Scattering Theory for Second Order Elliptic Operators (*),

MARTI~ SCItECHTI~ (New York, ~-. Y., U.S.A.) (**)

Summary. - We find suf/ieient conditions on a perturbation o/ the Laplaeian to insure that the wave operators exist and are complete. Our method allows us to obtain new results o~ this topic while recapturing results previously known.

1. - I n t r o d u c t i o n .

This paper is concerned with scat ter ing theory for the Sehr6dinger operator

(1.1) - ~ + V(x) ,

where d is the n-dimensional Laplaeian and V(x) is a potent ia l function. In part i- cular we are in teres ted in finding conditions on V(x) which will guarantee existence and completeness of the wave operators, the invarianee principle, the inter twining relations, etc. (see Theorem 1.1). This problem has a t t r ac ted considerable a t t en t ion in recent years (cf. [1-13, 16-19, 23, 24] and the references quoted in them). To obtain these objectives, various authors have made different assumptions. We quote a few of the more recent ones here. IKEBE [9] made the assumption V(x) = O( Ix] -~-~)

as Ix] -> c~ for some e > 0. REJTO [11] weakened this to 0(Ix[-~-4/3), while KATO [5] allowed

(1.2) V(x) = o(Ixl -~-~) as Ix l -~ ~ .

KURODA [6] used a slightly different type of condition. He assumed

(1.3) sup f tV ( y ) ] I x - - y l~ - - "dy -+O $¢

and 1~-~[<~

(1.4/ (1 + lxl) ~ v (x ) E z • ,

as 8-->0

for some ~, p satisfying l < p < c~ and ~ > 1 - p-1 (note tha t (1.3) is unnecessary when n < 2p). When p = c% (1.4) is the same assumption as (1.2). For n ~ 3, KATO and KURODA [1] showed tha t V e L ~ is sufficient, while SL~O~ [13], using a

(*) Entrafa in Redazione i l 16 gennMo 1974. (**) This research was partially supported by a N.S.F. Grant.

314 MA~TI~ SCHECHTE~: Scattering theory ]or second order elliptic operators

result of KA~O [3], ex tended this to V satisfying

(1.5) .f~ir(x) r(y)lI~- yt-2 d3xdSy ~ c~ .

(He calls this the Rollnik class.) ALSHOZ~ and Sc~I)m)~ [10] assumed

f iv(y)]lx- yi ~(~ -') dy

bounded and tend ing to 0 as Ixl-> 0%

Ix-~l<i

bounded for some a ~ n - - 4~ and

for some b > 1 - - i n .

sumed tha t

(1 + lxl) ~ V(x) e L ~

GttEIFENEGGER~ JSRGEI~S~ WEIDMANZN ~nd WINKLER [12] aS-

( l + !Xt) ~ f tV(y)121x--y]-~dy I~-~I<l

is bounded for some a > n - - 4 and b ~ 2. SCIIECtITEI~ and BULK~ [16] assumed (1.3),

(1.6) f tv(y)l~y-~o as I x i - ~ ,

and

(1,7) s u p f tg(y) l ( l -~ ! x - - y l ) - ~ d y < c<)

for some fl < ½(n--1). Another me t h o d due to KA~o [3] works when V e L ~ n Z ~ for some p < ½n and some q > ½~.

Some authors s tudied higher order operators which include (1.1) as a special case. We describe some of thei r results briefly as applied to this operator . BIRlVIAiX [18] assumed tha t ( l Jr [xl)~V is in L 2 for some a>½n, and BnALS [17] assumed

f Iv(Y)llx-Yi~-'dY<V(l+ Ixt) -b

for some b > n. AG~ION [19] and KUROD.& [7] assumed (1.2). SCI-IECtITER [23] as- sumed (1.3), (1.6) and

( l ~ - I x t ) ~ f IV(y)[dyeL ~ (1.8) Ex-vI<l

MAlVrI:N SC~CHTER: Scattering theory ]or second order elliptic operators 315

for some c¢,p such that ~>0 , l<p_<oo and

(1.9) a > l 2n

(n+ 1)p"

Note that (1.6) is unnecessary when p : co. Note also that (1.4) implies (1.8) for the same ~ and p. THoMPso~ [24] makes the first two assumption of Alsholm and

Schmidt. I t should be mentioned that some of the authors cited obtained additional results

as well, such as eigenfunction expansions, higher order peI~urbations, etc. We have merely stated the results as they apply to the operator (1.1) and so the question of existence and completeness of the wave operators.

The purpose of the present paper is to prove the conclusions of scattering theory under hypotheses not covered by any of the previous authors. Moreover, our method recaptures all of the results cited. Our results will be obtained under any one of the

following hypotheses. (A) (1.3) holds,

(1.10) f ]V(y)Iix-yl~--'*dy~O [~'-yl<l

and

(1.11) f IV(y)I]x--yI½(1-")dy is bounded and -~0 as ]xI-~c~.

(B) (1.3) and (1.10) hold, and

(1.12) / . / , | | iv(x) V(y)Ilx- yil-"d~du< ~ .

(C) n = 3, (1.11) holds and

(1.13) I | IV(x) r(y)llx- yl-~#xd~y < ~ , {x--y]<l

(D) n = 3 and (1.5) holds.

(E) (1.3) and (1.6) hold, and

(i.14) f iV(y) idyeL~Lq Iz-~l<l

for some p < ½n and some q > 1

316 MARTI~ SCHECI{TE~: Scattering theory for second order elliptic operators

(F)

(i)

(ii)

(iii)

(1.3) and (1.6) hold, and (1.8) holds for some a, p such tha t

c¢~> 0 if l < p < 2 n / ( n + I)

~> 0 if 2n/(n -~ 1 ) < P < ½ n

~ > l - - ( n / 2 p ) if ½ n < P < o o .

Hypothesis (D) merely recaptures Simon's result [13], while (E) is only a slight gene- ralization of tha t of KATO [3]. They are included only to show tha t t hey are also consequences of the same method. The reason for the restriction n ---- 3 in (C) and (D) is tha t for n > 3 the only reasonable function satisfying

ff Iv( )V(y)j1 - yl --'odxdy< ]~-vi<:

is V(x) --O. Our main result is

THEOREI~{ 1.1. - / ~ e t V(x) be a real valued function satis/ying any one o/ the hypo- theses (A)-(F). Let H and H1 be the Hamiltonian operators corresponding to -- d and (1.1), respectively. I~et R+ denote the interval [0, oo), and let E()O, Et(]J denote the spectral families o / H and H1, respectively. Then we have

(1) There is a closed set ecR+ of (Lebesque) measure 0 such that the restric- tion o/ HI to E~ (R+--e)L ~" is absolutely continuous.

(2) The absolutely continuous part o/ H~ is unitarily equivalent to H.

(3) The wave operators

(1.15) W±u = l im exp [itH1] exp [-- itH]u ~ ~t e L ~

exist and are complete.

(4) The invariance principle holds.

(5) The intertwining relations

H~ W± : W±H

hold.

(6) The operator S = W* + W is unitary on Z 2 and commutes with H.

Completeness of the wave operators means t ha t their ranges coincide. The india - riance principle states tha t

(1.16) W ± u - : l im exp [i@(H1)] exp [-- i t~(H)]u t -~±oo

MARTI~ SCI~ECHTEg: Scattering theory /or second order elliptic operators 317

holds for any funct ion 9(~) satisfying

(ln7) as

0 I

for any compact in terva l I conta ined in R+ -- e. F u r t h e r explanat ions can be found in KATO [4, Chapter X].

Theorem 1.1 is the consequence of an abs t rac t theorem to be described in the nex t section. The mMn work is to show tha t each of the hypotheses (A)-(F) implies the hypotheses of the abs t rac t theorem. This verification is carried out in Sect- ion 3 for (A)-(D) and in Section 4, for (E) ~nd (F). The details of the proof are given

in Sect ion 5.

Notation. L 2 will denote the Hi lber t space of square integrable funct ions over n dimensional Eucl idean space E ~. For any t t i lbe r t space H, B(H) will denote the set of bounded operators on H. C o will denote the set of tes t functions, the infinitely differentiable funct ions wi th compact supports . The surface of the uni t sphere in E" will be denoted by S "- ' . The norm ]I ]l~ will be tha t of L~(E"), while a norm wi thout a subscript will be tha t of L ~. November 26, ]973

2. - T h e abstract T h e o r e m .

We now s ta te the abs t rac t theorem which is used to prove Theorem 1.1. A proof of i t can be found in [23]. I t is based upon the work of KA~o-KuRoDA [1-7] which evolved over several years. The theorem proved in [7, I] does not quite mee t our requirements , bu t it exhibits most of the main features.

Le t H be a selfadjoint operator on a Hi lber t space H. Le t E(2) denote its spec- t rM measure, and let R ( ~ ) = ( ~ - - H ) - t denote its resoNent operator for $ in its resolvent set ~(H). We assume

1) There are an open set A of the real line, a Hi lber t space C and a un i t a ry opera tor F f rom E(A) H onto L~(A, C) such tha t

(2.1) FE(I) F -~ = Z±

holds for any subinterval I of A, where Zx is the characterist ic funct ion of I .

2) There are a Hi lber t space K and closed l inear operators A, B f rom H to K such t ha t _D(]HtO)cD(A), D(IHII-o ) c D ( B ) f o r some 0 4 0 < 1 .

3) A is injective.

4) There is a point ~o in ~(H) such tha t the closure of BR(~)R(~o)A* is a com- pac t opera tor on K for each ~ in ~(H).

2 1 - A n n a l i di Matemat ica

318 MAt¢~I~" SCHECH~E~: Scattering theory ]or second order elliptic operators

5) For each ~ in Q(H), the opera tor BR(~) A* is bounded on K, and its closure

converges in no rm as ~ - + P e A to cont inuous funct ions Q±(2) f rom A to B(K). ( In general , the l imits f rom above and below will be different.)

6) There is a funct ion M(~) f rom A to B(K) which is in tegrable over a n y com-

pac t subset of A and such t h a t

d (E(2)A*u, B 'v ) = (Pg(2)u, V)K, u e D ( A * ) , r eD(B*) .

7) H~ is a selfadjoint opera tor on H with spectra l fami ly E~(~) such t ha t

and

(2.3)

D( tH~I °) : D(IHl°),

(Hlu , v) : Hu, v) + (Bu, AV)K ,

D( ?-°) = 1)( IHI

u~D(IHt°) , v~D(IHl~-°) .

The basic resul t is

THEOIgElg 2.1. -- Under assumptions 1-7, there is a closed set e c A o] Zebesgue mea- sure 0 and unitary operators W± ]rom E(A) H onto E I ( A - e) H such that (1.15) holds /or u E E(A)H, and the conclusions of Theorem 1.1 hold (with L ~ replaced by E(A)H).

A proof of Theorem 2.1 can be found in [23]. See [1-7] for r e la ted results , the

p ro to types f rom which the t h e o r e m was taken . We shall also use the following im-

porta.nt observa t ion due to KATO and KVnoDA [2].

LEPTA 2.2. - Assumption 5 is implied by

5') There is a funct ion N(~) f rom A to B(K) which is locally H61der cont inuous

and such t h a t

d (E(~)A*u, B'v) -= (3~(~)u, v)x, ueD(A*), r e D ( B * ) . d2

A proof of this l e m m a can be found in [2, 7, 23].

3. - Continuous boundary values.

We now show the hypotheses of Theorem 2.1 are satisfied when any one of the

assumpt ions (A)-(D) of Sect ion I hold. We p repa re the way with a series of Lemmas . I n the first two we do not a s sume a n y of the hypotheses of Section 1.

L ~ A 3.1. - Suppose d, p are nonnegative numbers, and assume that

(3.1) f IV(y)]lx-yi- dy O as

~IAI~TI~ SCtIECltTER: Seattering theory for second order elliptie operators 319

For R ~ 0 put

Then

(3.2)

V~(x) = o, Ixl < R

= v(z) , !zI> t¢.

/ ,

sup | [VR(y)l]x--yI-~dy-->O ~S R--->c~

P~OOF. - L e t e > 0 b e g iven . T h e t h e r e is a R~ such t h a t

(3.3) f tV(y)iix-yI-*ay<~, ixt>R1.

I n add i t i on , t h e r e is a n R~>~d such t h a t

(3.4) f tV(y)llyi-~dy<2-*s, I'~l </¢,

s ince t he i n t eg ra l (3.1) ex is t s for some x. T~ke R---- m~x(2R~, R~). I f ]y] > R a n d

IxI<½R, t h e n fyl<2tx--yt . T h e n we h a v e

f FV~(Y)]lx-yi-~ay<2~f]v~(y)I]yL-~ay, [x-~I<a

a n d th i s is less t h a n s b y (3.4). On t h e o t h e r h a n d , i f IYl > R a n d Ixl > ½R>~RI, t h e n

f tv(y)II -yl-,dy< f lvl<-~ i.~-vl<a

b y (3.3). []

L E P T A 3.2. -- Suppose d> 0 and that

(3.5)

Then

(3.6)

l "

J t~7(y)ilx-yI-~ay-~o as I x l - ~ .

sup ~ IV~(y)fIx--yl-,dy-+O as t R i - ~ .

PROOF. - L e t s > 0 be g iven . T h e n t h e r e is an R1 such t h a t

(3.7) f IV(y)llx-yl-~dy<~, Ixl>Rl:

320 ML4mmI~ SCItECHTER: Scattering theory /or second order elliptic operators

Take R : 2 m a x ( R l , d). I f lYl> R and [ x - - y l < d , then Ix]> R~. Thus

f IV"(y)11x--~l-'dy<~ i~-vl<d

by (3.7). [] Le t H denote the selfadjoint operator de termined by -- A on ZS(E"). For Im ~ > 0,

let G~(x) denote the Green's funct ion for the operator (us_~ A)-~. I t is well known

that

(~t2~I~1)" ~r~>(~Ixt)

(1) Z where v ~ ½(n-- 2) and H~. ( ) is the Bessel funct ion of the th i rd kind (cf. TITCH- ~AI~SH [25]). The following est imates hold (cf. WAmSO% [26])

(3.8) IG~(x)l<Clxt ~-~, I~xl<x

< cl~l~-~lxt-'-~ exp { - l:m ~lxl}, E~<xl> 1.

For ~ in a bounded set this implies

(3.9) !G~.(x)l <cons t . lxl 2-~ , Ix I < 1

<eonst. IxT -~--~, Ixt> 1.

P u t $ - ~ u s and R ( ~ ) = ( ~ - - H ) -~. Then

(3.10) R(~)u=G~,u , ueL ~.

Let V(x), W(x) be funct ions satisfying ~ny one of the hypotheses (A)-(D) of Sec- t ion 1 ( they are to sat isfy the same hypothesis) . Le t A(x), B(x) be functions such tha t

IA(x)l ~ : IV(~)], IB(z)l ~: lW(x) l .

For any funct ion we wri te h~(x )= h ( x ) - hR(x). YVe ha~e

L]~:~A 3.3. - At~(~)B is a bounded operator in L ~ /or each ~, and

llAR(~)BII <K~ ÷ K2 ,

where K~ can be taken as the square root o] either

(3.11) sup f tV(y)O.(x--y)tdysup f IW(Y)e.(x--Ylldy l ~ - v l < l "~ l z - v l < l

MAlCTI~ SCHEOttTER: Scattering theory /or second order elliptic operators 321

o r

(3.12) ff !V(x) W(y)]IG~(x-- y)12dxdy ,

and K~ can be taken as the square root o/ either

(3.13) s~p f IV(y)~(x-y)Iaysup f IW(y)V.(~--y)ldy x ]~-vI>1 x i~-vl>l

o r

(3.14) ff ]V(x) W(y)]iG~(x--y)]~dxdy.

PRO0~ ~. - For u, v e L = we have

ff ÷ ff l*-~1<1 I~-~,l>1

By the Schwarz inequality, the square of the first te rm is bounded by

i~¢--yI<I IX--y]<1

in the case K~ is (3.11). A different application of Schwarz's inequal i ty gives as a bound

ff tv{ )W(Y)t1<{x-Y)t a dY ff lu(y)v(x)i a dy<-K 11 11 tlvll l~-~,1<1 [~-vl<l

in the case when K~ equals (3.12). Similar reasoning gives (3.13) and (3.14). The lemma will be proved if we can show tha t K1 and K~ are finite under the various hypotheses. By (3.9), the expression (3.11) is bounded when (1.3) holds. This takes care of hypotheses (A) and (B). By the same token, (3.12) is bounded when (1.13) holds, as assumed in hypotheses (C) and (D). Thus K1 is finite under any of the hypo- theses (A)-(D). Similarly, (3.13) is bounded when {1.11) holds (as in (A) and (C)), and (3.14) is bounded when (1.12) holds (as in (B) and (D)). Thus K2 is finite under any of these hypotheses. []

For each M, let ~M denote the set lu t<M, I m z > 0 . We have

COROLLARY 3.4. -- For each ]inite M,

sup IIAR(~)BIt < oo.

322 MA~rI~ SO~CHT]~R: Scattering theory for second order elliptic operators

L E ) ~ A 3.5. -- For each ]inite M

sup IIA~2~(~)B!I -> 0 as N -~ oo . ~ M

P~oo~'. - By L e m m a 3.3 it suffices to show t h a t the expressions (3.11)-(3.14) t end to zero as N -+ oo un i fo rmly in £2 M when V(x) is replaced b y VZC(x). For (3.11)

and (3.13) this follows f rom L e m m a 3.1 and 3.2. For (3.12) and (3.14), i t follows f rom (1.12) and (1.13). []

Fo r the nex t l e m m a we shall need the following e s t ima te (cI. [27])

(3.15) Ivy(x) - eA~)I <cons t . [~-: ~'tl~i ~-'~ , /xl < 1

< c o n s t . ] ~ - z'[lx] ~(~-"), Ixl > 1 .

LE~wA 3.6. - Por each £V, the operator ANR(~)B N can be extended to a conti- nuous /unction ]rom ~ to B(L~).

P~ooF. - I t suffices to consider it in ~M for finite M. Set

d(x) = lxi "-" , Ixt < 1

= I~l ~<~-~) Ixl > 1 .j

I f we set $'~-z~ '~, we have

!(A~c[R(~) -- R($')I BNu, v)[ < const, lu -- u'l

½^ f f lvN(x) l o (x- y) tW (y)I*l (y) (x)ldx dy

As in the case of L e m m a 3.3 we m u s t show t h a t the expressions

(3.16)

(3.17)

(3.1s)

and

(3.19)

sup f t V N ( y ) ! ~ ( x - - y ) d y s u p f iWN(y)i~(x--y)dy,

j'f IV ( )wN(y)l d(x- y) a ay ,

f ^ f ^ sup IVN(y)tG(x--y)dysup IWN(y)lG(x--y)dy g3 J,~-uI>l I,~-~I>1

~zJj I V_~(x) WN(Y) I G(x-- y)~ dxdy

are bounded. By (3.9), the first two are bounded when (1.3) and (1.13) hold, respec- t ively. Each of the hypotheses (A)-(D) assumes one of then. The expressions (3.18)

MARTISi SCIIECI-ITER: Scattering theory ]or second order elliptic operators 323

and (3.19) are bounded by

respectively. Both of these are finite since V is locally integrable. Thus A~R(~)B~ is uni formly continuous in /2 M. []

IJE~:~A 3.7. -- For each ]inite M

sup I]AYR(~)B~II-->0 as N - - > ~ . r2~

PROOF. -- Follow the proof of L e m m a 3.5 ~nd make use of the f~ct tha t IBN(x) I < < [B(x)[.

LEM:M:A 3.8. -- For each ]inite M

sup IIANR(~)BN -- AR(~)B II -~" o ~M

as N--> ~ .

PROOF. - We have

IIAR(~) B -- ANR($)B N II : tlANR(~) B + A~vR(~ ) B NII •

Apply Ibernmas 3.5 and 3.7. []

THEOREM 3.9. - The operator AR(~)B can be extended to a continuous ]unction ]rom ~ to B(L~).

PROOF. -- For each N, ANR(~)B N can be so extended (Lemma 3.6). Moreover,

it converges to A R ( $ ) B uni formly on f2~5 for each finite M. I:2

LEI~MA 3.10. -- I] Im ~ > O, then AR(~) is compact on L 2.

PROOF. - I f (1.3) and (1.10) hold, then this follows from Theorem 4.1 ch. 6 of [14].

However, the following a rgument covers alt of the cases. Firs t we note tha t it suf-

fices to show tha t A R ( - - 1 ) ~ is compact . For if I m p > 0 , then ( I + t I ) R ( $ ) is a

bounded operator on L 2, and AR($) ~ -- AR( - - 1) ( I + H ) R ($). Moreover, since

lIAR(-- 1)½tt~= lIAR(-- 1)AIt , we see t h a t A N R ( - 1) ½ converges to AR( - - 1) ½ in norm as N--> c~. Thus it suffices to assume t h a t A has compact support . For each ~ > 0,

define the operator

W~u---- f A (x )O~(x - - y )A(y )u (y )dy .

By (3.9), G~(x) is bounded for Ixl > 6. Since A is locally square integrable and has

compact support , it is in L 2. Hence Wo is a Hi lber t -Schmidt operator for each ~ > 0.

324 MARTI~ SOn~EC~r~rER: Scattering theory ]or second order elliptic operators

Moreover it follows from Lemma 3.3 tha t it converges in norm to AR(-- 1)A under any of the hypotheses (A)-(D). Thus AR{--1)A, and consequently AR(--1) ½, is compact (cf. Zemma 5.5, ch. 8 of [14]). E1

4. - The remain ing cases.

We now turn our a t ten t ion to potentials V(x) which do not satisfy any of the hypotheses (A)-(D) of Section 1. For them we must use techniques which differ from those of the preceding section. Following an idea of KA~o [3] we use the group U(t) = exp {--itH}. Fi rs t we shall need several lemmas.

I~E~A 4.1. - ~or any l< r~<2 ,

(4.1) ](V(t) ], g)l < (4~t?(~-1/~)Ill II~[]gll, •

P~ooF. - See [3, p. 277].

Tex t let A(x), B(x) be localty square integrable functions, and put

(4.2) ~(x)= (fexp [--Ix--yt~]IA(y)I~dy) ~

(43) ~(~)-- ( j" l~(y)f~dy) ~,

with similar expressions for B. Define the operator T~ by

f exp [-- I x -- Y]2]A(Y) u(y) dy. (4.4) T~u(x)

We have

L ~ I ~ A 4.2. For any q>2,

(4.5) Ii T~ ~T(t) ~.li < citl - '~ i I~ li~iiB iI~.

P~ooF. - Pu t ] l r = ½ - ~ l / q . Then 1~<r~<2, and

I(U(t) T.u, TAv)]<(4z~t) - '~/~l iT, uH~IiT~vll~ .

:[flow

l(T~v, w)l <ff o ~ [ - I~- yl2]lA(y) v(y) /(~)] dx dy

< (ff cxp ~- i~- yl'llv(y)l:d~dy)'(ff exp ~-I~-yI~Jl~(y) w(~)l~d~dy) ' < c]Iv/] [l~u lI < Oily ][ ]I~lIolI~ 1[~,,

MARTI~ SC~EO~T]~R: Scattering theory ]or second order elliptic operators 325

since ½- -1 /q -~ 1/r'. This shows tha t

(4.6) ]1 T~ v]I~< q]lAIlo]tv 11,

with a similar es t imate for B. We now make use of the fact tha t

(4.7) ~ " If A ilq< (?ll ~ If~

for any q [23~ L e m m a 4.7]. []

LE~:M:A 4.3. - I] there is a ~ > 0 such that

(4.s f ( l -~ tt])~IITB U(t) T~] I dt < c~ , - - o o

then there is a mapping M(2) ]rom [0, c~) to B(L 2) which is locally H61der continuous with exponent ~ and such that

(4.9) d (E(2)Au, Bv) -: (M(2)u, v) a.e. . dA

PROOF. - I f $ = ~-~ ie, we have

Thus

(4.1o)

c o

R(~) --~ i f cxp [i~t] U(t) dt o

o

R(~) = - i f exp [i~t] U(t) at

([R(~) -- R(~)] f, g) ~-- i f exp [i,~.t -- sttl] ( U(t)], g) at .

Let t ing e-> 0~ we have

(4.11)

~'ow

(4.12)

d (E(2) ], g) = -- 2z f exp [-- i2t](U(t) ], g) dt . - - o o

(E(~)Au, B y ) : exp [24] d~ (E(),)Tu, Tv) d2

326 MARTI~ SCttECt~TER: Scattering theory ]or second order elliptic operators

(cf. [23, Lemma 4.5]). Thus (4.9) holds with

(4.13)

Hence

M(A) ----- 2~ exp [22] f exp [-- i2t] T~ U(t) T A dt .

iiM(;O!) <2~ exp [2;.] f i t~ ~T(t)T~il dt.

To show tha t M(~) is HSlder continuous, it suffices to consider M~(;t)= e-2~'M(~). But

It M~(,~) -- M~(Z')II < 4~ f lsin ½(), - - ~') t t II Tl~ U(t) T A I1 dt

c o

<2~lz- ,~'1" f Itl~ll r~, u(t)f~]I at , - -oo

showing tha t it is indeed HSlder continuous with exponent 6. [] We now give two impor tan t consequences of L e m m a 4.5. Pu t e : @(x) :

= (i + txl).

COROLLARY 4.4.. - I] A and JB are in. L~ ~ L~ ]or some p < n and q > n, then the conclusion o] Lemma 4.3 holds.

PROOF. - By Lemma 4.2

IIT~ [( t) 2 ~]i < e[tl - ~

< C l t l - ' / o .

Take 6 : (n--p) /2p. Then the left hand side of (4.8) is bounded by

C f (1 + Itt)~ltI-"l~dt-~ - C f (1 -~ ]tt)~ltt-"/~dt, t t t < l ] t l > l

which is finite. []

COt~OLLAI~¥ 4.5. -- I] o~A and ~ are in L" ]or some p > n and ~ > 1 -- (nip), then the conclusion o] Lemma 4.3 holds.

PROOF. - Le t f i < ~ be ~ number satisfying 1 - ( n / p ) < f l < n ( ½ - - l i p ) , and put ]/q ~-- fi/n~- liP. Then 2 < q < n. Moreover, by HSlder's inequality,

A

NIA~TI~ SCHEC:gI~ER: Scattering theory [or second order elliptic operators 327

Since (p/q) '= p/(p -- q), t h e r i gh t h a n d side is finite. H e n c e the h y p o t h e s e s of Corol-

l a r y 4A are satisfied. []

LE:~I~A 4.6. - I] ~ A and e~J~ are in L ~ for some ~ > ½, then the conclusion o]

Lemma 4.3 holds.

P R O O F . - [23, L e m m a s 4.5 and 4.8].

LEMMA 4.7. - I] ~ A and 9 ~ are in Lp ]or some p > 2 and zt > 0 such that > ½ - - 2 n / ( n @ l~p, then the conclusion o] Lemma 4.3 holds.

P~ooF. - [23, T h e o r e m 4.9].

THE0~E~ 4.8. - Assume 9~.4 and ~ B are in L ~ for some p>~2, and that

(a) ~>~ 0 i] 2 < p < 4n/(n @ 1)

(b) ~ > 0 i/ 4n / (n@ 1)<-~p<n

(c) ~ > ½ - - n / 2 p i] p > n .

Then the conclusion o] Lemma 4.3 holds.

PROOF. - The case (a) follows f r o m L e m m ~ 4.7. Consider case (e). The re is a

0 ~ 1 such t h a t 2 ~ - - l @ ( 2 n / p ) ~ 0 ~ n / p . P u t q = O p . T h e n q ~ n a n d a = ~ - -

1 @ (n/p) -- ½0 is pos i t ive . Set 5 = ½ @ a and ~ = 1 @ a -- (n/q). Fina l ly , define

{M(~)} = L!~I(,~)H + s u p !~ - 4 ' l -~HM(),) - i ( ~ ' ) Lt •

B y L e m m a 4.6 t h e r e is a b ~ 0 such t h a t

{M(),)} < Cil ~ AII ¢~ ]tg~/}l] ~ •

B y Corol la ry 4.5 t h e r e is a b > 0 such t h a t

{M(~)} < Clt e" ~ l],He'B/to.

Since ~ = (1 - - 0) (~ @ 07, i t follows f r o m an in t e rpo la t ion t h e o r e m due t o STEI~ a n d

WEISS [20], t h a t

{M(~)} < C iI d 3 ]1, II d/} ]l ~.

This gives (e). To o b t a i n (b), p ick q > n so t h a t

> 1 -- (n/q), a.nd 0 := (½ -- lip)l(½ -- l t q ) , fi = ~I0 .

328 5IA~TLN SCI~C~TEI¢: Scattering theory for second order elliptic operators

Then fl > 1 -- (n/q), and l ip ~- (1 -- 0)/2 ÷ O/q. By Lemmn 4.7,

{M(;.)} < ctlX !I I!/~l!,

while by Corollary 4.5

Hence by the Stein-Weiss in terpola t ion theorem

{M(,~)} < C ]1 ~ X ]1~ II ~ ~B[I~ ~ . []

5. - The loose ends.

We give the proof of Theorem 1.1 based on the results of the preceding two sec- tions. For H we take L ~. The operator H corresponding to -- A is easily defined. One way is to consider it defined on tes t functions and then take its closure. A simple a rgument via Four ier t ransforms shows tha t the closure is selfadjoint (cf. [4, p. 299]). For A take the in terval (0, c~), and pu t C ~ L~(S~-~), denot ing its scalar p roduc t by ( , ) s . By g well known formula

(5.1) d2

where 5 is the Four ier t r ans form and y~ is the mapping from L 2 to C given by

(5.2) 7~h(~) = 2-*)J'~-2)/dh(Po~), ~ ~ S " - L

Thus if we take F = y~. 5 , we have

(5.3) (E(I)) ], g = f(F], Fg) dZs, I

and assumption I of Theorem 2.1 is satisfied. To define the operator H~, consider the bil inear form

(5.4) h~(u, v) = (Vu, Vu) ÷ (Vu, v).

It is well defined on the tes t funct ions since V is locally integra, ble. Moreover, it is easy to show tha t this bil inear f rom determines ~ unique selfadjoint operator Ha such that .D(IH1] ½) ~-D(IHI ~) and

(5.5) (HI*, v) -~ (Hu, v) + (Vu, v) , u, v eD(iHl½),

MARTIN SCIIECHTEI~: Scattering theory ]or second order elliptic operators 329

provided there ure constants a ~ 1, b such tha t

(5.6) (Ivlu, ~) <a][W[l= + b[l~l] ~ , ~ Co

(cf. [4, pp. 320-323]). We shall see present ly tha t (5.6) holds under any one of the hypotheses (A)-(F). Assuming thi for the moment , put A(x)~- tV(x)l ~ and B(x)

IV(x) 1½ sgn V(x). The mult ipl icat ion by A(x) and B(x) represent closed operators A, B whese domains contain D(]H] ~) (cf. [14, p. 72]). Moreover, (5.5) reduces to (2.3) wi th 0 ~ ½ and K ~ L ~. As we have defined it, the operator A need not be injec- rive. However, this can be remedied as follows. Let W(x) be a positive function sat isfying the same hypothesis as V(x). P u t

A ( x ) = ]V(x)l ~, V ( x ) ¢ O

= W ( x ) ~ , V(x) = 0 .

I t is now clear tha t A is injective and the val idi ty of (2.3) has not been affected. Let us now prove (5.6). We first note tha t it is guaranteed by (1.3) (cf. [14, p. 140]).

This is assumed in hypotheses (A), (B), (E) and (F). I f (1.5) holds (as assumed un hypothesis (D)), (5.6) was proved in [13~ p. 21]. I t thus remains to check it under hypothesis (C). We have

(5.7) (AR(-- Z~)Au, v)= ~ f f A(~)lx--y1-1 exp [-- ,~Ix -- yj] A(y)u(y)v(x) dxdy .

Let e > 0 be given, and take 6 > 0 so small t ha t

(5.S) / t Iv(a) V(y)Lt~- yp'-a~dy < (2~)~

(this can be done by (1.13)). We break up the integral in (5.7) into the sum of two integrals, one over the set Ix-- y] < 6 and the other over the set Ix-- Yl > 6. The first is <½elIullI[vll by Schwarz's inequal i ty and (5.8). The square of the second is bounded by a constant t imes

ff ff

t r (y ) j ix - y 1-1 exp [ - ~ Ix - y i]Iv(x) I" dx @

I V (y) l lx - - y1-1 exp [ - )dx - y t]lu(y) l~ dx dy .

Thus the second integral is bounded by

C exp [-- ;~i)]liulI tlvll SUp 1~_i> tv(y)i Ix- yt-ldy •

330 ~IARTI:~ SCHECH:r]~n: Scattering theory ]or second order elliptic operators

This tends to 0 as ~-~ c~ by (1.11). Thus the operator AR(--~2)½ tends to O in norm as y -~ c~. Thus

for ), sufficiently large. This gives (5.6). So far we have verified assumptions 1, 2, 3 and 7 of Theorem 2.1. Turning to

assumption 4~ we note tha t it is implied by BR(~) being compact . By Lemma 3.10~ this is t rue under any of the hypotheses (A)-(D). On the o ther hand, it is also implied by (1.3) and (1.6) ([41, p. 111])~ and these were assumed in hypotheses (E) and (F). Thus assumption 4 of Theorem 2.1 is verified.

Similarly, assumption 5 follows from Theorem 3.9 in the ca.ses (A)-(D). For the other cases we use Corollary 4.4 ~nd Theorem ~.9, and apply Lemm~ 2.2. Note t h a t

X ( x ) ~ = / } ( x ) ~ = f IV(Y)1 dy ,

so tha t the ~, p have to be halved in the application. F inal ly we t u rn to assumption 6. For the cases (E) and (F) we use Corollary 4.4

and Theorem 4.9 again. For ~he other cases we note tha t

2zi ~ (E()~)Au, Av) = ,-~01im (A[R(2 + is) -- R(~ -- is)]Au, v)

where ~ ( ~ ) is the continuons l imit of AR(~ ~= is)A as guaran teed by Theorem 3.9. We now merely take M(2)-----[0~+(~)--~_(~)]/2~i. Thus all of the assmnptions 1-7 of Theorem 2.1 are verified. Theorem 1.1 is an immedia te consequence. []

RE~[ARK. -- I t should be no ted tha t (1.3) can be weakened. I t was used only to obtain (5.6). As no ted in [14~ p. 168] there is a constant c depending only on n such tha t (5.6) is implied by

liminfsup~__~o ~ f IV(Y)I ]Y--XI2-~ dy < c"

~ote added in proo]s.

We wish to call attention to the very interesting paper of R. A. REJTO (0~ a limiting case o] a theorem of Kato and Kuroda, J. Math. Anat. Appl., 39 {1972), pp. 541-557) which weakens (1.2). (We would have described this result in our introduction had we known about it then.)

5 £ ~ SC~ECH~E~: Scattering theory for second order eltiptiv operators 331

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