero; i.e., (5.22)

K \ 0 ; ? o ) l i - | < ; ( 0 ; / o ) P = 0.

In other words, the energy contained in

M^'^^O)

i^ equipartitioned at f = 0.

By construction of k^^\ kjj^ and ki}\ k{s - to) - 4'^(^.0 + f^lf\s,to) + k''{s-

to).

Hence the same relation holds for the data represented by these functions, and for the solutions corresponding to these data: (5.23)

u{t + to) = ujj\t; to) + u'<^\t; to) + M<^V + ^)-

It follows from (5.16) that (5.24),

1"''('O)IE<^-

Relation (5.21) implies that, for to large enough, (5.24)2

Wrf\to)\E<^.

Finally, by (5.22), energy is equipartitioned in u^fj\t\t^ at / = 0; Therefore, we conclude from (5.23) that IIK^O)IL~I^(^O)I|<2€.

This proves Theorem 5.3.

191

354

P. D. LAX AND R. S. PHILLIPS

6. Invariant Integral Operators Invariant integral operators were introduced by Selberg in [4], These are operators which commute with motions. An integral operator K is of the form (6.1)

g{w) =^ Kf^

f

K{w,w')f{w')dw\

where dw' is the non-Euclidean volume element. Commutation requires that, for every motion r,

where r(>v)=/(T-'w). It is easy to show that K commutes with motions if and only if its kernel satisfies the relation

Such a kernel is called a two-point invariant function by Selberg and is simply a function of the non-Euclidean distance d(w, w'). Recall that cosh d(w,w') = 1 + '—

^

,

where | • | denotes Euclidean distance. We can therefore write (6.2)

K(.,.') =

k(t^^

SO that (6.1) can be expressed as

(6.3)

^(,)=J^A:(lli^)/(.')*v'.

In view of the above, the right side of (6.1) is like a convolution. In the Euclidean case the Radon transform of the convolution of two functions is the convolution of their Radon transforms with respect to the scalars; see [2] for an application of this observation. We now prove that in the non-Euclidean case the Radon transform of (6.1) is the convolution of the so-called Selberg transform of K with the Radon transform of/.

192

TRANSLATION REPRESENTATION FOR THE WAVE EQUATION. II

355

The simplest way to compute the general form of the Radon transform of Kf is to evaluate it at /? = oo; in this case the horospheres are the planes: (6.4)

z = e\

Hence using (6.3) we get

S{'^^)^^^,SW

dxdy

where r^ = (x — x')^ + (7 ~ 7')^- Introducing r^ + (z ~ z'f V =

Izz'

as new variable, we get

(6.5)

i(-)=i,(H:.>v./''')*7)/(-)^.

Performing the x\ /'-integrations, making use of (6.4), and setting (6.4y

z' = ^^

this becomes (6.6)

1(5, 00) = 277 f^

f f"^

k{x))dv\j

J-00 \J{z-z')^/2zz'

}

Now

2zz'

2 \ z'

z /

= cosh(5 - r) - 1.

193

{t,Qo)e'-'

dt.

356

P. D. LAX AND R. S. PHILLIPS

If we define the Selberg transforai of K to be (6.7)

ß(5) = 27rr°°

k{v)di,,

Jcosh s — 1

then we can write (6.6) as a convolution: e'g{s,<x^)^r Q{s-t)e'f{t,cx,)dt. By invariance this holds for any ß: (6.8)

e'g{s, ß) = r

Q{s - t)ef{t, ß)dt.

Finally differentiating with respect to s and integrating the resulting expression on the right by parts, we obtain

We define the action of the operator K on data F = {/i,/2} by letting it act independently on each of the component functions/j a n d / 2 . Denote by R the operator which assigns to F its, say, outgoing translation representation. From the definition (3.27) of R and formula (6.9) we deduce the simple relation RKF-^ { Q{s- t)RFdt (6.10) -^Q{t){RF){s-t)dt. Since /? is a translation representation for the unitary group U (see (3.28)), we deduce from (6.10) that (6.11)

RKF=^ f

Q{t)R{U{t)F)dt,

which can be stated as: THEOREM

(6.1 ly

6.1.

For the operator K we have the relation K^^Q{t)U{t)dt,

where the integral is to be taken in the strong sense.

194

TRANSLATION REPRESENTATION FOR THE WAVE EQUATION. II

357

Thus every invariant integral operator can be expressed as a superposition of the operators U{t) with the Selberg transform as weight factor. 7. Correction to the Proof of Theorem 3.8 The correct version of the argument goes as follows: Set (3.47) into (3.46); then, since v{w) = 0, we get instead of (3.49) (7.1)

0 = d^v{w%^j ^jk{s,

ß)Q^{d,, ß)<^{'-s)dsdß

and instead of (3.51) (7.2)

0 = 9^t;(p>v)|_^. = J/:(^,py)ÖN(a„y)<>(-^)rf5^y.

Expand k into spherical harmonics: (7.3)

k{s,ß)^^kp{s)Yi,{ß).

For any rotation p about y we have (7.4)

>';(py) = SC(p)>^r(Y). m

Combining (7.3) and (7.4) gives us the spherical harmonic expansion of k(s,py). We set this expansion into (7.2), multiply by eß^{p) and integrate over all rotations p. In view of the orthogonality of the functions elJ^(p% we get (7.5)

0=jk;{s)Y„\y)Q^{d,,y){-s)dsdy,

Carrying out the y integration, we obtain (7.6)

0^jk;{s)P„{d,){--s)ds,

where

(7.7)

P„{d,)^JY!;{y)Q^{d,,y)dy.

Because of (3.48) the leading coefficient of P/^{d^) is

jy^Y„\y)dy;

195

358

R D. LAX AND R. S. PHILLIPS

if we choose A^ with \N\ = n so that this integral is not 0, then P^^idJ ^ 0. We then proceed in the way it was done at the bottom of page 652 of [3]. Acknowiedgement. The work of both authors was supported in part by the National Science Foundation, that of the first author under Grant No. NSF MCS-76-07039 and that of the second under Grant No. NSF MCS-77-04908.

Bibliography [1] Lax, P. D., and Phillips, R. S., Scattering Theory, Academic Press, New York, 1967. [2] Lax, P. D., and Phillips, R. S., The Paley-Wiener theorem for the Radon transform, Comm. Pure Appl. Math. 23, 1970, pp. 409-424. [3] Lax, P. D., and Phillips, R. S., Translation representations for the solutions-^f the non-Euclidean wave equation, Comm. Pure Appl. Math. 32, 1979, pp. 617-667. [4] Selberg, A., Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, Jr. Indian Math. Soc. 20, 1956, pp. 47-87.

Received October, 1980.

196

Reprinted from JOURNAL OF FUNCTIONAL ANALYSIS

Vol. 46, No. 3, May 1982

All Rights Reserved by Academic Press, New York and London

Printed in Belgium

The Asymptotic Distribution of Lattice Points in Euclidean and Non-Euclidean Spaces* PETER D . LAX Courant Institute of Mathematics, New York University, New York, New York 10012 AND RALPH S. PHILLIPS Department of Mathematics, Stanford University, Stanford, California 94305 Communicated by the Editors Received October 22, 1981 DEDICATED TO ATLE SELBERG WITH ADMIRATION AND TREPIDATION

The asymptotic distribution of orbits for discrete subgroups of motions in Euclidean and non-Euclidean spaces are found; our principal tool is the wave equation. The results are new for the crystallographic groups in Euclidean space and for those groups in non-Euclidean spaces which have fundamental domains of infinite volume. In the latter case we show that the only point spectrum of the Laplace-Beltrami operator lies in the interval (—((m— l)/2)^, 0]; furthermore we show that when the subgroup is nonelementary and the fundamental domain has a cusp, then there is at least one eigenvalue in this interval.

L INTRODUCTION

Early on Gauss observed that the number of integer lattice points in a circle of radius s is equal to the area of this circle its'^ to within an error of no more than the circumference, that is, 0{s). Since then lattice problems of this sort have been extensively studied. The lattice points are the orbit of the origin under the action of the group of integer translations. This group can be replaced by any discrete subgroup of motions, for example, the group generated by any two linearly * The work of the first author was supported in part by Department of Energy under Contract DE-AC02-76 ERG 3077 and the second by the National Science Foundation under Grant MCS-80-01943.

280 0022-1236/82/060280-71 $02.00/0 Copyright © 1982 by Academic Press, Inc. All rights of reproduction in any form reserved.

197

ASYMPTOTIC DISTRIBUTION OF LATTICE POINTS

281

independent vectors or the crystallographic groups; and the plane can be replaced by a Euclidean space of any dimension. The number of these orbital points inside a ball of large radius can then be computed asymptotically. In the non-Euclidean analogue, we count the number of orbit points in the (non-Euclidean) ball of radius s about a point w which are generated by a discrete subgroup F of the motions of hyperbolic space: N{s; w, Wo) = #[^ ^ ^; dist(>v, zw^) < s].

(1.1)

If r has a fundamental domain F of finite volume |F|, the main term in'N{s) is again equal to the volume of the ball of radius s divided by |F|:

2'^-\m-l)\F\ However, since the surface of the ball is also 0{e^^~^^^), a surface error approximation is useless. Moreover when \F\ is infinite, the above way of even estimating N is useless. However, it turns out that both N{s) and the error term can be expressed in terms of the eigenvalues and the eigenfunctions of the Laplace-Beltrami operator, even when |JF| = oo. The study of the counting number in the non-Euclidean setting was initiated by Huber [5, 6], who treated Fuchsian subgroups with compact fundamental domains in 2-dimensions. Somewhat later Patterson [12] treated all discrete Fuchsian subgroups in 2-dimensions with fundamental domains of finite area. Selberg (see[l]) solved the same problem in real hyperbolic spaces of arbitrary dimensions, again for fundamental domains of finite volume, Selberg's error estimates are significantly better than those of Huber and Patterson. In the present work we extend the above non-Euclidean results to all discrete subgroups with the ßnite geometric property, that is, to all subgroups for which the polygonal representation of the fundamental domain has only a finite number of sides; the volume may be finite or infinite. For F a discrete subgroup with the finite geometric property, the Laplace-Beltrami operator has at most a finite number of eigenvalues Mi^Mi^ ''' ^MN lying above the continuous spectrum: that is, ju^ > —({m — l)/2)^. Denote the corresponding normalized eigenfunctions by {(Pj\j= 1,..., A/^}; (pi{w) > 0 and is constant if and only if | jp| = vol(F) < oo. Our main result requires the existence of at least one //,.; it can be stated as follows: THEOREM 1.

Set

J=\/i'j+{^)

««^ 198

^ = ^1-

(1-2)

282

LAX AND PHILLIPS

Define Z = E{s; w, WQ) by

'^'^=I^^'

^ (A. X-my. ''^''^ "^'''^ ^""' '"""• ^'-'^

summed over the Xj > X(m — l)/{m + 1). Then as s-^ oo \N(s;w,w,)-i:{s)\

exp[(m-.)(! + ^ ) . ] )

for m > 2 ,


for m = 2.

^, ^^

In the case where | F | < oo our error estimate differs from that of Selberg by a power of s. Selberg's proof uses a particular class of integral operators that commute with non-Euclidean motions; we use a different class, derived from the wave equation. This enables us to base our error estimate on the principle of conservation of energy for the wave equation. It is clear from definition (1.1) that N{s;w,WQ) — N{s,yWjywQ) for any isometry y for which y^^ry = r. In the non-Euclidean problem the motions do not commute. It is not surprising therefore to see, when | F | = oo and hence when cpy is not constant, that the leading term in N{s; w, WQ) depends on w and WQ . In order to familiarize the reader with our method in a simpler setting, we prove the Euclidean analogue of Theorem 1 in Section 2: THEOREM

2. Let F be one of the crystallographic groups and set N{s; X, XQ) = #[r G T; |x - r^ol < s].

(1.5)

The number of orbital points in a ball of radius s in f(^ about a point x is Nis;x,Xo)=V^{s)-\-0{s'^^'^-'^/^"'^'^)

for m > 2,

(1.6)

where 0)^

ymis) = m\F\ CO^ = surface of the sphere of radius 1 in R*", \F\= volume of the fundamental domain F of the group F. When m = 2, Nis; X, ^o) = 1 ^ + 0(s^/^(log sY^'). 199

(1.6)'

ASYMPTOTIC DISTRIBUTION OF LATTICE POINTS

283

As far as we know, Theorem 2 is new for the crystallographic groups, except for groups of translations. In the latter case (L6), without the logarithmic factor when m = 2, was proved by Landau [7,8] in 1915. For integer (or "near" integer) lattices in dimensions >4, considerably better estimates have been obtained by number-theoretic arguments; see the book [15] by Walfisz. It might be of interest to know if better error estimates than the ones derived in this paper hold for the distribution of orbits of crystallographic groups whose translation subgroup is integral. The same question can be raised about arithmetic subgroups of non-Euclidean motions. The organization of this paper is as follows: Section 2 contains the proof of Theorem 2. Sections 3 and 4 deal with automorphic solutions of the nonEuclidean wave operator. The main result here is that if F has the finite geometric property, then the energy form E is positive definite on a subspace of finite codimension. This is equivalent to the statement that the spectrum of the Laplace-Beltrami operator in the interval [—{{m — l)/2)^, 0] consists of a finite number of eigenvalues. We also show in Therem 4.8 that if the fundamental domain has infinite volume, there are no point eigenvalues <-((m-l)/2)^ In Section 5 we prove Theorem 1, which requires that the LaplaceBeltrami operator have eigenvalues in the interval {—{{m — l)/2)^, 0]. If this is not the case then, as shown in Theorem 5.7, N{s\ w, H^o) = oise^^"^-'"^'^^').

(1.7)

Section 6 contains examples of groups for which the energy form is positive definite, nonnegative and indefinite. We prove in Theorem 6.4 that if the fundamental domain contains a cusp of maximal rank and if F is nonelementary, then E is indefinite and the Laplace-Beltrami operator has eigenvalues in the interval (—((m — l)/2)^, 0]. This extends to higher dimensions a result of Patterson [13].^ Section 7 presents the derivation of explicit formulas for the solution to the non-Euclidean wave equation that are used in Section 5. In a first reading of this paper the reader may find it helpful to limit himself to the three dimensional case.

* We thank Dennis Sullivan for acquainting us with this result of Patterson. A proof for m > 2 due to Sullivan is soon to appear [14].

200

284

LAX AND PHILLIPS 2. THE DENSITY OF ORBITS FOR A CRYSTALLOGRAPHIC GROUP IN EUCLIDEAN SPACE

The main tool in our approach to the lattice point problem is the wave equation, which in Euclidean space of dimension m is m

u,, = Au = Y,d%u.

(2.1)

I

This equation has solutions of the form (see [2, pp. 223, 226]) /I

x('«-3)/2 1

u{x,t) = c„{—dA

—

f{x-\-y)d(j

formodd, (2.2)„

dy

for m even, (2.2)^

M(X, t) = c^{ — dA where ' " = 2<»-'«((m"-2)/2)!

^°""°^^' (2.2)'

2(m-2)/2^(^_2)/2)!

for m even.

The initial values of u as given by (2.2) are M(X,0) = 0,

u,{x,Q) = (D^f{x)',

(2.3)

here ct>„ = 2n'"^^/{{m — 2)/2)! is the area of the unit sphere in IR'". If/ is a function of x automorphic with respect to the group T, then so is u{x, t) for each value of t. We shall write the automorphic function / in the form f{x) = Y,h\x),

h\x) = h{T-'x).

(2.4)

We choose h of the form *W = i - ' . , ( ^ ) ,

(2.5)

where Äj is smooth, nonnegative, supported in the ball {|A:| < 1} and normalized as {h^{x)dx=\. 201

(2.6)

ASYMPTOTIC DISTRIBUTION OF LATTICE POINTS

285

It follows from this that j h{z)dz=\ hxz-x\
if IJX: —r^oK r — a (2.7) if | ; c - T X o | > r + a

and that the value of the integral lies between 0 and I otherwise. We shall study the expression I{T, a) = c ; f (T' - f2)<'«-3)/2 tu(x^ t) dt,

(2.8)

where 2((m-2)/2)! ^((m-3)/2)! LEMMA

for all m.

(2.8)'

2,1. The following identities hold: do dt Jo

\ [

m—3 ^2(—3)/2_^f I

/

fi^y^^dy

I

J\y\=t

form odd,

(2,9),

\y\
T

/ 1

m — 3

\ <'"-2)/2 ^

r i

= 2('"-'>/' —-—! V/TT (

|y|
Proof

f( ,A

/ ( ; ; ) dy

for m even.

(2.9)e

Making use of the substitution s = t^, S = T^, I^ can be rewritten

as / , = 2<'«-^>/^ r^(5_^)('«-3)/2^(m-3)/2^-l/2 r

f{y)dGds.

Integrating by parts (m — 3)/2 times, this becomes

/„ = 2^-^>/^^! Cs-'" { _ 2(^-3)/2

m— 3

202

f{y)dads

286

LAX AND PHILLIPS

We treat I^ similarly; setting s = t^, S = T^ and I, = 2""-''^' f {S - s)""-''^' d["^-'"^ {

r=y^, {^^^

dy ds

,,.„-.. ((»-3)/2)!.^l_. fjy) .^... yjn •'0 yJS — s h^<s yjs — y'^ Interchanging the order of integration, we get

\Jn

V<5

\ V V ( 5 -s){s -

y^)i

The inner integral is easily transformed into the Beta function J5(^, ^) = 7r and hence

This concludes the proof of Lemma 2.1. Setting (2.2) into (2.8) and making use of (2.4) and (2.9), we get /(r)=f

f{y)dy

= Y.\

= Y.\

h\y)dy

h{^)dz.

(2.10)

•I\TZ-X\
It follows from (2.7) and the definition (1.5) o{ N{S\X,XQ)

A^(r-a)
that

(2.11)

which is equivalent with I{T-a)

< N{T) < I{T + a).

(2.11)'

In order to prove Theorem 2 we need an independent asymptotic evaluation of I{T). The main term is obtained by splitting off the mean value of M. Define ' " W ^ T i r f "ix,t)dx;

(2.12)

here \F\ is the volume of the fundamental domain F. Making use of Eq. (2.1), we see that for automorphic solutions of the wave equation

203

ASYMPTOTIC DISTRIBUTION OF LATTICE POINTS

287

Thus m is a linear function: m{t) = at + b.

(2.13)

From (2.13) we see that a = j^j

u,{x, 0) dx,

b = — I u{x, 0) dx.

(2.13)'

For a small, the sum (2.4) has only a single nonzero term in F; so from (2.3) u,{x,0) = co^h{x),

u{x,ö) = 0.

(2.14)

Setting this into (2.13)' and using (2.6), we find that

and hence (Or

m{t) = j^L

(2.15)

Next we decompose u as u{x,t) = ^ t + vix,t).

(2.16)

Clearly v satisfies the wave equation Vft = Av,

(2.17)

and the mean value of v is zero for all t: f v{x,t)dx = 0.

(2.17)'

v(x,0) = u{x,0) = 0.

(2.17)"

Note also that We set (2.16) into expression (2.8) for / and after a straightforward calculation we obtain

where F = F(x, r) = c'„f {T^ - f^)^«-^)/^ ^y(;c, 0 dt. .'o 204

(2.19)

288

LAX AND PHILLIPS

We shall estimate V with the aid of the following version of Sobolev's inequality:^ I nx)\ < c II V\\li;_,,,, II V\\ll„\,,/, a.e., (2.20) where || V\\^ denotes the H^ norm of V over F. For automorphic functions V whose mean value is zero \\Vl = \\A^^'Vl (2.21) where || || denotes the Lj norm over F. LEMMA 2.2.

For m>

2,

\\V\\in.,iy2<0{P--'^^'a-"'% lini(.-i)/2<Ö(r<'«-»/^a<^-'">/^).

(2.22) (2.22)'

Combining this with (2.20) and making use of the fact that V{x) is smooth, we obtain I V(x)\ < OiP^-'^/^a^'-""^/^). (2.23) Setting this into (2.18) we get / ( r ± a, a) = - ^ r'" + OiT'^-'a) + o(r<^-^>/'a<^"'">/'). m\F\ Combining this with (2.11)' gives N{T)

^^T"^ < OiT'^-'a) + 0(r^'"-n/2a
(2.24)

We now set a = 7^(i-^)/(i + '")

The resulting inequality is (1.6). This proves Theorem 2 for m > 2. The proof of Lemma 2.2 eventually splits up into four cases depending on the residue of m mod 4. We set m = 4q-\-r,

r = - 1 , 0 , 1, 2.

(2.25)

We begin deriving some identities. Applying A to (2.19), using Eq. (2.17) and integrating by parts gives ^In the original version of this paper we used a slightly different form of Sobolev's inequality, which led to less precise results (see the case m = 2 discussed at the end of this section). We thank Haim Brezis, Louis Nirenberg and Jaak Peetre for suggesting the use of (2.20) in its stead

205

ASYMPTOTIC DISTRIBUTION OF LATTICE POINTS

Jo

289

h

= -f[dXT^-ty"-'^^^t]v,dt = -{d^T^

for m > 3

- /2)<'"-3)/2 t]v \l + f [^^7^' - ty"-'^^^

t]v dt

for m > 4.

•'0

(2.26) Recall that v{0) = 0. Hence JF=f

[d^{T^~ty"-p^^t]vdt

for

m>5.

(2.26)'

•'0

Applying A to (2.26)', using (2.17) and integrating by parts

-f

[dXT^-ty"-'^^^t]v,dt

for

m>6.

(2.26)"

Notice that ( r ^ — ^2ym-3)/2 ^ j ^ ^^^^ jj^ ^ ^^ ^.j^^^^ ^j^y ^^^j^ number of differentiations will vanish at t = 0. Consequently A^V=-\

[d]{T^-t^Y'"-^^'''t]Vtdt

for

m > 7.

(2.26)'"

•'0

After q applications of A the following expressions hold:

-([d]''-\T^-t^Y'^-'^"'t]v,dt = - ( [dY-\T^ -t^Y"'-'^"' t]Vtdt

for for

r = -\, r = 0,

•'0

(2.27)

^_j^2<,-l(7^2_^2ym-3)/2^jy|r

+ ( [^]''{T^ - t^f"-'^^^ t]v dt = C [d^%T^ - t^Y""-'^^^ t]v dt JQ

206

for r = 1, for

r = 2.

290

LAX AND PHILLIPS

Next we derive some inequalities for the function v{x, t). LEMMA

2.3.

The ineqality ||J^i;,||<0(a-'^-'"/')

(2.28)

holds for all t and —1 < p < 1

when m > 4,

—i ^ /^ ^ 1

when m = 3 and 4.

(2.29)

When m = 2, (2.28) holds for 0 < /? < 1; whereas for —i < /? < 0 ||J^i;,||<0(a-2^-' |loga|-P).

(2.28)'

The inequality WA^vW^Oia-^"^'-'"^^)

(2.30)

holds for all t and —5 ^ P ^ 1

when m > 4,

0 < ;? < 1

wÄe« m = 3 a«^ 4,

^ < /7 < 1

ivÄe« m = 2.

(2.31)

Proof In the proof of this lemma we shall use the conservation of energy principle: For any automorphic solution w of the wave equation, the energy in F is conserved; that is, E{w)=(

{wl + w',)dx

(2.32)

'F

is independent of t. Since v is an automorphic solution of the wave equation with mean value zero, so is A''v. From the definition of energy and its independence of t, we deduce that WA^v^f ^EiA^v) = E{APv{0)) = \\APV,{0)\\^;

(2.33)

in the last step we have used the fact that v{0) = u{0) = 0. Inequality (2.33) shows that it suffices to prove (2.28) at / = 0. At f = 0 we know v^ explicitly. Combining (2.5), (2.14) and (2.16), we see that in the fundamental domain/' uXx,0) = a-"'hi{x/a),

(2.34)

v^{x, 0) = a-'"hiix/a) - const.

(2.34)'

207

ASYMPTOTIC DISTRIBUTION OF LATTICE POINTS

291

Consequently in F \uXx, 0)1 < const, a""*

for \x\ < a,

=: 0

elsewhere in F.

(2.35)

Since v^ has mean value 0 and differs from u^ only by a constant, we deduce that

il^XO)il< II «,(0)11 = 0(0-/^).

(2.36)

This gives inequality (2.28) for p = 0; (2.28) for p = 1 follows immediately from (2.34)' by differentiation. To prove (2.28) for intermediate values between 0 and 1 we appeal to the following convexity principle: Let A be any nonnegative operator in Hilbert space and/any vector in its domain. Then for 0 < /? < 1

iMyii
(2.37)

This is easily proved using Holder's inequality on the spectral representation for^. Next consider negative values of p; we remark that the operator A~^ is defined on functions whose mean value is zero. Nevertheless it is convenient to introduce the invertible operator M: M=\-A.

(2.38)

Since 0 is an isolated point of the spectrum of A with constant as eigenfunction, and since v^ is obtained from u^ by removing the constant component, it follows that \\A-^v,\\ < c \\M-^v,\\ < c \\M-^u,\\.

(2.39)

Now M~^ is an integral operator whose kernel has a mild singularity along the diagonal; in fact the kernel is bounded by const. - ^ for m > 2

and

const. | log | A: — j ; | j for m = 2. (2.40)

M"^ is obtained by composing M~^ with itself; an easy calculation shows that the kernel for Af "^ is bounded by - ^ ^ ^ \x- y

for m>A.

Using expression (2.34) for uj{x, 0), we deduce for |j>c| < a that 208

(2.40)'

292

LAX AND PHILLIPS

I »^-2 /mi/ x/const, r "

dy

•'iyi
<0(a'•-'")

|A/-.«,(0)IW<^J <0(a'-'")

for

m>4,

r ^ ^ for

\M-\{0)\{x)^^\

(2.41)

m>2,

\\og\x-y\\dy

<0(|loga|)

for

m = 2.

According to (2.6) [ \u,{x,0)\dx = (o„. Consequently \\M-\{0)r

=

{M-\{0),u,{0))

< const, max \M~^Uf{0)\ (x) \ \Uf{x, 0) dx Jp

\x\
<0(a'*-'") \\M-"''uSO)f

for

(2.42)

m>4,

= ( M - ' M , ( 0 ) , U,{0)) < 0(a2-'«)

||M-'/2M,(0)f = ( M - ' M X O ) , «XO)) < 0(|log a\)

for

m = 3 and 4,

for

m = 2.

Finally, applying the convexity principle with A—M~^ for m > 4 and ^ = M~^^'^ for m = 2, 3 and 4, we obtain inequalities (2.28) and (2.28)'. The proof of (2.30) is now an easy matter. Integration by parts, together with (2.32), gives II jJ/2w||2 = {A'/^w,A'^''w) = -{Aw, w)=\

wldx^E(w).

^F

Apply this to w = A''~'^''^v and we get WA'vf = \\A"''AP-'/Hf <,E{AP-''''v). Inequality (2.33) then gives \\A^v\\<^\\A^-^f'vM\209

ASYMPTOTIC DISTRIBUTION OF LATTICE POINTS

293

Thus (2.30) is a consequence of (2.28). This completes the proof of Lemma 2.3. We turn now to the proof of Lemma 2.2. We treat the four cases in (2.25) separately. Case (i): m = 4q—l. (m+l)/2:

To prove relation (2.22), we use (2.21) with s =

\\vim,w2=\\^""'''''n=\\^'n

(2.43)

Next we use formula (2.27) for J T , with r = — 1: A'^V= [5,2-^-2(72 - ^2)2'7-2 ^] j^^ |7- _ r'" [52^-1(7^2 _ ^lyq-i ^1 ^^ ^^

(2.44)

Clearly 1^2^-2(7^2 _^2^2,-2 ^1 ^ const, r^"-» = const. r<'"-'>/2 and 1^2^-1(7^2 _ ^2)2a-2 ^1 ^ ^^^^^ pm-3)/2^ So we obtain from (2.44) the estimate

|M'^ni/2||^j|), which by (2.28), p = 0, is OiP'^-'^^^a""^^). Combined with (2.43) this yields inequality (2.22) of Lemma 2.2. As for (2.22)', we see by (2.21) that II ni(.-i)/2 = \\A'"'-''^'V\\ = IM-^-'/^Fll.

(2.43)'

We now apply A ~ ^''^ to (2.44) and, arguing as before, we get IM-'-'/^Fii < o(r<'"-*)/2 p -i/2yj|) which by (2.28), p = - 1 / 2 , is o(r<'"-^>/2^<2-m)/2^ Combined with (2.43)', this proves (2.22)'. Case (ii): m = 4^+ 1. By (2.21)

11 ni(«.,)/2 = IM^'-^'^^'^rll =^ \\A^''''V\\.

(2.45)

We now use formula (2.27), r = 1: Z, ^=-[52a-I(7^2_^2^2„-l^J^|r^|-'"j^2<;(7^2_ ^2)2^-1 ^Jy^^ (2.46)

210

294

LAX AND PHILLIPS

Applying J*^^ to (2.46) and arguing as before, we get

which by (2.30), p = i is 0(r<'"-»^/'a"'"/'). Combined with (2.45), this proves (2.22). To prove (2.22)', we again appeal to (2.21):

lini(.-.)/2 = IM^'"-^^''^^ll = IM''^l!.

(2.45)'

Arguing as before we see from (2.46) that \\A^V\\^0{P'"-''^'\\v\\), and using (2.30), /7 = 0, this is ^0{P'"~^^^^a^^-'"^^^). Combining this with (2.45)' we obtain (2.22)'. Case (iii): m = Aq. In this case

!l^ll(..,)/2 = IM^'"-^'^^'^^ll = IM''"'^'ni-

(2.47)

We apply A^''^ to formula (2.27), r = 0:

h = \\f^W^+W,, .'o -'s

(2.48)

where w = A^'^v\ here S is some parameter 0 < 5 < T to be specified later. We shall estimate W^ and W2 separately. In the case of W^, we integrate by parts, obtaining W, = -[d^,'^-\T^

- f2)2'J-3/2 ^ ] ^ |5 _ (^ |^2<7(j.2 _ ^2)2<7-3/2 ^ | ^ ^^

Now [d^f''~^{T^ — ^2^29-3/2 ^j ^^^ jjg expanded as a sum of terms of the kind

The largest terms comes from Ä: = 2^ — 1 for which a suitable bound is const. T^i-in(T-t)-^^\ Likewise

211

ASYMPTOTIC DISTRIBUTION OF LATTICE POINTS

295

Using these bounds, an easy calculation shows that \\Wi\\^0{P'"-'^^\T-S)-'^^)\\A'^'v\\. Applying (2.30) with p = j and setting T-S

= a,

(2.49)

we obtain ||H^,||<0(T^"'-*>/'a-'"/2). We estimate PFj directly, using (2.28) with;? = 5: II ^2ll < O l(p'"-'^/^) \\A'^'v,\\ f {T-t)-''^

dt\

which by (2.47) is 0(7^'"-'^^'a-'"/'). This proves (2.22). To prove (2.22)', we write

II^^IU-i)/2 = IM^'"-'^^^nNIM'^-'^'^llWe apply A'^''^ to (2.27), r = 0, obtaining an expression for zi^~'^'*rsimilar to (2.48) with w = A~^'*v on the right. To this expression for A^'^'^^^^V we apply the same argument as above. This time we use (2.30) and (2.28) with p = —\. The resulting estimates for || W^\\ and || fFjIJ yield (2.22)'. Case (iv): m = 4^ + 2. By (2.21)

linU.,)/2 = IM^'""'^^^J^II = IM'^"'^''ni and

II^^IU-i)/2 = IM^'"-'^^''f^ll = IM'^"'^'^l|. We therefore apply A\ s = \ and | , to (2.27), r = 2:

Jo S

T

= ( + f =w, + W,. •^0

(2.50)

Js

We now rewrite W^. Using the fact that v satisfies the wave equation and performing an integration by parts, we get 212

296

LAX AND PHILLIPS

Wi = f {d]\T^ - /2)2'?-i/21] A'-' Av dt

= f [&^'(T^ - ty-'""

t] A'-' V,, dt

- f [d^'' + \T^-ty-'^^

t] A'-' v,dt.

(2.51)

Estimating as before, using (2.28) with p = —j when s = ^ and p = — | when 5 = 1, and choosing 5" according to (2.49), we obtain for m > 6 ||fr,||<0(7^'"-'>/'a-'"/')

when

5 = 1,

||fr,||<0(7^'"-*>/'a*-'"/')

when

s= i

Making use of (2.30) with p = s, 5 = | and | , we obtain directly identical estimates for || W21|. Combining all of the above inequalities we can verify (2.22) and (2.22)' for all values of m = 2 (mod 4) which are >2. This completes the proof of Lemma 2.2 and thereby proves Theorem 2, except in the case of m = 2. To treat m = 2 we use a different form of Sobolev's inequality: |F(x)|
e>0,

(2.52)

where the constant c satisfies c(e) = 0(e-'/').

(2.52)'

linii.2e = IM''^-^^^^l|.

(2.53)

By (2.21)

This leads us again to (2.50) with ^ = 0 and 5 = | + e. We proceed as above, using (2.28)' with /? = —5 + e to estimate || Wi || and (2.30) with p = j-\-eto estimate || W^jlj. This gives

IIPFJK o(r»/2(r-5')-^/'a-2^ liogap/'-^), II W4 < 0{T"\T- Sf"" a-'-""'). Combining these estimates with (2.52) and (2.53), we get T'^^ « " ' ' /a'/^|logap/^ I V{x)\ < const. - ^ - ^ ( ^T_SY'2 + 213

{T-Sy^^\ ^1/2 )•

(2.54)

ASYMPTOTIC DISTRIBUTION OF LATTICE POINTS

297

In this case the optional choice for S is T-S

= a\\oga\^^^.

With e = |loga|~', (2.54) becomes | F ( x ) | < c o n s t .«^i /|2l o g a | ^ / ' * Setting this into (2.18) we get 2^2

I{T ± a, «) = — - + 0{Ta) + OiT'/'a''^' Combining this with (2.11)' gives 7lT' < 0{Ta) + 0{r^^a-'^^ N{T)

\\oga\'^').

|log a\'^').

Finally, choosing a = r~'^^(log T)^^^ yields <0(r'/3(log r)'/2).

N{T)

(2.55)

Relation (2.55) is inequality (1.6)' and with this we conclude the proof of Theorem 2 in its entirety.

3. THE ENERGY FORM FOR AUTOMORPHIC DATA

In the rest of this paper we study the distribution of lattice points in a nonEuclidean setting, using methods analogous to those of the previous section. For purposes of exposition we shall limit our considerations in this and the next section to three space dimensions; however, the results hold and the methods of proof work for real hyperbolic spaces of any dimension. The model IH3 we shall use consists of points {Xi,X2, y}, y > 0, with the Riemannian metric

The isometrics for IH3 can be described by means of the 2 x 2 complexvalued matrices I \c

j, d)

where

214

ad — bc= 1

(3.2)

298

LAX AND PHILLIPS

in the following fashion: To each point iXi,X2,y) quaternion w with three components

of IH3 we assign a

where i^ = ß = —l and ij= —ji. Every isometry of IH3 can then be written as zw = (aw + b){cw + d)~\

(3,3)

The invariant volume element is ^^^dx.dxyy^^

(3.4)

the invariant Dirichlet integral can be written as jy'\du\'dw;

(3.5)

and the invariant Laplace-Beltrami operator is LQU= y^ d — du.

(3-6)

Automorphic functions are defined by means of a discrete subgroup F of the isometrics by the property u{yw) = u{w)

(3.7)

for all y in F. Such a function is uniquely determined by its values in a fundamental domain: F=\HJF, We shall limit our considerations to subgroups for which the fundamental domains satisfy the finite geometric property. This means that F may be chosen as a polyhedron with a finite number of sides. On the other hand F may have either finite or infinite volume. The fundamental domain F can be regarded as a Riemannian manifold; if F contains elliptic motions, i^ is a manifold with a finite number of singular geodesies; these are the fixed points of the elliptic motions in F. This manifold can be described by a finite number of conformal charts. We divide the charts for such a manifold into three categories: (1) Interior charts mapped onto the ball: dist(w, (0, 0, 1)) < 1 or, if they correspond to points on the fixed geodesic of an elliptic motion, onto a sector of the ball: idin"^ xjx^ < 2K/n^ n an integer >2. (2) Regular charts at infinity mapped onto the half-ball: {x^-f JC2 + J;^ < 1, j ; > 0} or, if they correspond to a fixed point at infinity of an elliptic motion, onto a sector of the half-ball: tan~^^2Ai < ^'^/^-^ ^ an integer >2. 215

ASYMPTOTIC DISTRIBUTION OF LATTICE POINTS

299

(3) Charts of neighborhoods of points ß at oo which are fixed points. The shape of the fundamental domain near ß depends on the structure of the subgroup r^ leaving ß fixed. If T^ is generated by a single parabolic motion, the chart is mapped onto the portion of the slab: {—1/2 < ^2 < 1/2} • (or the half-slab: {—1/2 < ^2 < 1/2, x^ > 0}) exterior to the sphere {x] + ^2 + j^^ = 1} in the upper half-plane. For a doubly parabolic point the chart is mapped onto the portion of a prism with vertical generators lying above y = a. Such a prism may have as cross section a parallelogram, a triangle or a hexagon. Since all charts are conformal, where two charts overlap the mapping from one to the other can be realized by a fractional linear transformation of the kind described in (3.3). We shall treat the lattice point problem in non-Euclidean space by means of the wave equation Uf^ = Lu = LgU + u.

(3.8)

With the zero order term u included on the right, the solutions satisfy Huygens' Principle. We denote the initial conditions:

by the data p a i r / = {/i,/^}. One can either suppose that the automorphic wave acts on automorphic data defined on all of IH3, or on data defined on the fundamental domain F with suitably periodic boundary conditions or simply on data defined on F considered as a manifold. We shall take the latter point of view in what follows. We need a suitable Hilbert space framework for our analysis. Our starting point is the energy form for (3.8), that is, E{Ag) = -{LA,g^)-\-(f2,g2)-

(3.9)

Here («, v) denotes the LjCF) inner product. It is easy to see that E is constant in time for solutions of the wave equation. For / , g in C^(F) an integration by parts brings (3.9) into a more symmetric form: E{f, g) = \ {y' df, dg, - / i f, + /2 gl) dw.

(3.10)

As such E is obviously locally indefinite and it will be shown by examples in Section 6 to be globally indefinite for some F. We now construct a related locally definite form G = E-{-K 216

(3.11)

300

LAX AND PHILLIPS

which differs from £" by a form K which will be shown to be compact with respect to G. As we shall prove, all such G forms are equivalent. Our Hilbert space ^ is then obtained as the completion with respect to G of C^{F) data. The following calculations involve only C^iF) data. We first choose a finite C°°(F) partition of unity for F subordinate to charts of F; by F we mean F plus the points at infinity with the Euclidean topology of the charts. We denote these functions by {coj}, {cpj} and {ii/j} according to whether they are supported on interior, regular infinite and parabolic charts, respectively. We require that the support of each of the ^'s and \j/s overlap with that of at least one of the co's. We also require that the cü's, 9>'s and ^ s be nonnegative, that their supports have connected interiors and that the i/s be identically one near the parabolic ßxed points. Clearly, the indefiniteness of E occurs only in the first component; the contribution of the second component is positive definite. Therefore in what follows up to Theorem 3.4, we shall restrict our attention to data whose second component is zero. This being the case, we can omit the subscript 1 and let/stand for the first component. We now set

with similar expressions for EJ and Ef. We also set

Ky(f) = 2Jcoj\f\'dw

(3.13)

and we define GyW) = Ef{f) + Kr{f)=^coj{y^ \df\' + l / n dw.

(3.14)

Clearly Gf is >0. Next we use the local chart coordinates to bring E*^ and E"* into a more convenient, but no longer invariant, form. We begin by writing

\^yk(^)\

dj

f

dy=\^yVy-yr

dy

i,{^_^,^).„ An integration by parts gives

. J)/ .l '. . | . ^^y\f\' . . = -...| ( . ,( L. ^ - I/I' 2.i^ 217

ASYMPTOTIC DISTRIBUTION OF LATTICE POINTS

301

Combining the above and inserting the resulting expression into £"* we get

£;(/)=}«'>/^(Q

dw-KJ(f),

(3.15),

where KJ{f) =

j{d,(pj)y\f\'dw;

(3.16),

in each of these formulas the integration is over the corresponding chart- We also define GJif) = Ef{f) + KJ{f) = j(pj/

dw.

(3.17),

Finally we define (3.18) and (3.19) It is evident from (3.14) and (3.17)^^ that G is a locally positive form. We need an estimate on the local L2 norm of the data: LEMMA

3.1. For any compact set S in F there is a positive constant c^

such that i

\f\'dw^CsG(f).

(3.20)

Proof. The following is an easy variant of the Poincare inequality: Let Z) be a connected domain in R^, whose closure is compact, and DQ a nonempty subdomain of D. Then there exist constants a and b depending only on D and DQ such that for every smooth function g defined on D \ \g\'dx^a(

\dg\'dx + b(

\g\'dx.

(3.21)

Using this fact, we now show that for ö small enough there is a constant c^ such that or every ball B^ radius S and center a point of S,

f l/l^rfw
(3.22)

JR.

Since S is compact, it can be covered by a finite number of balls B^ of this kind; so (3.20) is a consequence of (3.22). As a preliminary step we enlarge S to ensure that, while still compact, it has the following additional properties: (i) 5 is connected, 218

302

LAX AND PHILLIPS

(ii) S contains an interior point in the support of one of the functions coj. We then choose Ö so small that in each ball B^ at least one of the functions coj, cpj or y/j is >fc > 0. This is possible since

is a finite partition of unity and since each of these functions is uniformly continuous on the compact set S. By property (ii), for d small enough, there is at least one ball B\ on which COJ'^K. It follows then from (3.14) that for this ball (3.22) holds with c^ = \/K. Next let B^ be a ball in which, say, g>j^K. It follows then from (3.17)^ that

I" \dg\'dx^^G{f),

(3.23)

KI

JBS

where g=^-

(3.24)

and Y= min y m B^, Suppose B^ intersects another ball B'^ for which (3.22) holds. Applying (3.21) with D = Bg, DQ = B^r\B'^ we conclude, using (3.23), that (3.22) holds for B^ as well, possibly with a larger constant c^. Since S is connected, each ball B^ can be linked by a finite chain of balls to B\. This proves (3.22) and thereby Lemma 3.1. LEMMA

3.2.

Let S be any compact subset of F\ then

[i/r•' dw

(3.25)

>s is compact

with respect to

G{f).

Proof. As before, it suffices to prove this for S = Bg. This follows from inequalities (3.20) and (3.23), by means of the Rellich compactness criterion. The following is the basic result of this section: THEOREM

3.3.

Let K and G be the quadratic forms

defined in (3.18),

(3.19). Then (i)

K is compact

with respect to G,

(ii)

E is bounded with respect to G.

Proof Since G = E -\-K, (ii) follows from (i). To prove (i) each term in the sum K is compact. We note that all terms Kf, associated with a multiple parabolic point is of the form (3.25) compact by Lemma 3.2. T o show the compactness of the remaining ATj" and of the

219

we show that and each K^ and therefore Kf we break

303

ASYMPTOTIC DISTRIBUTION OF LATTICE POINTS

each up into two parts, depending on a parameter a: In the first part the y integration is extended over y>a/m the second over y
K,{f)<eaG{f\

where e^ tends to zero as a tends to 0. It follows from (3.16)^^ that K^ is bounded by a sum of quadratic forms

M/r lo r

dydxi

(3.27)

dx2.

the integrals being confined to the support of d