Limiting Absorption Principle for Partial Differential Operators (Memoirs of the American Mathematical Society)

Memoirs of the American Mathematical Society

Number 364

Matania Ben -Artzi and Allen Devi natz

The limiting absorption principle for partial differential operators

Published by the

AMERICAN MATHEMATICAL SOCI

ETY

Providence, Rhode lsland, USA

March 1987

.

Volume 66

.

Number 364 (end of volume)

MEMOIRS of the American Mathematical Society

suBMlssloN.

This journal is designed particularly for long research papers (and groups of gog.natelapels) in pure and applied mathematics. The papers, in general, aie longer than those in the TRANSACTIONS of the American Mathematical Society, with which it shar-es an editorial committee. Mathematical papers intended for publication in the Memoirs should be addressed to one of the editors:

ordlnary dlfferentrar equarlons, partlar drfferentlar equatrons, and appiled

mathematlcs to JOEL A. SMOLLER, Department of Mathematics, University of'Ulcnigan, Ann Arbor. Ml 481O9

complex and harmonlc analysls to LINDA pREtss RorHscHlLD.

Departmenr of Mathematics, University of California at San Diego, La Jolla, CA 92093 Abstract analysls to VAUGHAN F. R. JoNEs, September 19g6-July 19g7: lnstitut des Hautes Etudes Scientifiques, Bures-Sur-yvette, France 91440 classlcal analysls to PETER w. JoNEs. Department of Mathematics. Box 2155 yale Station, Yale University, New Haven, CT 06520

Algebra' algebrarc geometry, and number theory to LANCE w. sMALL. Department of Mathematics, University of California at San Diego, La Jolla, CA 92093

Geometrlc topology and general topology to RoBERT D. EDWARDS. Deparrment of Mathematics, University of California. Los Angeles, CA gOO24

Algebralc topology and dlfferenrlal topology to RALPH coHEN. Department of Mathematics, Stanford University, Stanford, CA 94305

Global analysls and dlfferentlal geometry to TILLA KLorz MlLNoR, Department of Mathematics, Hill Center. Rutgers University, New Brunswick. NJ 09903 Probablllty and statlstlcs to RoNALD K. GETooR. Department of Mathematics. University of California at San Diego, La Jolla. CA 92093

comblnatorlcs and number theory to RoNALD L. GRAHAM, Mathematical Research center,

sciences

AT&T Bell Laboratories. 600 Mountain Avenue, Murray Hill, NJ 07974

Loglc' set theory, and general topology to

KENNETH KUNEN. Deparrment of

Mathematics, University of Wisconsin. Madison, Wl 53706

All other communlcatlons to the edltors should be addressed to the Managing Editor, wlLLlAM B. JoHNsoN, Department of Mathematics. Texas A&M University, college Station, TX 77843-3368

PREPARATIoN oF coPy.

Memoirs are printed by photo-offset from camera-ready copy prepared by the authors. Prospective authors are encouraged to request a booklet giving d'e_ tailed instructions regarding reproduction copy. Write to Editorial Office, American Mathemitical society. Box 6248. Providence, Rl 02940. For general instructions. see last page of Memoir.

suBscRlPTloN lNFoRMATloN. The 1987 subscription begins with Number 35g and consists of six mailings. each containing one or more numbers. Subsiription prices for 1987 are 0227 fist, $182 institutional member. A late charge of TOyo of the substription price will be imposed on orders received from nonmembers after January 1 of the subscripiion year. Subscribers outside the United States and tndia must pay a postage surcharge of ti5: su-bscribers in India must pay a postage surcharge of $43. Each number may be ordered separately; ptease specify number when ordering an individual number. For prices and titles of recently released numbers, see the New Publications sections of the NorlcES of the American Mathematical society. BACK NUMBER INFORMATION.

For back issues see the AMS Catalogue of Publications.

Subscrlptlons and orders for publications of the American Mathematical Society should

be

addressed to American Mathematical Society, Box 1571. Annex Station, providencel Rl 029019930. A/ orders must be accompanied by payment. Other correspondence should be addressed to Box 6248, Providence, Rl 02940.

MEMOIRS of the American

M-athematical Society (ISSN (x)65-9266) is published bimonthly

(each volume consisting usually of more than one numbLr) by the

Lmeriiln-ilaltrematlcaf Societi

at 201 Charles Stre€t, Providence. Rhode lsland 02904. Second Class postage paid at provi_ dence, Rhode lsland 02940. Postmaster: send address changeq to Memoirs-of'the American

Mathematical society, American Mathematical society, Box 62-49. providence, Rl o2g4o. Copyright @ fOaZ, American Mathematical Society. All rights reserved, lnformation on copying and Reprinting can be found at th; back of thls journar. Printed in the United States o[ America. The paper used in this iournal is acid-free and falls within the cuidelines established ro snsure

Tesr,e oF CoNTENTs

Section

page

1. Introduction

2.

.

..

..

..

Preliminaries....

...

1

..,...5

3. The Limiting Absorption Principle for

Il :

Ho

*V

_

.,..... . .

L2

4. Stark Hamiltonia.ns with Periodic

Perturbations

5. The Schr6dinger Operator

-A

...... +

y

.

.. .

22

Bz

6. Simply Characteristic Differential

Operators

... ....

45

...... ..

60

7. Some Further Perturbations of

8.

-A .

References

..

ul

.....

..

..

69

ABSTRACT

Let rr be a self-adjoint operator in a H'bert space ,v. It is said to satisfy (limiting the absorption principle, (l.u.p. in t/ c R if the limits 13a(,\) : ) lh"*o+(11 - ) afe)-l, ) e u, exist in some operator topology of B(r,!),

,cX,!cI.

The paper presents a unified abstract approach to the l.a.p. for operators of the form H : Eo + z. The spectrar measure associated with f/e is assumed to satisfy certain smoothness assumptions which yield immediately the r.a.p. . The perturbation r is assumed to be oshort-range' with

respect to .I{6 (a concept which is introduced in the abstract setup) and the l.a.p. for 11 is proved, along with the discreteness and finite multipricity of its eigenvarues embedded in

[/'

various classes of diferential operators are studied as special cases, incrud_ ing schr6dinger operators, generalizations of the stark Hamiltonian and simply characteristic operators. In each case, the verification of the abstract assumotions imposed on IIe is simple and straightforwa^rd.

AMS (Mos) subject crassifications (1080; Revision 1985). primary 81c12,

81F05; secondary 85p25, 47A40.

Key words and' phrases. Limiting absorption principles, wave operators, scattering theory, differential operators.

Library of Congress Cataloging-in-publication Data Ben-Artzi, Matania. l94g_ The limiting absorption principle for partial differential

operatots.

(Memoirs of the American Mathematical Society, ISSN 0065.9266; no. 364) "March 1987.',

Bibliography: pl. Partial differentiat operators. 2. Scattering (Mathematics)

I. Devinatz, Allen. II. Title. III.

911:As7 rsBN

Series-

no.364 tQA32s.42l 5r0s

0-8218-2426_0

lv

lsts.7'2421

87-180?

INTRODUCTION

our aim in this paper is to present an abstract unified approach to limiting absorption principles for self-adjoint operators sshort-ranger with perturbations. specificallS let I1o,

.Fr

such that

be self-adjoint operators in a separable Hilbert space x,

E:Eo*V.

(1.1)

In very general terms, the limiting absorption principle can be stated as follows. Let R(z): (E - z)-r,Im z O,be the resolvent operator and let I, I ! be Hilbert spaces such t'hat r is densery and continuousry has a stronger norm). Then one says that

principle in an open set U C R if the limits

Er())

:

embedded in )/ (and thus

ri

satisfies the limiting absorption

"It!a()*ie), )eu,

(

1.2)

exist in the norm topology of

from

'f

into

!'

B(I ,!), the space of bounded linear operators Naturally, one takes Lr to be co'tained in the spectrum of r/.

The importance of the limiting absorption principle lies in the fact that it implies irnmediately some significant spectral properties of 1r. Thus, for example, if a dense subset of .V can be identified with elements of the dual space !* then I/ is absolutely continuous in I/. Furthermore, it was shown by Kato and

Kuroda [g] the limiting absorption principre (in the same setting), then fr and r/6 are unitar'y equivarent over u and this equivarence can be realized via the existence and compreteness of the associated wave-operators. In concrete cases (namely, differential operators) this approach corresponds to the so-called nstationary methodo in scattering theory. It was used by Agmon [1]

that if r/o

Received

arso satisfies

ty tt'"

't-,fr 29, 19g6 and in revised form June 29, "ait-r Partially supported by USNSF Grants MCS 8200896 and DMS 8501520.

19g6.

3

M,

III:N.ARTZI AND

A.

DEVINATZ

fr* *;';iri*plrte eiurly of ficlrnidinger operators with short-range potentials (more !*rulally, {,p.t',rt,,rrrr pri'cipal type). This was later extended by Agmon and

'f

iJbtrl*rrrler lz,ll to the study of simply characteristic operators. As an exampre rif e llarniltonian 116 which has non-constant coefficients one can consider : 116

'''a -' sr,

the quantum-mechanical Hamiltonian of a free pa.rticle in a uniform clecfric field. When adding a Coulomb potential V(r): _lrl-t, it represenrs the well-known "stark Efect.o rhe limiting absorption principle for this case was studied by Herbst [6] and in somewhat more generality by yajima [12].

All of the above mentioned exampres will be shown to be special cases of the general method presented here. At first (sections 2 and B) we shan construct an abstract framework from a minimar number of assumptions, which nevertheress are suftcient to guarantee the limiting absorption principle for fle. In doing this we shall focus our attention on the most fundamental idea of this work, namery

the srnoothness properties or ,t'e spectrar measure of Ho. As we shalr see, these properties are extremely easy to verify in all of the concrete cases. Furthermore,

it

has been shown

in a previous work [5] that under rather generar conditions they are "transmittedt through sums of tensor products. More expricitly, if Ho : Hr & Iz * Ir @ Ez (where 11, 12 areidentity operators) and if H1, E2 possess these properties

it

means that if

rle

then so does fre. Turning back to differentiar operarors,

has sepa^rated variables then we need onry study the spectrar

structure of its elementary components. Given the appropriate abstract setting for .616, the study of rr is carried out by perturbation-theoretic arguments. Thus we start out by introducing the concept of a (short-rangeo perturbation I/ in this setting (see Definition

2'1) and proceed to derive the limiting absorption principle for r/. As is wellknown, one cannot rule out completely the possib'ity of a point spectrum oo(rr) embedded in the continuous spectrum. However, to prove its discreteness (and

the finite multiplicity of eigenvarues) we must impose an additional assumption on I/ (Assumption 8.2) which "intertwineso the smoothness properties

of the

spectral measure of r/6 and the short-range character of z. In the applications this condition is satisfied by imposing a rapid decay condition on y (e.g., (1

+

THs Lrl\,rrtlt{c AnsonprroN pRrNcrpLE

lrl)-s/z-c for the schr6dinger operator). when ,,optimal' decay rate is desired (u.s., (f + lol)-t-e for the Schr6dinger case) we use a .bootstrap, argumenr based on elementary interpolation techniques (this is analogous

to upgrading

the decay rate of possible eigenfunctions). However, we note that in many cases the restrictive assumption imposed on z leads, to the best of our knowledge, to the only available proof of a Iimiting absorption principre (see sections 4, z). Following the abstract presentation we discuss various classes of difierential operators in Sections 4-7.

In section 4 we discuss a generalization of the stark Hamiltonian of the form -f1 -zt*s@r)+7.,+V(a), where s: (a1,r') € RxR.-1. Here nAZ

q(,'1) is a periodic one-dimensional perturbation of the uniform electric field, e, is a self-adjoint semibounded operator in 12(n;,-1) and v(a) is a short-range

(with respect to c1) potential depending on all coordinates. The results obtained here are straightforward applications of the general theorems. Remarkabry, the properties

"f -#

- r + q(a)(x

€ R.) are such that very little is required of the

part depending on the remaining coordinates. The verification of the abstract assumptions is therefore reduced to very elementary (one- dimensional) asymptotic estimates and properties of tensor products. In particular, for the case

Ts,

: -[.', g:0

our results are identical to those of yajima [fT].

section 5 deals with the schr6dinger operator -a+v and provides yet another example of the reduction procedure to the one-dimensional

case. It

turns out that our definition of nshort-range" perturbations coincides precisely with Agmon's definition [1] in this case. section 6 extends the stuily of

-A*I/

to the class of simpry cha.racteristic

operators (a class that contains, in particular, alr hypoelliptic or principal type operators)' our results here are in general similar to those of H6rmander [z],

except that we are working in a weighted Hilbert space framework which allows us to derive convergence in operator norm (in (r.2)) and smoothness of

a+()).

In Section 7 we have

chosen two classes of operators (see (7.1), (7.2)) to

illustrate the broad applicability of the abstract theory presented here. In both cases a suitable limiting absorption principle has been proved in the literature.

M. f{rrwugrr,

rli'ar:tly

*tr ltrirrl

lltr:w-Atn'ilr rttrlr

A.

DuvINnrz

lrpre is {,hilt whil* the abetract theory may not always yield

l,lre rir;rrperr, poaaiLrl*

ral,e {rf por[urlrnl,iorrr

rasults (in terms of the weighted spaces used, decay tnd ao on), it can still be applied to a wide variety of cases

wil,lroul tho need for a very detailed study of special properties. Finally, it shoulil be noted that, among others, Kato and Kuroda subsequently Kuroda [10,

ul

[9] anil

have developed abstract theories to deal with short-

range scattering. However, the first named authors present their formulation in rather broad terms while the latter relies on a number of abstract assumptions

of which several crucial ones seem to be modeled on perturbations of constant coefficient elliptic operators by short range potentiars without locar singularities. These assumptions do not appear either to cover a number of concrete case which are presented here, or at best they would be difficult to

verify.

we also note that the mentioned papers use the sometimes convenient method of factorization of the potentiars. In some cases certain ancilary re_ sults which can be obtained from a limiting absorption principre for short range potentials, e.g. , the existence and completeness of wave operators, can often be effectively dealt with by the use offactorization (see e.g. [14j, [1S]).

2. PRELIMINARIES

In this section we set up our basic notations and assumptioas, and recall briefly some of the results and definitions of [bl. In this work we shall

use onry sepa'abre Hilbert spaces.

spaces (with norm usually denoted

(B(T

,T): 8(7))

lv ll llr,

If r, s are such

etc.), we designate by B(T,S)

the space of all bounded linear operators from

operator norm designated by ll

T to s with

117,5.

As we mentioned in the Introduction, we ret

rle be a self-adjoint operaror in a separable Hilbert space .v, and let r be a space which is densely and continuously embedded in ,v (thus, the norm of ,f is stronger than that of )/). clearly

,v can be considered as densely and continuously embedded

inner product. we denote by

with

116, and by

{Eo(r)}

in .L*, via its the canonical spectral family associated

Eo(d^) or d.E6(.\) the corresponding spectral measure.

Limiting values of the resolvents Rs(z) as

z

:

(Hs

_ 21-r

and

B(z)

:

(H _ z)_t

approaches the (real) continuous spectrum from above and below,

B(r,,f *)'

elements of

operator

v

will

Actually more general singularities of the perturbation

may be allowed

if

one restricts further the range of these operators.

Thus we may assume that these limiting resolvents have range in a subspace r;{o c.f", generally dependent on rre, which is densery and continuously embedded. However, we do not assume

,

normed domain of the closure of rre in

Ifi".Typically .ffio will be the graph .r* (provided such a closure exists) or a

C

subspace of functions for which certain distributional derivatives are

in

r*.

In

the former case our theory becomes particurarly simpre and incrudes a number of important applications.

DEFINITIoN

Let 0


2.1

U c R be open such that.E.(R\U) : 0. Hs is said to be of type (X,Iito,a,U,) if the following

Gee

1,. then

,

be

[5]).

Let

6

M.

co nd it

(i)

ions are

B0N-ARTZI AND

A,

DpvINerz

sa tr'sf ed..

The operator valued function

)*Eo(r) eB(f ,f*1,

^eu,

is weakly difrerentiable with a IocaIIy H6lder continuous derivative in

B(X,Ifi");

that is, there exists an operator valued function

) *,40())

e

B(X,Xiq), ^

e(J,

such that

ft{"ol^)r,v)

: (AoQ)",v|, a,y € r,

(where the right hand pairing compacb

K

CU

6.here

il,ag(r)

(ii)

For

is

exjsfs an

that of

Ms

x*

and

A

t ),and

e

u,

(2.1)

suc.h that for every

satisfying

- Ar(p)lft,xi,o S Mxl\_ p1,

),,p€

K.

(2.2)

*-" u-",0: r,*"",J),,::::r'function

&*:u

;"" A-z Ju\K takes values in

B(x,rfi)

and

is locarly Hirder con,inuous in the unilorm op-

erator topology. The classical Privalofi-Korn theorem now yierds a nrimiting absorption propertyo for the resolvent of f/6 (see Theorem Z.B of [S]).

THEoREM

2.2.

Let

Hs be of type (f

,?d()) exist

in

,

f fi",a,(I);

: ,\p no() + ie),

the uniform gperator toporogy of

then the limibs

[r,

(2 s)

B(r, rft") and are rocarly

H6lder

€ ^

contiauous with the same exponent a. Rgrure'Rxs

2.3.

(a) Note that our hypotheses, in conjunction with the rast theo-

rem' imply that the spectrum of rls is absolutely continuous. These hypotheses

TXP LrIVrrrrNC are usually satisfied

ASSORPTTON PRINCIPLE

in applications to differential operators and simplify

the

statement of subsequent theorems.

(b) If in Theorem 2.2 we assume that U sup ll,{6(.\)111,';o "

ren

.

:

R, and.

-, t/p ,* ]119rr39]]gi ll - pl.

( oo,

(2.4)

then also the limits 8o*(.\) are uniformly bounded and uniformly H6lder continuous in R and the convergence in (2.8) is uniform in

R

(see Theorem 2.3

of

tsl). For future reference we now note an important (for applications) case when

.f;io is taken in the special way noted previously. THEoREM

its domain

2.4. Let Hs be closable in f*,with closure F,6, and let X,iro be 6" f-) equipped with the graph norm, ll"ll7 ;

suppose fur6-h er that

4 (r) :

"

: ll"ll"r. + llE o"ll"r

"

(2.5)

.

dEo(I) I d exisrs in the weak toporogy of

in the uniform

and is IocaIIy Hiilder continuous

topology

of that

B(r , r")

space..

Then all of the conditions of Definition 2.1 (and hence the result of rheorem

2.2) hold true.

Proof

Note that the present assumptions imply that the limits (2.3) exist in the

uniform topology of

B(x,.L.)

and are locally H
foreveryc€X,,

and

"LT! "o(^ tim FoRo(,r,

€+o+ Since f/-6 is closed

*ie)x:

*

ie)r

rft*(.\)c e

.f",

: r + )R*(,\)r

e

f..

in .f * we have

Eo,?on()):r+)xf()). It follows that af,()) are locally H6lder continuous in the uniform B(f , f fi). The estimate (2.2) now follows from /4o(.\)

: fr t+t^t - u; ())).

topology of

(2.6)

M. IlrN-ARTzt

AND

A.

DEVTNATz

Firrally, using the same notation as in Definition 2.1 (ii)

'olurnflo^:

it

follows that

Iu,*g'

is a Lipschitz continuous function in the norm topology of

B(I , Xfi"). I

2.5. For future reference we shan note the folowing formura. Let p € If where K C (I is a bounded open interval. Then as was shown in RuvraRx

[5],

a#(p):

\

a

,, l*ffiar *izrAsa4 * luro$^

Next, let I1e be of type

(I ,Ifto,o,t/)

(2.7)

and

H:Eo*V.

(2.s)

we want to consider limiting absorption for fr in u, with respect to the norm topology of B (T, I fi ), for I/ a "short range, potential with respect

to,FI6. Thus we shall define such a potentiar which is nothing more than an abstract version

of the definition of short range given in various concrete situations. Among other things, a short range symmetric i:otential will ensure the self_adjointness of r/ on a suitable domain. In the next section we shall be adding an extra hypothesis on the potential in order to ensure the discreteness of the eigenvalues of rr' However, in the concrete apprications which we shalr consider, this extra abstract hypothesis will be provable simply from the hypothesis of short range

on V. In what follows we denote bV De) the domain of an operator ? (specifying the underlying space if necessary). If not specified, the domain is taken in )/. DEFINTToN

(i)

2.6. An operator V : Ifi. I -

wiII be

called.

short, range withrespec6 to Hs

if it is compact; (ii) synmetric if D(Eo) n Ifto is dense jn X and V D(Uo) n I in X.

.f;" is symmetric

2.7. If Ho is of type (X , Iiro, a,()), V is s.horf range wibhrespecr to Hs and symmetric, and there exis's a non- rear z and a linear space C THEoREM

D,

Txn LrurrrNc AssoRprrou pRrwcpr,p D(Hs)nLi.o so

thab

the range of (Hs-

defined on D, is essentially self-adjoint.

")l

D, is dease in

I,

then

H : Hn+V

Proot. Clearly, I/ is a symmetric operator in )/. Further r € D, implies (H Hence, ;r

g +vRs(z))

- z)t : (I + v Ra(z))(86 _ z)x.

is invertible in

.f, it

follows that

r/-z

has dense range

in )/. On the other hand, for any non-real

u, Rs(w)(Hs_z)D":

D_ is in D(Hs)n

(H6-w)0. : (Hs-z)0.. Thus (IIe-ta) | D- \asdense range contained in .L. Thus, tf (I +v R6(w)) is also invertible in x , H has zero deflciency indices. Let rl be any non-real number. using the compactness of I/, it follows

Iiro

and

from the Fredholm alternative that either (I +VBo(u)) is invertible

in.I, or ( e X sothat (I +VRs(w))Q: 0. Let Ht : Ho *V be the symmetric operator with domain D(Hs) n Xfio and. set ry' : Rs(u)g e D(Hs) n.Lfio; then (h -.)r/t: 0. But a symmetric operator cannot have a non-real eigenvalue. I else there is a non-zero

Conor,r,env 2.S. (to the proof). If Hs has aclosureI.o in D(Eo)

*itn

the graph norm, then

f"

if [fro: D(nd g Ifio and E : Ho+V with D(H): and

D(Ho) is self-adjoint.

Proof. Clearly D(Ho) g f;{" : D(Iz). Thus H : Ho*

tr/ is well defrned and symmetric on D(H6). But in this case, for any non_real ut,VRa(w) is compact from )/ to ,V, so that the same proof as above shows that (1 Izr?6(u)) + is

invertible in .v. Reviewing the first part of the previous proof shows that the range of H - z is all of ,V. I

Rpuenx 2.9. Another proof of the previous corollary fo[ows from the relative compactness of v with respect to r/6 and the Rellich-Kato theorem. we shall continue to designate the serf-adjoint crosure of rr by the same symbol, and let R(z) be its resorvent. Let us now consider the resorvent equation

R(z)(I +V Rs(z)) =

Ro("|,

Im z 10.

(2.e)

M. BnN-Anrzr

10

AND

A.

DEvTNATZ

rls and I/ this equation is certainly well defined. from I - Xfio. As we have already noted in the proof of the last theorem, for Im z 10, (/+ VA"Q)) is invertible in .f . Thus (2.9) leads to Fbom the hypotheses on

E(r)

:

ft6 (z)

(/ + V Rs(z))-L

where the equality is certainly valid from suppose now that

) e u.

(2.10)

,

I * tft".

By Theorem 2.2

and.

assumption (i) of Definition

2.6 we have

*ie)

Y,%()

:

v,Ro+(.\)

"1T.

Thus,

if (f + yE'*(,\))-1

n*(r)

BW).

exists, then (2.10) implies the existence of the limits

:,\t. r(^ *ie):,?"+())(/+

in the norm topology of Let,

in

y.B*()))-1

,

Q.Lt)

B(I , Xfi").

p € Lr be a point at which, say, t +V

The compactness of v,RS (p) in

.r

and the Fredholm alternative imply that this

is possible only if there exists a non-zero d €

6

fit) is not invertible (in B(.I)).

na+

:

-v nt

.f so that (2.r2)

0")d.

Set

,l' bt,,

d)

:

:

at

_

"rip

0,)d e

(&(r *

Xfi";

then

ie)g,v R"(p + ie)O).

By conilition (ii) of Definition 2.6, i.e. the symmetry of

in

.V,

(2.13)

r

the right side of the last equation is real for every e

t-:

on a suitable domain

)

tj:,

0, so that (r/,

/)

is

real. Thus

I^(Rtfu)$,d)

:

o.

(2.14)

Flom (2.14) and (2.7) we immediately conclude rhar (Ao(p)6,d)

By (2.1), the form (,4e())o, y) on

I x.f

:

o.

(2.15)

is Hermitian symmetric and positive

semi-definite. Hence the Schwarz inequality and (2.15) show that

Ao(P)6:

o.

(2.16)

,a

Trrp LuvrrrlNc ABsoRPTIoN Observe

PRINoIPLE

11

that (2.6) in conjunction wirh (z.rO) yietas At0")6: Ri1id.

Thus, instead of (2.12) and (2.13) we actually have 4

:

-v RiQ')6,

{:

(2.L2')

n"'U')4.

(2.13t)

Using conventional terminology [1], (2.13r) means that

ry'

is both "outgoingn and

"incoming.t

2.10. The equivarent conditions (2.15) and (2.r0) may be considered

REMARK

as a generalized ozero traceo condition, since in concrete cases, where a Fourier

transform is available, this is equivalent to the trace of 6'u"ios zero on a certain manifold determined by 116 and p. we also note that we used (2.7) to conclude (2.15) from (z.ta). Now, (2.2) is valid because of the H6lder conrinuity of ,,{o(,\)

which is needed for the existence of the Hilbert tra.nsform. Indeed, (z.rs) will follow from (2.14) rt we only assume that (/0())d,/) is continuous in a neighborhood of where o

(,4e

:

p.

To justify this remark let

Im(Rt 0t)4, d) :Im

: Let

K C [/

be an open interval about p,

()) 6, 6) i" continuous. Then,

ls

(.Bs( F

+ ie)$,41

"hm. ar lim o"1?*

I I Uo(t)6' t't ; I A:Wiha^:"(Ao(P)6,6)'

I

be the set of points in U where (Z.tZ) (or, equivalently, (2.12'))

holds. As we have seen, we have limiting absorption for.Fr in t/\rg in the norm topology of B(t,rfio). Thus the problem is to identify the set Ds in terms of more familiar and classical numbers associated with the operator

oo(Hl n u, the the next section.

eigenvalues of

ff,

namely

H in tl. we shalr carry out this identification

in

3. THE LIMITING ABSORPTION PRINCIPLE FOR H

: Ho.tV

As we observed at the end of the last section, the (rimiting absorptiono

limits (2.11) exist in

u\xr, where Es is the set of points in Lr where (2.12) (or, equivalently, (2.12t)) holds. ln this section we shall identify !g with the

eigenvalues of

rr in u,

a.nd indeed we shall be able to establish the discreteness of this set. In our abstract context this requires an extrahypothesis on the potential

7

which combines its short range nature with the smoothness properties of the spectral measure of rre, when this spectral measure is restricted to a subspace r. This relationship will become clearer when we discuss va.rious concrete exampres, where actually it will be shown that this extra hypothesis may be proved under known optimal hypotheses for the space .f and the potential

Z. In order to deal efectivery with the above questions it is first necessary to construct a space unitarily equivarent to ,f which wilr not only diagonalize , 116, but in addition will take into account the role of the space r. In various concrete cases such a technique is equivalent to the use of the Fourier transform, or various modifications of it depending on the differential operator 116.

ForleUand.s€f.set r()) :,,{6(.\)c. For such ,1, define a linear space

I{

G

.ffi"

(3.1)

by

Hi:{t()) :reI}. For z,y

(3.2)

€ .I, the form (;(.\),

where the right side is the ,f

i(.\))l : (,{s(l)c,

y),

", .f pairing, is a scalar product for ,Vi.

(3.3)

Indeed,

it

is a semi-definite, Hermitian symmetric form so that schwarzrs inequality shows T2

i $.

Trrn LruIrrNG ABSoR"PTIoN PR]NoIPLE that (/()),t()))^ :0 implies A()) :0.

by.tfi the complerion of )(

Denote

under the given norrn. Thus )/r is a Hilbert space for every Next, consider the subset S c f|,V1 given by S

: i;:;()) :,4s())c

clearly s is a vector

space over

for some (fixed) r €

c

13

I

) € U.

and

) e t/).

(by component-wise addition). Furthermore,

if try €.f , then one can define a scala.r product on S by

(i,ilx, ,: [@{^),i()))^d) : [{oo{^),,v)d): (,,c)x. J"J This sets on

r unir*r"map r + ; u"trr,""l L $s

adense subspace

(3.4)

of X) and

s. Let )/o be the completion of s under the norm defined by (e.a). Then.v is unitarily equivalent to ,vo and we continue to denote by r - i this unitary map. Note that if 7,i e X@, then there are sequences {r.), {yr) C .f so that a.e. -Lebesgue,

i*(^)::{6())r,

* i()),

i*(^):,{o())y, - i()) in }/.r.

(7(,\),i(,\))r : lim(,4e(.\)c.,v.) is measurable and (3.a) holds for i,{ and i(l), !(,\). Thus i,! e X@ may be considered as elemenrs of fl .V1. Hence

Suppose now that

A C U is a Borel measurable set and a,g €

I . Then

by

(3.4), (.oe(A)c,

d

:

IQ+o{t)x,y)d,): [@{^),i()))^d.\. Lr"

(3.5)

This equality may now be extended by continuity to all z, y e )/. Indeed, (8.5) shows

that

LEMMA

,v@

3'1.

diagonalizes Es, which is the content of the next remma. Let

g be a complex valued

Borel

i"."ur.b]"

function on

(1.

Then I

J lo0ll'lw(t)ll?

d) <

oo

U

->

xe

D(s(H6)).

(3.6)

In the event that either side of (3.6) is true

s(H6b0): e());(.\).

(3.7)

14

M. BEN-ARTZI AND A.

DEVINATZ

Proof. The equality (3.5) implies that for r € ,V, f^^f

I b/))nw(\ lli d): JUI lg(t)1,(so(d))a,a). JU Thus the left side of (3.6) is finite if and only if

it

r e D(g(Hs)). In the latter

case

follows from (3.5) that for every g € X,

b(Ho)*,Eo(a)y)

: I o@@,(il)x,yl i

: /I -'" n661^r,i()))^d) A

: |f- (c(Ho)"(.\), r(r))rd). t" Hence, by the separability of N, (3.7)

follows. I

We now shall impose an additional assumption which combines the short range nature of

V with the structure of the spectral measure of I/6. This as-

sumption will enable us to prove that" oo(H)nU

:

Es., as well as the discreteness

of this set of eigenvalues. ASSUMPTIoN

with

pr

3.2.

Let

K C U be compact

e K. Then there exist positive

and

constants

$€

C

I, a solution of (2.I2'l

and

e (depending only

on

Hs and K) so that

(,{o())d,4)


(3.8)

The following simple lemma gives a sufficient condition which implies the

validity of this assumption. LEMMA

3.3.

Let X1 C X be a continuously embedded space and assume 6hat

in some neighborhood

(i)

N,

of

p.,

The operator-valued functions

\-I2ftf,()), have values in B(X , 11), and

l€N,,,

Trrs LIMrrrNc AssoRprIoN PRINCIPLE

(ii) in N*,

,4e

())

is weakly differentiable in B G b

X

15

i) with the derivative Ato(\)

Hiilder continuous in the norm topology of I)(Xr, XI). Then the condition (g.E) is satisfied.

Proof. By assumption {i) aX solutions of (Z.tZ,) are in .L1 ancl satisfy lldllr, < Ullfllr. Now, (,{6(p)/,4): o while (,ae(.\)/,d) > 0. Hence (,4i,(p)d,il: o. Thus by the H6lder continuity assumption of (ii) we have in .A{u, l(A!ot)4,d)l < cl) _ priltil?, < cM2l^ _ pf Integrating this inequality we obtain

116ll"r.

(8.8). I

observe that the conditions (i), (ii) of the lemma reflect, respectively, prop-

erties of

I/

and

{Ee(.\)}.

However,

in concrete

cases these

conditions do not

yield the best results under which short range scattering theory can be developed. For example, 'tf Po(D) is a real constant coefficient elliptic operator of order rn, the condition (ii) is satisfied with

r : L2,"(R."), tr_r,2,r+r1g';, s> tf2, (see

Appendix in [5]). But then for condition (i) to be satisfied we need a decay

rate for

lcl-t-"

v

of

at infinity. This is to be compared with the decay rate of which is optimal for the short-range theory in this case.

lrl-z-'

As we mentioned in the Introduction, in a variety of concrete cases we will show that the condition (3.8) follows by using a .bootstrapo argument based

on suitable interpolation theorems. using such a technique, we will be able to obtain proofs of the limiting absorption principle for, say, the stark

Hamiltonian

or short range perturbations of simply characteristic operators. These proofs are considerably simpler than those extant in the literature. However, instead of trying to develop this in the abstract, we feel the arguments will be clearer and more meaningful if.they ale carried out in concrete cases. Recall now that we had denoted the set of points in [/ where (I+yEo+(p))-1 does not exist by

LEMMA

3.4.

Es.

We also denoted the point spectrum of

Under the hypotheses of Theorem 2.2

oo(H) nU

c Ep.

I/ by

oo(H).

M.

16

BEN.ARTZI AND

A.

DevIH,c.Tz

Proof. If p 4 Ds, (I + V Ri 0t))-1 exists, and by the continuiry of lzg* ()) as a function of .\, the inverse exsists in a neighborhood of pr. Thus g+()) exist in a neighborhood of p so that rr has an absolutely continuous spectrum in this neighborhood. Ir follows that p, $. oo(H) n U. I For the rest of this section Assumption 3.2 on Hs and

I/

\n'e assume

the hypotheses of rheorem 2.7 and,

and proceed to show that or(H) O U

: Xrr and

the discreteness of this set. THEoREM

3.5.

U

nor(H) :8".

Proof. Flom the last lemma the left hand set is contained in the right hand set. Conversely, suppose p e Es and / satisfies (Z.tZ,). Set ry': 8$(p)d; lr" claim that p e or(H) with eigenvector /. To see this, let .8( be a compact interval in tI whose interior contains p. Set

,I(r) =' xot"tj '' ""{o())d _ 1t + tr

,t1''10(l)d xr((^,)i;.

: G- y *(l)) /(l-

The function g(.))

p) is a bounded Borel measurable function so that by Lemma 3.1 the second term on the right belongs to Ha. As for the first term, using Assumption 8.2, we have

qo{d6f l^_.q-

II K Hence, taking g(,\)

that the first term

d^ <

M

ol,,-lldtt? [ -, v_pr_,

*

<

"".

: Xr(f)lA - pl-t we may again invoke Lemma also belongs to IIo. Thus, r/belongs to.Vo.

we claim that under the unitary map from N to N@ we have ,b suppose t/ ir tttu unita,ry map of {o e X. Then we have

ll&(p +

iu)6

-

,toll,x

8.1 showing

$,

Indeed

."t^lo - 4.(r)d ll' a1. - J/ llll^-tL +xe ,r-p ll,l

clearly, the right hand sicle goes to zero by Lebesgue's dominated. convergence theorem, so that Rs(p+ie)$ + ry's in X as e A fortiori this convergence is in

.f*. But r?s(p *ie)$ *

norm, we have

r/ts

/-

E.

-O+.

rtt

io X;r", and since f;r" g.f*, with a stronger

TTTB Ln,ITTTNG ABsoRPTIoN

Next, we note that

,/)

€ D(Eo).Indeed,

PRINCIPLE

17

we have

/'\)'{o())d-t tr" Xr((^Jrl;. r,/()): -' xK\^, - , rlrrllo())d Again,

it is clear that both

^=

p

terms berong to x@, so by Lemma g.1,

€ D(Ho). € or(H), since / : -V$, if we show that (fle _ dnifu)Q - 6, we will be done. To see this, we have for every e > 0, Finall$ to show that

(Ho We have shown that

-

ry'

pt

*

p)Ro(p

no(p*

ie)g

:

g + ieRa(p

+

ie) S.

- A#fu)6 in )/, so thar e,E6(paie)/ + ,V. Since I/e is closed, we have our result. I ie)g

CoRoLLARy 3.6 (to rhe proof). If g e X, then

Ao(p)d:0 andif

nifu)d € D(Ho) and moreover,

if

4:

-VRil(p)6, tt "n

t:

(3.8) is sarisfed,

Iiro,

^ n*fu)d

(s.e)

safr.sfes

(H-t)rl':0.

(3.10)

THEoREM

3.7. Unoo(H)

Proof. By

the last theorem, this is the same as showing that

So suppose

{6*} c

I,

a i1

isdr'screte.

!s

is discrete.

that {p7"} C Xg and pk + p € t/. Then there exists a sequence ll{rllx = 1, so that /y" : -V Rtfur)Oo.

: l implies that there exists asubsequence (again denoted {d*i) t" that gp + / weakly in .L, and hence {/p} is weakly Cauchy in ,f . Now'

1116111

by

Let us now write 6

- - 6 n : v l&t 0d - nt

0414

"

- v [Rt 0,i - n*

(p)]4

^

+v

Rt 04lo * _ 6 ^1. (3.11)

Since

{/-}

is a bounded sequence in .L and VRotfu*)

first two terms in (e.ff) go to zero in f as n, m zero in BG) by the compactness of y.R$(p).

+

-VRo+04 in B(,f ), the

oo. The last term goes to

Without loss of generality we may assume that all of. the y.1" and pr are contained in the interior of a compact interval K g u. Let > 0 be given; 4

we

M.

18

BEN-ARTZI AND

A.

DnvIN,q,Tz

may suppose, again without loss of generality, that

K

is taken sufficientlv small

so that

I

lp"

- )l-r+"d) < 4,

n: Lr2r, ., .

I( Let us set

6*$)

:

Ao())6*

A-Po

and

$*,x(A): x-,, 1'l;4e-0)d', a_ Fn

e u. ^

From Assumption 3.2 we have

116^,o-$*,*llx,<4Cq' Now take ,l e

(3.12)

(U\If) n [-,?,.t?], where lt > 0 is fixed.

Then

lleo(r)d,ll s_ c

ll;;1,^

p -11lld"ll*,

(3.13)

> 0 is a suitable constant depending only on dist{{p,}, U\I(} and on B. Thus from the facts that Q^ 6 in ,f and pn + p1 we get where C

xr-",or())(1 -

x*OD+9P*

xt-o,ol())(1 -

x*(^D+ry.

(8.14)

Finally, we have, for all sufficiently large n,

t

J1r|>n

| d^.. : 6- 2*1" l,^,ro@o{o^)d",6^)

ll1,rl06"lf ll s - r"^ ll:^

(3.15)

t2C '@-t'f' From (3.12), (3.14) and (3.15) we

see that ti.l ir Cauchy in ,Vo. As we saw at the end of the proof of the last theorem the element in )/ unitarily equivalent to ''l. is {t.: Rf,(p.1/.. Thus tn l) in ,v. From the last theorem,

each ltn p,. Without loss of generality we may suppose that pn * p* f.or n I rn. Thus {/, } ir an orthogonal sequence, so that

is an eigenvalue of

llrl'"

-FI at

-,1'^ll"

:

llrl'.ll"x

+ l\'^ll?

- o as '-,rn 1 @.

:1..

t: "i1: ,&,

Trrg Lrulrnvc ABsoRPTIoN PRINcIpI,E

19

llt/.llx + 0 as n. + oo so that ry' : 9. By letting m+ 6 in (3.11) we find that the limit

Hence

element / satisfies also : 6 -VRt!t)$. Further, since rBo+(p*) d* - R*(p)g in I", and {Bo+(p.)d.} ir u bounded sequence in )/, there is a subsequence which converges weakly to R{(p)g in )/. Thus O: {: Rt0t)6, so rhat d:0. But this contradicts the fact that 1 : lld"llr - 11411r. r

the equation

There does not appear

to be enough structure in our abstract setup to in u is finite. However,

prove that the multiplicity of each eigenvalue of 11 under various additional hypotheses, this is the case.

PRoPoSITIoN

3.8.

the multipliciby of

Ii,"

p,

E every eigenvector of H at p € U belongs to

In particular, if Hs hx

is lfuite.

Iiro,

Es in f.*,

then and

is D(F,s) with the graph norm, then this is indeed the case.

Proof. To prove the last statement that, D(Hsl

e Iir,,

so

that

D(H):

we simply note, as we did

D(Ho).Thus

To prove the first statement, let ,/t

a closure

e fit"

and for every e

-Vrlt + ier!

: 6a

)

0, upon setting

ie)g

By the short range assumption on rp

be

a^n

€ D(Ho) implies ,! e

fft".

H at p € U.

Then

eigenvector of

O: -Vr/t we have (Ho - p+iel{:

- n{ (p)/,

v,

:

aieRs(p.

$€

,f

+ ie)lt.

(3.16)

so that the left side converges in

while the right side is uniformly bounded in e , and hence has

a weakly convergent subsequence in ,V. Thus ,lt

n{fir){

2.g,

iery'; that is,

{ - nojr * X- to

ry'

ry'

in corollary

- nt Ui6 e .V, which implies

e )t.

R"(p+ie)/ is uniformly bounded in.V, and thus {Ao(p +ie")g} convergingweakly in.V to,Roa(pr)/. We

From (3.16) we see that there is a subsequencu thus have (Ho

weakly in

)/.

IIs is weakly

-

p)Ro(p

*

ie^)Q

:

6 + ie"Ro(p

* ie.)g -

g,

since 116 is closed, and since the strongly closed graph space of closed we see

that @o - dLt(p)d: 6, so thar (Ho - p)(t, -

M.

BEN-ARTZI AND

A.

DEVINATZ

Atlt)il : 0. Since p is not an eigenvalue of fI6 we see that g : R{(p,)g, or (I +V h+1t))d : 0. By Fredholm theory the space of solutions of this equation is finite dimensional. I Another condition, which is sometimes easier to apply for differential operators, is given by the following Proposition.

Set.ff6: Ilo I D(Hs)a.ffio;then

-ffs - ) is a map with domain in ,tfio and range in .V. \ h'as a closure h ffi" to ,{, if the closure in ,Ifro x X fro- l is again a graph. We designate this closure by I/-0,1.

for any real or compl"* ,\, We shall say that fro

of the graph

"t

3.9. Suppose pr is an eigenvalue of H : Ho *V, and fro - p, .has a closure in f ;r" to X. Then the eigenspace of p, is frnibe dimensional and every eigenvector at p. belongs to D(Hs) n f fro : D@s) n D(V). PRoposITIoN

Proof. Let {/r.}i c I be a basis for the solutions of (.I+ VRt0i)6: By Fledholm, r is a finite number. Set ry'i : R*(p)di; then we claim * {rlti}\ g f fi, g f is a linearly independent set. To see this, suppose n#(p) f f,*,v,: 11 Setting

6: L;a'{i, (Ho

,,4,

:

o.

rhat

o.

we have

- p)Ro(p+ie)g: 6+i"Ro(p+ie)$.

As e varies in (0, oo), eRa(p'*ie)$ is bounded in .v. Thus there exists a sequence

€n

+

that e"Ro(p t ie^)g

- t+ (i.". weakly) in )/. On the other hand, eRa(p+ie)S * 0 in .f* as e + 0. Thus, if g e I, (5,"-RoAr*ie,)S) + (s,f*) :0. Since .f is dense in .V we have gt :9. O so

- p)Ro(p*ie,)g - d in .V. On the other hand, Ro(prie)S - A"*jt)O in .ffio as € + 0, so that RA(p*i")O Rt01,)d i\ .ffr'. Since the strongly closed graph of Fs,, is also weakly closed it follows tltat R{ (p,)g € D(Eo,p) and Hs,rfiot p)d : d. Since r?$ (p)/ : 0, this implies d : 0, which proves our contention that the set {r/r.}i is linearly independent. Consequently we have (Eo

(

It is easy to see that the elements of {/r}i are orthogonal to the range of I +VR{(pl in the sense of the .f*, .f pairing. By the Fredholm theory this

THE LIMITING means

that rf r! e L*, and ry' is orthogonal to the range of

ry'

is a linear combination of the set

,l

:

at?t)d for

some d

{{i}i.

{

be any element of ,f

so that

e*Ro(t"tie")$ 0

Thus in fact

ry'

I +VR"*(p),

then

€ Iiro, and indeed

e X.

To complete our argument, suppose that

let

2l

ABSORPTION PRINCIPLE

.

ry'

is an eigenvector for

If at pt, and

As we have shown, there exists a sequence Ea

0 in .V. Since

-

0

R"(t"+ie.)Q e D(Hs)n D(Y) we have

: ((fl - p)rl',no (p + ie $) : ({, (Ho - p + V) RoA' + ie.) $) ") : 0b, Q * V R6(p, r i 6) + i(,1', e" Ro {p + i $). ") ")) e

e

* O, VRo(p*ie*)g - VRtUr)6 in .f and hence in )/ also. Hence (rlr,U +vnt?r))d):0, which by our previous argument shows that 4t e f;r"

As eo

and that the space of eigenvectors is finite dimensional. If we note the equation (3.16) we see that

t

e D(Ho) n

Xito. I

4. STARK HAMILTONIANS WITH PERiODIC PERTURBATIONS As our first application of the theory presented in the previous sections, we shall derive the limiting absorption principle for the operators of the type A2

H: Ho*V(r),r": -iA-q*q(tt)*7,,,r:

(t1,r'l € RxR"-1.

(4.1)

T.' rs a self-adjoint (from below) operator in .[2(n3,-1) and I/(c) is a real potential

Here q(c1) is a twice differentiable function with perioil 0, semibounded

depending on all coordinates which is short-range with respect to 116 (in the sense

of Definition 2.6).

The only case that has been studied in the literature is

[, : -A,r,

q E 0,

leading to the operator

H:-A-tt*V(t). Note that with

lz(o): -l"l-.

(4.2)

the operator (4.2) is the quantum-mechanical

Hamiltonian of a hydrogen atom in a uniform electric freld (the Staxk Effect). The additional potential g may be regarded as a periodic perturbation on that field. Herbst i6] proved a limitng absorption principle for (+.2), using the weight

(t+ a71-"/2, s > Lf 4. Yajima [17] used a more general weight function ndistinguisheso that between the positive and the negative sides of the c1-axis function

(see

(

.ZO)). Our weight function is identical to Yajima's and our results for the

special case (1.2) are identical to

his. However, our proof is a straightforward

application of the abstract method and appears to be much simpler. seems

.l:

It

also

:,.

-

..2,:

i, :i i.

t .,1,

'-;l-.

',a,:

:;

to us that the methods of [6, 15] cannot be extendeci even to the case of

a non-zero periodic potential q(c1) added

4.I2 the potential Iz(c) must decay as zr

,i:..

to (a.2)

+

(as we shall see

in Corollary

oo).

In verifying the assumptions imposed in the abstract we shall need some elementary properties of t€nsor products and eigenfunctions of the one-dimensional 22

:i

:i .il'

.t $',

&

$,

Tnn LrurrrNc ABsoRprroN

pRrNcrpLE

23

operator,

n, It

is well-known

o

: -ir, - x * q(r), q(a + 0) : cb), that

retain the notation

lV {fr(f)} the

in ,2(R,).

r11 is essentially seu-adjoint when restricted

frr for its unique

to

(4.3)

cff (R). we

self-adjoint extension. we shall designate

associated spectral family. For every s € R we define the space

X" by

r": {f , ll/ll?. ': J G+az)"lf(r)l2dx+ o

0

< J'I lf(r)lzdr

@\.

(4.4)

-€

In what follows we shall use generic constants with dependence on various parameters as indicated by suitable indices.

LEMMA 4.

1.

For every s

>

Lf

4, H1 is of type

(I",I_",c,

R), for some o >

0

depending oa s. Furthermore, for every b € R, t.here exist constants C6,", C6,",o

such that, witfi A1())

: d4t(^)/d^,),F 1b,

ct,",

ll,4'(l)llr.,r- . 1

lla'(r) - At(p)llr",r-" < cb,","lr

Proof. The proof is based on an eigenfunction erator

rlr

(the case q

necessary results, let, such that for every

)

:0

- pl'.

(4.s)

expansion theorem for the op-

being the Airy function expansion). To describe the

w(rr,\) be a real non-zero continuous function on R x R € R,

/,{2\ Hp(a,)) : (x dr, - + a@) ) w(x, )): (ii) ur(c,.\) decays exponentially as r + -oo. (i)

),a(c, )).

The existence of such a function follows from Lemma 8.2 in [B] and its proof (in fact, it suffices to note in the proof that Q(c) : r C(4 satisfies gt,(c)(l+

-

lzll-s1z + Q'(x)2(t + ltll-stz € rl(R)). Replacing r by a* 0 and recalling that q(c) : q(r +d) we have H1u(x, + 0, : (# - r + q@)) u@ a 0, \) : ^) (d + ,\)tu(z + d, ,\). The exponential decay property determines tu up to a scalar factor, so that

u(r * 0,f)

:

B())tr(c,d +.\).

M. BnN-AntzI

24

AND

A.

DPvrNerz

Now, by Theorem 4.2 of [3], there exists a real function

u(r,.\), continuous

onRxRandsuchthat (a) u(c,

)) : {(.\)tr(z'.\)'

)

€ R,'

(b) The transformation

: f@)u(",\a", /€cf (R), J

(//)())

(4.6)

R

extends as a unitary map from

(c)

/

L'(Rr)

onto .L2(R,1).

diagonalizes .EI1, namely,

)

f HJ-L:

(multiplication by .\ in ,2(R)))

.

(4.7)

We now have

u(r

* 0,I) : {())u(c+ d,.\) : f(.\)B())u(c,d +.\) : €(r)p())€(l + d)-1u(o, d + .I).

But by (b) above, the transformations

ff

I- Jf@)u(a+0,))dr, f - J I@),t(x'o+))dz are both or,rruo

l*o* L2 (B..)onto .[2 (It; )), rl rnt necessarily, u(t*0,)) :r(r,d+l)'

or replacing d by an appropriate multiple rnd,

u(c,)):u(o*l-4,4),

\-q:mo'

011<0,

(4'8)

From the considerations of [3] (see Lemmas 3.1 to 3.3 a.nd Theorem 4-2),it

follows that 7uf interval

K

exists and is continuous for

) € R,, and for every

there exists a constant Css so that 4 €

(

l,(",,i)l

E\

<{

,"(t 't

lcx,",

x)-

K

compact

implies

t/+ '

lK"-,n)l={":\n')'/-n'to^\ "" - lcxe'' c(o'

'=o'

Tgp

LTUITING ABSoRPTIon

PRrncpIE

25

so that by {a.8), with some consta.nt C,

(4.e)

It follows from (4.7) that for every pair f , g e L2(R),

fr{o,ft)t,d Nowlet s > 1f4,

f e I".

= 7f (^)'7s(\, for a.e. .\ e n,.

€R

FixD

and let

(4.10)

) < D. Using (a.O) and {4.6) we get,

by the Schwarz inequality and (4.4),

(\ lfrl

{ I

l//(r)l <

.

*

L>'-^

,u''*'0")''"

= C(It + Iz-t Is

(+.rr)

r+\)-ttz(r+*t-"a")''" * (r!-^(1+c+ \-'r"0")'/'

" {[.-"{,-^,,'* [=-,1,-^

| ",+^171"11a"1 ,!-^ )

0+ a+ ))-1/+171r)ld,r+

*(.1

-^'

+

\ t/'l '2)-"

e2("+^\ da

/)|

|

ilrrrr.

* td .llf \ft,.

Now, by the H6lder inequality,

\'l'o | \'/'o r,={( |, tt+a+))-o/za,l ( t. 0+,,y-"o4,1 L>^"J(',-^) ,/ L>*"ito,-^r ) where

1/p*1/C:t.

It follows that

Since s

l11l 1Ct,". l12l

<

>

Lf 4we canchoosep

)

2

suchthat also sq>

Clearly,

(max(o

,b))'/',

r"tr*"^ =*"u, llnl

s I [r+(v - ))21-"e2vd,y 1 cu", rSs
,

tl}.

M. BEN.ARTZI AND A.

26

DEVINATZ

so that we have

(4.r2)

17f(^)lsco,,lltllx". It follows now from (4.10) that for s ) If 4, and f , g € X",

ln

e)l< c,,"ll/llr"llgllr", ) e (-oo,D)'

lfitt't^ti,

This proves the existence of ,41()) as an element of estimate

in

B(I",X-")

Taking f € X,, ) (

7l(^+ Assume lhl

and the first

(4.5).

To prove the second estimate (4.5), we take first s before.

(4.13)

h)

-

7l(^)

< 1. Using the

|

and

fixb€R,as

obtain

6, we

:

>:/4

rcfrtr,)+o,h)dt. h,

estimate (4.9) for 0of

0)

and,

o

< 0, <

L.

proceeding as in (4.tt),

we get

a. t.:

(r l7f(t+h)-

7f(^)l sclhl

.1,

i{V2**i1o,-r1 t (r+r+^)1/2(r+,')- "Or)'''

i:.

I

(

\'i' *t,+r.fll I +[ |, tt*r+\)L/24,1 f llr" L>">-.. J ) :

clhl(Il + Ii + Is + Ia,)llf llr",

where 13, 14 a.re as in (4.11). Now

forc) ' max(0, -))

so

,\ <

b implies

that

'i ,t

(1

+,

+ .\) < cb(r + r)

that

V'tl < cu

'j;,

i ,:.

[=,"J,-,,

VL |

(t + a1t/z-2"

;i

4')''''

.i

3 Ca(r + lbl)r/2max(0, 6).

'a:

IIence,

l7f(x+h)- 7f(^)l
)(6,

s> zf4.

(4.r4)

THE LIMITING ABSoRPTIoN PRINCIPLE

27

The estimates (4.12), (a.ta) can be viewed as norm estimates for the operator 7

f (^ + h) -

7

f (^) : X"

+ tr-(-oo, D).

Hence, we obtain by interpolation, for every ,y

f (s + h)

l7

- 7 f (^)l I

> 0, lhl <

co,",tlir1zs-t 1z-, ll/llr",

1,

) < b, rf 4 < s <

Clearly, this estimate is valid for every lr such that

3

f

4.

(4.15)

) + A < D in view of (aJ2).

Noting that

((a'(r)

-

A,(p))f ,s) = (71(^)

(where the left-hand side is

-

7f

7l04.enT:7i6

A'0.7i(\+

the I-u,I" pairing)

we obtain the second estimate

in (a.5) from (4.12), (a.fS). The proof is complete if the fact thar

X-, : /;.

we note Theorem 2.4 and.

I

In view of Theorem 2.2 and Remark 2.3(b) (applied to (-oo,

D)

we have

nolv,

4.2. Let fu(z): (I/r - z)-r,Imz lO. De R,and s>Ll4thehmits CoRoLLARY

Bi()) exisC

:

"\p

+ie),

ftr()

uniformly in the norm topology of

)

B(Xr,I-")

and uniformly H6lder continuous in (-oo,

(

Then, for every fixed

(4.16)

D,

and are

uniformly bounded

D].

Also, we shall need the following

4.3. Let s > 112, b e R and assume p, 1 b and f € f". Then for sonre 6 > 0 and all ) < D, CoRoLLARY

(,{r())/, "f) < ca,, !} Proof. By (a.10) we

that

plt+u llf

ll1l^".

f (^)

7

AlQt)f :0,

have

(,a,())/, f)

:

17

| (^)1"

:

17

-

f (p)1",

where

(4.r7)

M. BEN.ARTZI AND A.

28 so

that (4.17)

(4.15). I

foUows from

We now turn to the operator 116 in

(a'f)'

Ho: Ht@Iz*h&Hz where

I[

DEVINATZ

is given by (a.e) and' H2

in

- f]''

it

Clearly,

can be written as

12(R,)8.L2(R,"-1),

(4'1E)

In what follows we denote by 'B;'

E;())

corresponding to the resolvent operator and. the spectral family, respectivelS

i:0, 1,2. Similarlv, A;: dqt{^Jld'\ (when it exists)' (fr). SV our assumption I I (a,oo) f'" some 4 > -oo'

We set

l:

IIi,

Spectrum

Weshallnowshowthatelementarypropertiesoftensorproductsimplythat -8I1, without any further assumptions

uinherits" the "spectral structure' of 116

on H2. To this end, we extend X" as a weighteil-'Lz space in

R'

by

s€R,

t":X"@L2(&*-r),

(4.1e)

so that the norm is given bY

?r

llsll?: JJ |

|

O+x?)lg(a)l2d'a'du+

b P"-l

LEMMA

4.4.

For any s

i,

J J -@ Rt-r

^ ls@)l2d't'dt1'

> tf4, Ho is of type (i",i-",o,R)

tlre same as in Lemma 4,L. Furthermore,

in

w"here

(4'20)

o > 0 is

analogy to 4.5 we have, for every

b€R, lleo(r)ili,,r_ ll.4o())

t: t,

,1 cr,",

(4.211

< 'co(t{llZ.,7-, cb,,,.ll

-

'.:,

- Pl",\,p3b' a

Proof. For e > 0 we have (see [5, (2.17)])'

'i 'j

Ro(,\ 'r +

i")

f

: I Br() + *ie -

v)

I

dE2(v).

(4.221 :-a

By Corollary 4.2 we can let e in order to get ,4o(,\)

+

0 and use 'at

: /f er(r I

v)

I

())

:

dE2(v).

.i:

(Bf (I)

- nr $D lzri (4.23)

ia

*

THE LIMITING ABSORPTION

if I is infinite the integral

Note that

PRINCIPLE

exists in the strong sense of

29

B(f",f-)'

The estimates (4.21) now follow directly from (a.5) and standa^rd estimates for

products. I

tensor

Flom Theorem 2.2 and the previous lemma we now obtain the following generalization of Yajima's theorem ([17], Prop. 3.2, i). THEoREM (Eo

4.5.

Let Ho

be given by (a.\

.Roi(r):,\pao(,\ in

i"

bv (a.19). Set r?s(z) =

Then 6he limifs

- ")-' , Im z 10.

exisf

and'

+4"),

)

€ R,,

(4.24)

B(i,",i-"), t > lf 4, unifotmly on every set of (--, b). Fhrfhermore, there exists a constant a ) O, depending only

the norm topology of

the form

on s, and constants Cb,r, Co,r,o such that

."p llfto-(r)llg" ,7-" 3 l
lf,ao+(.1)

Cu,",

- n*\')llf ,,f_"<. c6,",.1\- p,l.

(4.2s)

Next, we prove a result analogous to Corollary 4.3.

e R and assurne that As(p)f :O where P'1b and f e X.". Then for some 6 > 0 and all ) 3b,

Lol,tlvtil

4.6.

Let

s>

Lf2,

D

(,{o())/, /) < Ca,"ll

- pl'*ullfll?'

(4.26)

Proof. Let g € L2(nr-t) and denote by 5o the (closed) subspace of ,2(R,'-1) generated by {82(K)g: K is a Borel set in R}. As is well-known [16], the restriction of H2to So is unitarily equivalent to multiplication by z in L2(R",do), where da

-- (d,82(u)g,9). Also, it

was shown in the proof of Lemma 4'1 that

I/1 is unitarily equivalent to multiplication by.\ in.L2(R,d))(df : ordinary Lebesgue measure). Thus the restriction of 116 to its domain in ,2(R) I So is unitarily equivalent to multiplication by av

Vo

I +v n L2(e7,,,d'\do).

We denote

t I2(R) a So - L2(R?x,.,d do) the corresponding unitary map. Given

M.

30

BEN-ARTZI AND

L2(&.l,let ho : (11 o P)h, projection. It follows that he

(Ao(ilho,hr):

II

A. Drvnetz

where Po , L2(R"-L)

Vrnr(B

*

'Ss

-v,u)lzd,o(v), he

is the orthogonal

f,'

V'27)

Indeed,(4.27)foUo]vseitherfrom(a'23)orfromadirectanalysisusingthe diagonal form of

Tngrg P)Hot;t for continuou"

Tohn' and then extended by

the continuity of .46(P). Now, we have by assumption AoAt) f

itofu- v,v): o

:

0, hence Ao(tt) f o

:

0' so that

.

a'e' - do(v)' v €l'

(4'28)

t;' l:,

Let b

-

)(

6

andlet

z€f

satisfy (a'ZA)' Since

f

G [o,oo)

it

follows

that ]-v (

:;

o. We may therefore apply Corolla'ry 4'3 to obtain

l7rtr\-r,r)12 < cb,"l) - pl'*ollfn(',')ll'r",

a'e'

-

*

do(u)'

€

yields Integrating this inequality with respect ro do(v) and noting (4'27) (Ao(p) f o, f s)

3 co,l\

-

€.,

pll+6 ll /o ll3.

€

Theproofofthelemmaisnowcompleteinviewofthefactthatwecantakea I sequence {gr"} such that SooJ-,Sn, , i f k, and [Jo Sc*: L2(R-)' we conclude our discussion of the unperturbed resolvents mining their asymptotic behavior as I

.Bo+

+ -oo or as + -oo' 'l

PRoPosITIoN 4.7

'

The limiting values 'Ro+()) ((4'24)) satisfv

n*(r)

:

o

in

the norm t'opologv of

^!1-

()) by deter-

B(i'",f -)'t >

t1t'

(4'29)

Proof,By(4.22)(ase*0)itisobviouslysufficienttoprovethecorresponding claim for nf ()) in B(X",f-"). However, as in (2'7) we have

ni(,.)

:P.Y. IHo"*;o.4,(r)+ lr-lrlsr

IT*

lr-r'l>N

$

Tue

LTUTTTNG

ABSoRPTIoN

PRINCIPLE

31

.[-r. Thus, we of r{1(l) vanishes as ) * -oo, or using

The second integral can be estimated (in B(r'?(R)), in fact by need only to prove that the H6lder norm

the notation in (4.5), that

ulT-(cn' *

c6,o,o)

:6'

But this fact follows immediately from the estimates for 11,12,Is, in proving (4.13), (4.15).

I!,{

used

r

4.8. Let 6 € R,, s > of (-oo,0). Then for f e f",

PRoPosITIoN function

Ia,

LfA, and let

y{a)

be the characteristic

)(6,

llx("t)(t+l'1|)no*(^)/llr"1p"y
- ltl'll/||",

11,.\2

(4.30)

( b.

Proof . By (4.22) ir suffices to prove both estimates for ^Bf

f4.31)

(I) instead

of

,Bo+

()),

i" by Xr. Also, we may suppose that / is compactly supported. Now f : /, and since u is squareu : 8f (.\)/ implies (-fta2 - nL + c(rt)' - ,\ ^)" integrable over (-oo,0) it must decay exponentially as 11 + -oo (see (a.e)). Multiplying the equation bv (1 - r)i@j, integrating by parts over (-oo,0) and noting that -c1 - ) > lrrl- 6 and q(r1) is uniformly bounded we get replacing

!1, =

- r)2lu(al'

+ (t

-

,ilffrf g

(o" ro

{(-t |

0,,

lf

)

{",)l'0",+

| l"@,)l'a",|.) -'-

(We have used the obvious fact that u(O), du(O)ldcl can be estimated in terms

of the right-hand side.) Now (a.30) follows from the fact that

j

lu(r1)12 d.a1

< ll"ll?_" S co,"ll f ll7 ".

M. BnN-Anrzr

32

AND

A.

Dpvrrv.lrz

To establish (4.31) we take again a compactly supported

j : t,2. Thus u : ur

/

and set

ui

: R! (tr) f ,

u2 satisfies

C#,- sr t c(,'l -),) , : (A1- ),2)u2Proceeding as before we get

9r _{

-

L,t

-+ (r _

z1)2lu(c1)1,

s c,

{ti,

rr,,t"lor,

",)l#l

- ^,f !tu2@1)12

da1

I

n

I

,,ouf 0,,\

.

But according to Corollary 4.2, 0

I -;

l"("r)12a",

s c6,"ll(fif ()r)- nf

1rr;; fll"r_"

tdb,,,"ll1 - \rlr.llfll'x.,

n

_J

l"z('.)l2 aq

s

co,"llf

ll'r".

:

REMARK. observe that the above proof yields also estimates for the boundedness and H6lder continuity of 1(c1)(r

We now proceed

+ fu11)llz6p;(\l lAq.

to study the operator fI of (4.1), using the framework

provided by the preceding sections. clearly r/s has a closure Eo

* t"

for any s €

R.

We designate by D(Hs)" the domain of Fe in i", equipped with the graphnorrn (but we write D(Hs) for D(Ho)o).Also, let 1(c1) be the characteristic function used in Proposition 4.8 and set, for any s ) 0,

Iito,": {u: u € D(Hs)_",y(rr)(t + lcrl)u € .[2(R,')],

(4.s2)

normed by

ll"ll?;",.

:

ll"ll1" + lltro"ll1" + llx(:cr)(r + lrrl)ujl!,1.",.

Theorems 4.5,2.4 and Proposition 4.E imply

(4.33)

Tue THEoREM

4.9.

LIMITTNG ABsoRPTIor.I PnTNcTpLp

For every

33

\ € R', s > I/4, the operators.Bs())

are in

B(f f fi","). ", The abstract theory yields now

CoRor,r,eny

4.10.

Assume that for some

s > Ll4 the real potentialV e : Ho * V is self-ailjoint on

B(Xiro,", I"+r/+) is compact. Then the operator H D(Ho) g

L'(R')

and the limits

A-()) lim R() + ie) \', : e]O-+--r.'---rt exist

in B(f",Xfr","),

for every

R(z):(H-")-',

I € R except

(4.34)

for apossible discrete

set,

oo(H)

of eigenvalues of frnite multiplicity.

Proof. 7 is short-range and symmetric in the sense of Definition 2.6. Note that here D(Ho) g .$o,, (sue the proof of (4.30) with l : i) so Corollary 2.8 can be applied. Thus, the assertion follows from Theorem 3.7 if we ca.n verify

the validity of Assumption 3.2 in our case. so assume that for some

p € R, on Z it

6: -Vnt\tl\, O e i". Thun by Theorem 4.9 and our assumption follows that / e i"+t1r, and that ll6ll"*rtn < Clldll". The esttunate (3.8) follows from (4.26), noting that s > lf 4. I The condition imposed on

rate of (L+ or)-3/4-', e

(1+

lo1 l) as

-l"l-'.

rr + -oo.

)

I/ in the last corolla,ry implies + *oo, and a growth rate

0, ds x1

In particular,

However, we can improve

it will

it to get

now

(roughly) a decay (roughly) of o(1)

.

take care of the coulomb potential

a decay

rate of (1+11)

-t/2-c

as

,l +

*oo.

Also, we shall show under stronger assumptions that the (discrete) set of eigenvalues is bounded from below. we begin with the following generalization

of Theorem 1.1 of [17].

TupoRnu 4.11. Assume that for some s > L/4 the real potentialV € B(Iir",",i") i" ro^p^ct. Assume also thaty e B(Ifi",",i2"). Then the op-

H : flo + V is self-adjoint on D(Hs) C L2(B) ani! the fimirs g*()) exist in B(t",Iiro,), except possibly for adiscrete set of eigenvalues, oo(H), with frnite multiplicity, In particular, H has no singular continuous spectrum erator

M.

BEN-ARTZI AND

A.

DEVINATZ

and the wave-operators

w+: t - ,lf"it* "-it*o exist and are complete in 6jre sense that Range Proof . The assumptions imply that

I/

(W+):,B(R\ao(.il))f'(R,').

is short-ra.nge in the sense of

Definition 2.6

and the self-adjointness follows as in the proof of Corollary 4.10. However, the estimate (3.8) is not immediate here, since we have only Q : s

> lf

4 (so that (4.26) cannot be used). So, noting that

-V nt04Q e i", Ao(lt)Q:0 (see (2.16))

we set

f:: Clearly,

i3 i.

f"nker.Ae(p),

u closed subspace

f -^

-^

1

[i,1 ,i"o,l

u:

of.

i".

s>

L/4.

Furthermore, we claim that

i(or-u)",*u"r,

0

<,<

1,

sL,s2> rf

4,

(4.35)

where the left side denotes the interpolated space. Assuming (4.35) for the moment we take

lf4 <

Vnt0i e B(f:,,i,;.

s1

< s.

Then obviously VI,.+04 €

B(f,,,i";,

h"r,."

Al.o, by Corollary 3.6, Lemma 4.6 and the assumption

on Y we haveVR{Qr)

e B(ij.+l 1e,i,+r1+).

Thus by (l.ss) ancl operator

interpolation,

vn"*0i e B(f:,+o/4,i"+e/a.), o<, < 1.

(4.36)

Q: -VA*040 € n.Taking 0 : a(s - s1) implies that / e I""-", and since AoAi6 : 0 we have in fact $ e i3"-",. This procedure can now Now, let

be repeated, yielding eventually O

e f:+Ll4.But then (l.ZO) and Theorem

3.7

imply the existence of ,R+()) and the discreteness of oo(Hl. The existence and completeness of W1 follow from a general theorem of Kato and Kuroda [9].

it is clearly sufficient to prove (4.35) when i" is replaced by L O Sg,9 € L'(R*-').In this case, we have that (4.2E) is a necessary and sufficient condition for As(p)Q, : g. Proof

of (4.35).

Using the notation in the proof of Lemma 4.6,

From i4.12) we see that

7xf

:(7f)(^), f €I", )€R,

Trro LruIrIruG ABSoRPTIox PRIxcTpIn is a continuous functional on

Since for do-a.e. y €

,f,.

the condition (4.24) can be written

fi,-u6o(,u)

I

35

we have Qo(,v) e

x",

as

: o,

do

-

a'e' v

el'

(4.37)

Thus the proof of (a.35) is reduced to proving that

['f,, n k"" 71"-,,I"" nker f,,-,le (Note that 1I",,

I",lo : I6-fisr+0o,

: [Ir' I"rls nker 71,-,.

(4.38)

is obvious.) But (a.sa) follows from

well-known fact on interpolation of subspaces with finite co-dimensions

a

([J.2],

Th. 13.2). r

: -Ar,, so that

In order to get a more concrete criterion for V we takeTr,

IIo

: -A - sr*

q(o1) in

L"(R).Using the characteristic function 1(21)

as

Proposition 4.8 we have the following result, which is identical to Theorem

in

[17] when

in

1.1

q:0.

CoRoLLARY

4.12. Let llo : -A - at * q(t)

and assume that the real po-

tential V (r) satisfes

v (")

:

[x("r) (r +

a2r1r lz

+

(1

-

1(c, )) (t + r!1- " tz1 (% (r) +

vr(r)),

(4.3e)

wheteo>ll2and

Z-(R'),, fim [(c) : o, lal+€ V2 e L?".(Fc'), for some 4> p )

Vy(a) e

""rrd

f

Iim

(1

lcl+€

Then

+

"?). JI

lvr(dl"l" - yl-"+udy

0,

:

o.

(4.40)

l'-cl< I

H : flo + V is seU-adjoint on D(Hs)

and all t.he assumptions of

Theorem 1.77 arc satisfied.

Proof.

Corresponding

to D(fIo)" let

X2,"

(R") be a weighted

normed by

ll"ll'x",,:

ll"ll3 + IlAullS. (ll . ll" given by (a.20))

Sobolev space,

M. BoN-AntzI

36

AND

A' Dnvwerz

X?". We denote by )/2 the standard Sobolev space of order 2. Clearly, D(Ho)" I all s e R. Thus the fact that the part of I/ involving tr/1 satisfies the

for

and assumptions of Theorem 4.11 follows from the Rellich compactness theorem that it the definition of the norm (a.sz). As for the second term of v, we note

follows from

-Au : Hou* oru- q(c1) that

multiplication by (1 +

,?)-t/'i"

I

properties bounded from D(Irs)" to x2,", s € R. Thus the desired compactness of this term follow from a well-known condition on the compactness of maps

[13]. I

from )/2(R,') into .[2(R") REMARK

a.13.

(a) In the last corollary we could take any semibounded ?:,

whose domain is contained

in )/2(R,"-t).

(b) The remark following Proposition 4'8 allows us to add to some singularities for

cr S 0. For example,

I/ in (4'39)

we could add a term 1(c1) ' {1

+

into L"(R"l lrrl)'/'v"(r),where v3(o) is compact from [11(R)@ )/2(R'-1)ho' anil I/3(c)

+

0 as lzl

*

oo.

(c) If instead of (a.39) we take

r::

::

t,

v (") :

[x("r) + (r

-

1(o1)) (r

+ rl)-"

t2lv'1x1,

i:

by proposition

I/ € B{f -",7), t > tf 4, ar.d' 4.? there exists an o > -oo such that the operator I+VIif (l)

is invertible in

a(i,)

then oo(ff) is boundeil below' Indeed, in this case

for

)(

o. Theorem 3'4 now implies oo(H)

I

(o,

-)'

5. THE SCHRODINGER OPERATOR

_A+Y

In this section we consider the limiting absorption principle for an operator of the form

H: Ho*V, where

I/

I/o: -A in r2(R,"),

(5.1)

is a real short-range multiplication potential,

Our aim here is to use the abstract approach in order to give a very simple proof of the lirniting absorption principle for .EI, with the same class of shortrange potentials as that used by Agmon in his classical paper

[1]. We refer

the reader to [1] for earlier references related to the behavior of the oresolvent kernel" of

fl

on the spectrum.

Recall that our abstract method imposes certain nsmoothness' assump-

tions on the spectral measure of IIe (Definition 2.1), which yield immediately

:

z - 0. This is then followed by a perturbation-theoretic treatment of ff. In [1] Agmon emphasized the limiting behavior of Ra{z)

(Ho

- z)-t

as Im

Fourier transform techniques and properties of division by functions with sirnple zeros in Sobolev spaces. In fact, his method applies when

.616

is any constant

(real) coefficient differential operator of principal type (in pa.rticular, all elliptic

operators). This method has been generalized by Agmon and H6rmander

[2]

to include all simply characteristic polynomials. In the next section we shall see

that our method can also be applied to that class and Fourier transform

techniques

will

also be emphasized in verifying the assumptions of the abstract

setup. Thus, in this section we shall concentrate on the operator (5.1), where special features of the Laplacian can be used to advantage. Indeed, using some elementary abstract facts concerning resolvents of tensor products the study of (5.1) is reduced to the (almost trivial) one- dimensional case. 37

M.

38 Let

Hl

BEN-ARTZI AND

A'

be the oPerator

,,

H,: -i= d,zz It

DEVINATZ

in

,2(n).

is well-known that I/1, when restricted to

(5.2)

cff(Il), is essentially

self-

exadjoint. we continue to denote as I/r its unique non- negative self-adjoint

tension. weshalldesignateby{,gr())}and,B1(z) theassociatedspectralfamily weighted-'L2 and the resolvent, respectively. Also, we denote by L, t € R, the space

(rl

v ^E -

{

r, trtt?", : | 0+*)"lf(r)l2dz<

oof

[i)

'

(5'3)

As in the previous section, all constants are generic, depending only on the indicated inclices. LEMMA

5.1.

nition 2.1) for

For every s some

>

rf 2, Hr is of type

(see Defi-

o > 0 depending on s. Furffiermore, tiere exist constants

C", C6,r,n such that, with -41(.\) 1i,1

(I,,X-",d,R\{0})

: dEJ\ld\,

lllr(r)llr",r-. 3 c"\-112, ^ > o,

(ii) llAl(.\)

-

Ar(p)llx",x-"

1

(5.4)

ca,",o

(r-*tr+'l +r-i(r+'t1ll - pl',

l,p>d>0. (Note that in (5.a) we have taken \,

:

1.t.

)

0 siace clearly

I f@)r-'e"a, Htf :€'i, ,o that, for L g e L2(R), Proof. Let i(€)

fit",

t^)1, g\

Now let s

(2r)-L/2

= !7-t1z

>

Lf

2,

[it./})a;6i

f e r".

+

El(l) = 0 for ) < 0')

be the Fourier transform of

i(-,6)a(-tt]

, a'e' ] > 0'

Then

(5'5)

Then it follows from the Schwarz inequality that

forevery(€R,

li(€)l< (zr1-rrz

/'

(1,,. ,",-"0)

ll/llr. s c,ll/llr.,

Tsn Lrtvrrrlxc AssoRprIoN

PntNctpl,p

39

so that by (5.5),

lr

(E''(r)1, dl < c"^-'/'llfll',llsllt", ) > o. l* lo^

(5.6)

|

This establishes the existence of ,{1()) as an element of the estimate (5.4)(i). To prove (5.4)(ii) let 0 < a < min(s inequality

B(I",,f-")

- L/2,1).

and

Using the

la-ir' - ,-;ual 1 ,t-'l) - pl" lol" we get

|i(f)_fgtnt=,,^#|)-p|"([u-.,;_"*",")',,,,,,,, " (2tr)1/2(t/), + r/t)"' \*

)

(5.7)

The estimate (5.4)(ii) now follows from (5.7) and

(5.5). t

Combining Theorem 2.2 and, the last lemma we have

5.2. Let R1(z): (h* z)-r,Imz lo. 6 > 0 aad s > l/2 t-he limits

Then, for every fixed

CoRoLLARY

Ei()):,\p j?r()+ie), I ) exist uniformly in the norm topology of B(X",

(5.8)

6,

X-") and are uniformly

bounded

and uniformly H6lder continuous in (6, oo). We shall need also the following CoRoLLARY and

I e I".

5.3.

Let s

)

1, 6

Then for some e

> 0 and

assume that Ay@,)f -- o where

> 0 (depending only on s)

(,ar(r)/, f) 3 ca,, (.1-t-' + p-L-a) Proof. Note that in the proof of Lemma

lA

Ho

:

Ht

@

Iz

5.1 we can take o

Ho: -A

I Ir @ Hz in

where I11 is given by (S.Z) and H2

pl'*'"llf

-

Thus (s.9) follows from (5.5), (s.7) an
- -A,,.

i:

0,

1,2. Similarly, A;

6,

(5.e)

ll"x"

= tfZ+ eif s >

which can be written I'?(R,)

)

1.

s L2(n:;r),

as

(5.10)

In what followswe denote by

: dE;(^)/il

6

feJp) : o. I

{Sd())} the resolvent operator and the spectral famrly, ing to .EI;,

and for all ),

p)

-R;,

respectively, correspond-

(if it exists). As in the previous

M.

40

BEN-ARTZr AND

A.

Dpvnverz

section, we want to show, using properties of tensor products, that

fls

satisfies

the conditions of Definition 2.1. However, the weight function here will depend on all coordinates. So we set

^(r'l L''"(R"):

\l llfll?,: | 1t+1"1'1'11(t)l2d.r. - l, lJ"l

s€R.

Lnuue 5.4. For any s > Lf2, Ho is of type (L2'",tr2'-",a,R,\{O}), where a is the same x in Lemma 5.1. Furt,hermore, there exist constants C", C6.".o such that

(r) ll,lo(r)ll,

))0,

z,s.Lz,-, < C,^-L/2

(5.11)

(ii) ll,4o()) - Ao(p)l]2,,,,7.2.-" 1c0,,,. (,1-*(r+c) * u-|(r+")) lr - rl", .\,pc > 6 > 0. (Note again that Ee()) : 0 for ) < 0.) Proof. Given ) > 0, let / e C-(R,),

O

< 4 <1,

be zero on

(-oo,,\/B) and

one

on (Z)/S,oo). In analogy ro (a.22) we can wrire

ro() + i")

:

f f

I

I J ;;:*;g(u\du1(u) RXR +

I

dE2(v)

f aa2() JG-S(u))d.E1(w)

-w*ie).

R

Noting that d.O1(o)

:

At(u)du on supp

{

we get readily, using (2.2),

ff Ao(l): | 0$-v)A1Q-v)@dE2(v) + / (r- 6(v))dEL(v)eA2(\-v) JJ RR

:

Ir()) + /r()). (5.12)

(compare [5] Th. 3.6).

In both integrals above we have by induction. The case

n,

: I

)/3 1 ) - v ( ,\. We may now finish the proof

is precisely Lemma 5.1. Assuming the validity of

(5.11) for A2 and. using the standard estimates for tensor products we see that

Ir()),

12(.\) satisfy estimates analogous to (S.11) in the spaces

.f, O12(n'-t;

and .t2(R) A r2''(R,n-l), respectively. But clearly the topology of ,2,,(R,o) is stronger than the topologies in these spaces so the proof is

complete. I

TTTE LTUTTTNG ABsoRPTIoN PRINCIPLE

4L

Theorems 2.2, 2.4 and the last lemma now imply the standard limiting

absorption principle for the Laplacian, which we repeat here for the sake of completeness. Notice that

is closable in L2'" for any s

-A

€R

and-its domain

(equipped with the graph-norm) is the weighted sobolev space of order ,t where Xk'" TnnoRprr.r

5.5.

:

{u:llull2y*,.,: ll"ll3 +ll(-Llk/2ulll < -}.

Let IIo

: -A

in

L2(&),

and set R6(z)

:

(Ho _

:

2,

(b.13)

z)-t, Im z I

0. Tien the limits ftoi()) exist

:

.[p

no()

+;e),

in the norm topology of B(L2'", )12'-"),

pact subset of

(0,

(5.14)

> lf 2, uniformly on every

com-

oo), and are H6lder confinuous.

Next, we extend the result of corollary

PRoposrrloN

s

) > 0.

5.6.

s.i to the multi-dimensional case.

> 1,6 > 0aldassurne thatAs(p)f :O,wherep,) 6 and f eL',"(R.").Thenforsomee >0 (depeadingons only) andforallA) g, Let s

(,{o())/,

I) 3 Ca,"()-L-e + p-r-e)l^- plr*r,lltll?.

(5.15)

Proof. we use induction (on n), equation (s.rz) and the method of proof of Lemma 4.6- In fact, it foilows from (5.12) (with the notation there) that (IrAt)Lll: (Iz(p)L/) : o, since both forms are non-negative. Now {(}) is of the form (a.28) so that

it

follows from the proof of Lemma 4.6 and (5.9) that

(/t(r)/, f) < Ca,"(\-t-€

+

p-r-e)ll - pl,+r,llf

ll?.

I e L2"(E) c rr(R,) or2,r(Ru-1), we may view / as !(v,.) L"(&,;r2't(R,n-1)). Thus (12(p)/,/) :0 yields, as in Lemma 4.6, Also, since

(I - 4(v)) Az(u -

v)

1(v,

.1

: s,

By the induction hypothesis this implies, for

a.e. _ y € R.

) ) 6 and. a.e. v,

- 6{v)) A2(\ - v) | (u, .) , | (", .)) s co,"(l-t-" + p-1-.)l) - plt*r,llf

((L

(r,.)llr",

e

M.

42

BEN-ARTZr AND

A. Dpvruerz ..a

and integrating with respect to dE1(v) we get,

(r"(^)f ,.f) < co,"()-t-' + p-1-6)ll

- plt+"llfll". I

We are now in a position to derive the limiting absorption principle for

I/

from the abstract theorems and the preceding estimates. This will be done in Theorem 5.E. However, before doing that we pause for a moment to derive some

more precise estimates on .R$(,\). They follow as immediate consequences of Lemma 5.4 and Proposition 5.6. Even though such estimates are not needed

in the stucly of the Schr6dinger operator, they will be useful in studying more general operators (Section 7).

. (a).

,X"'-"), s > Lf 2, thefollowingestimates,withsome a > 0 dependingonlyon s. For), p,)

CoRolr,eRy 5.7

The operators rBi+()) € B(L"'"

(i) llas (r)11r,,,,, z,-, 1 c6,"\-r/2

satisfy 6

)

O,

,

(5.16)(i)

ll&+())

-

nf

(p)llr,,. ,rz,-e 1 c6,",o(\-tlz + p-L/2)l^

-

pl",

(,t lln*(l)llr,,e ,x,,-. 1ca,,, (s.16)(ii) il,?d

(r)

-

n6 (p)11r",,,x1,-. S C6,",ol^

(b) Given p. ) O, Iet L2;,i Then R{(p,)

:

o

.

L2'" nkerr{s(Ir) (a closed subspace for s

e B(L2;,i,X2,o) if s > uitp

-

pl"

1 and

llfto- (p)ll

in

> l/2).

this case,

1c ""*',1",x''"

(5.17)

",0'

Proof. The estimates (S.tO)(i) follow immediately from (2.7) and (5.11). The estimates (5.16)(ii) follow from (5.16)(i) and the fact that .l/1'-'is the interpolated space between L2'-" and.V2'-", where ll,?"*(r)/ll?,,_,

: ll4())/111" + ll- a.d())/ll,_" :

ll,?"*())/111,

+ ll/ + )Bo't(.\)/113,.

(or alternatively, by estimating directly the integral To prove part (b), we note that .R6+(p)

lary 3.6, since D(11o)

:

X2,o. Now,

if

f

If, "JHa

(5'18)

a").

e B(L2;.i,X''o) by (b.lb) and Corol-

e L"i.L, s ) 1, it follows from Lemma

Trrn LrIrartrNc ABSoR.PTIoN

PRINCIPLE

43

3.1 (as in the proof of Theorem 8.5) that

ll

n"t 0")

o f ll3: JI t1' t^u,p l^- p)"

The estimate (s.t7) now fonows from this expression and (s.rs) by interporation if we note (5.18) with s : 0. I

we now turn to the schr6dinger operator 11. Flom the abstract theorv

we

obtain

TnroRnu 5.8. Let the real potentialv(r) be compact from x2,o into L2,t+e for some e > 0. Then E : -A+V is se[-adjoint on X2,o. Let R(z) : (E _z)-t, Imz I O. Then the limics

B-()): exist

in the norm

"lB,B(rare),

topology of B(L2,",){2,-"), s

discrete (in (o' oo)) set oo(H) of eigenvalues

I

>

0,

> lf2,

(5.1e)

except possibly for

a

of frnite multipricity. Furthermore.

.R+(,\) are H6lder confiauous in (0,oo)\ao(If).

Proof.

Since multiplication by (1 +

lrl"),/" is bijective from L2,, fufio Lzt-, andfrom X2,'into N2,r-', it follows thatV: X2,-do + f,2,sorsg: (1 Ie)/2is short-range and symmetric in the sense of Definition 2.6. The self-adjointness of rr follows from corollary 2.6 (or simply from the relative compactness y of with respect to rlo). The proof will be complete in view of rheorems B.b, 3.7 and

Proposition 8.8

if

we can verify (3.g) in the present case. But this verification is completely analogous to the argument in the proof of rheorem

4.11. Let

review

it

us

briefly. So, let

/ e L2,,o1p7 satisfy { : _VRi(p)6, p > 0. Using the notation of Corollary 5.7, ir follows from (Z.fO) that / e ,ll,i.. Cftoose Lf2 < s1 < so. It follows from our assumption on I/ and rheorem b,E thar v

\"

(t') €

B

(L7.:;, L2," " ),

(5.20)

and from Corollary E.Z(b) also that v a"* fu) e B (L7,:;+r/2 , L2,"o+rtz)

.

(s.2 1)

M.

44

BEN-ARTZI AND

A.

DpvrN^erz

Assume for the moment the interpolation indentity I

rr,";*+u, )r-- "u,

r2,sttutt,Q r2,et+7/21

lur,o

0

<, < 1.

(5.22)

Itfollowsthat if { eL2;,L, sl < s < srlll2,then{: -VA"*!t)dq 72'et6 n ker.4s(p) : L\,io*o , where 6 : so - s1, and lldll,+o S Cll|ll". By iterating this procedure we finally obtain d € L2;|;+tlz ancl (3.8) follows from (5.15).

It i(g) .

remains to prove (5.22). Note that

L';.L, s

> lf2,

implies that

for lel : ,ttt (in the trace sense). Thus (5.20) follows from a well-known fact on the interpolation of Sobolev sp:lces with zero .V''o, and that

i({) :

f e

0

trace condition on a given manifold [12], REMARK

5.9.

$11.5. I

Note that our abstract definition of a short-range potential

(Definition 2.6) corresponds precisely to Agmon's definition [1] for this special case. More explicit conditions may be found in [fSl. We shall also list such conditions (for the more general simply characteristic case) in the next section.

6. SIMPLY CHARACTERISTIC DIFFERENTIAL OPERATORS In the last section we saw the application of our abstract methods to a perri

turbation of the Laplacian by a short-range potentiar. In this section we shall show that we can also apply our abstract methods to short-range perturbations of constant coefrcient simply characteristic operators (see [z] ch. 14). For general elliptic operators, it is possible to identify a reasonable crass of short-range perturbations by taking the space ri,o as in Theorem 2.4 and. an of our abstract assumptions are immediatery verified. However, for other types of simply char-

it is necessary to take the space f;r" This, then, requires some extra technical considerations. acteristic operators,

^

a diferent form.

we recall the definition of a simpry characteristic polynomiar. Let p6(() be a real polynomial in multi-index

",

{ e R'

a^nd

PJ')(() : d'po(€).

4(€) : D"

l.PJ")(6)1, where for every As in the previous sections an constants are

generic.

DEFINITToN

6.1.

Po(€) is called simplycharacteristicif p0(€)

*

oo as

l{l *

oo, aad

4(e) s

" (p,

rpJ")ror*

(6.1)

')

Note that the polynomials in the fonowing classes are simpry characteristic: Hypoelliptic polynomials.

(i) (ii) Polynomials

lfl)--t

of principal type; i.e., polynomials for which lVpo(€)l for { sufficiently large and m : deg ps.

In particular, all elliptic operators

as

welr as the wave operator

time dependent schr6dinger operator

a2

> C(l+

f ap- A, the

ialat- A, etc. are simply cha.racteristic. For any real porynomiar, pe(D) when restricted to cff(R") is essentialry self-adjoint. As usual we denote by 116 its serf-adjoint realization. In order to 45

M. BeN-Anrzl AND A.

46

DpvtN,lrz

study perturbations of fle we need to define suitable spaces .f , Xfio so that the abstract theory may be applied. Before we do this let us recall some basic facts about traces of functions in Sobolev spaces.

{ € R'.

Let Q({) be any real polynomial in

critical value of Q if there exists a €o € R,' so that We denote Uy ,t(Q) the set of

citical

values of

) e R is called a Q(€o) : I and VQ({6) : g.

Recall that

Q. It

is well-known that

lt(e)

is

finite.

In the next proposition ,v"

(R,')

: {f ' lVll? : I l+ J

l€l')'li(€)l

,

a€,

<

*]

is the usual sobolev space of order s. Note that the notation, ll llr, which we are using here, is different from that in Sections 4 and 5.

PRoposITIoN Q(€)

: \|.

6.2. Let Q be a real polynomial,

),

( A(e), and let f1 : {{

:

Let do be the Lebesgue surface measure of 11. Then the map

Cf (R') Ifr - tr2(Ir,do) l

extends to a bounded map of X,(R,')

* tr2(lr,

do), for s

> Llz. In particular,

t"r 7,0 € cfl(R.),

t,,to,"l

l/

j (6.2)

=',,u,",,u,,",

I

n,

and

e.

{{,1#l , tl#l,r< i ( n}, andlet rr,r:

Then f1,1 is a (possibly unbounded)

C-

rroM*.

manifold for which each component

can be represented as

€r where

lvif < 2\/;1.

:

.i rj

where C depends only on s,

Proof. Let M1":

j

h(€r,

..., €r-r, €*+r,.

..,

€,),

Thus the proposition is an immediate consequence of

the properties of .t2 densities on such manifolds (see [Z], Th.

2.8). I

i

THn Lnrrrrnc ABsoRprron pntncrpr,E

4T

From this point on we shal suppose that p6 is simpry characteristic. Recall that a real polynomial is said bo be weaker than p6, written < po, if for some

e

constant C,

la({)ls cn@.

(6.3)

As is well-known, this is equivalent with the condition

O(el < cFoG).

(6.3')

Let Q1,'' ' ,Qc be real polynomials which span the subspace of aI polynomials weaker than Pe. For a fixed real s, set

X with the norm on

,f

: Ir:

L2'"(F.n),

(6.4)

given by

ilrlt?

: ilf ll'r" :

J G+lrl2)"lf (r)l2dt,

(6.4')

RE

and

Xir"

:

Iiro,"

:

{f , eilD)f e f:,r

<

i

< tL

(6.5)

with the norm Ifio defined by t

llfll"r;"

: llf ll'r;" .:lllei{o)tll"r,. i=r

(6.5')

clearlS the space ,ffi,," with the given norm is the same as the

space defined by any other linearly independent basis of polynomials and the norms are equivalent.

THEoREM

(t,

6.8.

tr'or

s > I/2 there exists an d, >

O

so that Hs is of type

Xiro,",c, R\A(Ps)) (see Definition 2.1).

Proof '

As usual we denote

rv

(Eo(^)r,s):

I

{ao

(r)}

the spectrar family associated with

Clearly

Po({)
i(e)?(Tlae, f;3.e cy1n,1.

116.

M. BpN-Antzl

4E

Thus, if ,\ # A(Po)

A.

DEVINATZ we have

t"a Q(€) is any real polynomial'

rro{^)r,

fi

AND

a@)d:^,/*^

#h

i(€)dttlao,

where, as usual, do is the Lebesgue surface area mea"sure' 6,p'+6\nA(Po) Suppose p A(Po) and let 6 > 0 be taken so that [p-

f

Suppose

(6.6)

:0'

thai Q is weaker than Ps' Then lQ(€)l <

clvPo(€)|,

Indeed, from (6'1) and the fact that Fo(g)

*

lPo({)

6'

- pl< +

oo as l€l

oo' we see immediately

that

F.(g) <

clvPo(€)1,

lPo(6)

-

(6'7)

pl <

6,

(6'8)

which are sufficiently large. In this latter case (0.7) is an immediate (6'8) are clearly true consequence of (6.E). For { in a bounded sei (6'7) and

for all

{

since lVP6({)l does not vanish in the given range of P6({)' If we use Proposition 6.2, and (O'Z) in (6'6) we obtain

tt,t^tl,a(D)r)l l$ ld^' where C depends only on s,

<

cllill'llall"'

n,6, Q and Ps'

l)

-

pl <

6'

(6'e)

Using the elements of the finite

that for ) € R,\,t(Po)' {Q;} in (o.O), and using the norm (O'S') we find /o(,\) : dEo(^)/d^ exists in the wea'k topologv of B(I''ffio) and is locallv

set

bounded in the norm toPologY' Since supp

i

is compact, without loss of generality we may suppose that

: .\} n supp ican Ia(rr-6,p*6),where {ro(€)

be represented m

laihl

€' : h(€"))' €' : (€r' "'' €'-r)'

l#l=clvPo(€',h(€',r))l-t,

i:r'2'

Furthermore, using the equation

o2Po ae"a€-

a\*u3 :o, =o='1, - "' ar - a€' a€ka)

Ic:t,...,tu-L,

(6.10)

TNE

and also differentiating once more with respect to

|

ainl

^ #l I+ loek o^J

49

LTUTTING ABSORPTION PRINCIPLE

< GlvPo(€',h(€',1))l-''

l,

from (6'8) we get

i:r,2; L1k
(6.10')

I

In terms of Euclidean space coordinates, equation (6.6) can be written

(,{o(})/,

as

itet?lEltr +lYtn121rtz6q', (6.11) / Fffh I.u-

e(D)c):

r

{ : (€',h(€',.\)) and ) a (u - 6,P* 6). Ditrerentiating (6.11) with respect to l, using (6.7), and (6.10)' (6'10') for j :

where

both sides of 1, and again

invoking Proposition 6.2, we find that l)

lfi

t.r'trlt, a@dls cllill"+'ll?ll"+', ll |

pl < t.

(6.12)

Thus we see that ,46()) is weakly differentiable in B(X"+t,Xiro,"+r), with

a

derivative which is locally bounded in the norm topology of the latter space. We may now use the inequalities (o.O) anil (6.12) and interpolate the oper-

-.40(,\r)l between .f" and L+r for anv s arbitra.rilv close to 1/2. Recalling the definitions (6.S) and (6.5') for the spaces .ffi.," we find that there exists an d.> 0 so that for any )1, )2 € (p,- 6,pt* 6), ator Q(D)lAo()1)

lldo(rr)

-

Ao(^r)'lft,,ri",, 3

clll - )r1".

(6.13)

This establishes the first requirement of Definition 2.1. To establish the second requirement of Definition 2.1 to prove that for

f Ae Cf (n'),

is clearly sufficient

the function

r(z|: I lPe({)

it

-pl>6

(6.14)

"&i(e)?(dae

can be estimated as

lr(z)l s cllill,

llall,,

Reze(p.-612,p+612)

(6.15)

and l::.

V /..

krl - F ("r)l < Clr, -,,

l"

llfll, llAll",

Re

z; € (p

-

6

12,

t' +

6

I

2)'

(6.15')

M. BUN-AntzI

50

AND

A. DpvrNIrz

where C depends only on s,6, Q and P6.

We shall only establish (6.15') since the proof

$ eCfl(IL') with J $,G)ae:

mutand.is. Let

of (O.fS) follows rnutatis

1, and let 4

€

R..

Replace ,F(z)

by

FnQ): I W#i(e +r)?G+lIae. (€+4)-l,l>6

(6.16)

lPo

Clearly

F("): Let

,S

:

I

,,uror.

supp a; then there exist 1, Eo > 0 so that

lPo(€+?)l+lvPo(d+?)l> ",Foh), €e 8, irl>i?o.

(6.17)

4(g + d S Cnh) uniformly in : 4(C+rr * €) < CF1G*a). Using (o.r) and

Indeed, there is a positive constant C so thar

{e

^9

and ? €

R,.

Thus Fe(a)

the fact that Fo(e + 4)

Now let Xr

)

oo as lql

-

*

oo, uniformly

in { €

,S, we

get (6.1?).

<1f4for lal > ,B1. Flom (6.17)we get either lPo(€+ dl> h/4Fo(a) or lvp6({+?)l> h/4Fsb) tor every f €,9and |ril > r%. Given lAl )l?6 let Sr,n e Sbetheserof allf e ,S .Bo be such

that (lpl +6)lFohl

such that

lPo((+ Denoting by

p(r)

Q)

1,

dl> |F"@, €€sr,n,

I'rl >,to.

(6.18)

the characteristic function of B, let us set

:

F,h)-,

*',',{e) lPo(

Using (0.f8)

tFl'te,)-

it

,*{, -

o

(6.1e)

(i)-'

21,

Re z2

1",-""t11"",

€ (t, - 6, lr* 6) and for l4l > g1,

"q#

ir .

But the assumption that Q < Ps implies thut lXs.,, (€)A(€ + €, 4 € Ro. Thus for lr?l 2 Er,

lrJl)

(21)

-

Fltt

+ a)?(e + r)ae'

^,

is clear that for Re

Fl'|tp2)t=

ffiffifui(e

kil

<

cl", -

,,1116(

- ?)ill.lla(. -

rrll, ,?)

|

ro'?(

+r)r,.

S CFofu) for

z)ollo. (6.20)

Tnn Lrunnvc ABsoRprrox pnncrpln Next, let

,92,,?

--

,S\,S1,? so

that we

lvPo(€+?)l >

51

have

]F"{d,

€€ s2,,,

lzl > ,e,.

(6.2r)

: Fr(z) - rlt)p1and set ,Po,r(€) : po(€ +d/Foh), zn: z/Ab), pr: p/Foh),6,1:6/Ah) and ra,r : {€,po,r({) : )}. Th"n We now define Fj2)

we may write Fl2)

k)

:

Foh)-'

,^-!,,0,#;,{^""',"

tu)ffifffl

?G + n)iE

+itdo' (6.22)

Using (6.21), as in the proof leading to (6.13) the sesqui_linear form

o'o

-

defines a H6lder continuous

r x",."(€)frffido e{c)flO

^J

;;;r.r

of ,\ with compacr support, with operator

values in B(Lr,o , L2,-"), and with H6lder norm which is independent of l4l > rt1. Thus the range ofintegration with respect to ) in (6.22) is bounded independent

of 4. Using the Privalof-Korn theorem we have

IFr2tei where

c

on supp

- Fr2)p;r < cr", - ""rll$69^t:t,a -' Foh) rt *.,ri '"1i" li66q llvv'r''rtt)ll"' " *' rlll ll

is independent of l7l

)

df (R.) to be one to { of arf Of , *- rl&Al

.l?r and we have taken ry'€

6. Sirr." all of the derivatives with

respect are uniformly bounded, the last equality implies that lql2)

ki -

rlzt 1zr'11 <

for lnlZ Br,

cl,, _ "r,l"ll$( - n)fll"ilS( n)?11".

This inequality, taken together with (6.20) implies that for Rez1, Rez2 e (tr_

6/2,p+6/2),

lF,("r)

- r,("")l

<

cl", - ,,lll$(.- n)fll"ll!( _ n)?11".

Observe that by rhe definition (6.16) of

tainly true for lal

I

foe),

the

r?1. Thus (O.Zf) notas for all € 7

(6.28)

inequality (O.Ze) is

R,.

cer_

M. BpN-Anrzr

52

AND

A.

DpvlNerz

We now have

tt

trb)t:llr[ ,,aorl

I

R3

<

l

ct,,

-

"

Un$r

- ntLt?a,I''"

Uuot

-,)ou?.,\ (6.24)

If s: m is a non-negative integer, then u$t

- nl?u"^ < c'* D, llD"@(. -,ifll Iol<-

=

,*

!{ "p- tD',c-

Integrating with respect to 4, we have for s

I

:

,, ,'

} { "A

rr"

rr,

r,

o*

}

m

nlt -,t)Al'"an s c"llfll3.

Rtr

By interpolation this inequality is true for every non-negative s. Inserting this

into (6.24) we obtain TEEoREM

(6.15,). r

6.4. If Ps is simply characteristic,

and s

> tfz in (6.4) and (6.5),

then in R,\.[(Po),

ao*(r) exist

in

: .$

fio(r +;"1

the uniform operator topology of B(X,

Xfi)

and are locally H6ldet

continuous. Proof . An immediate consequence of Theorem 6.3 and Theorem

RsN,fA,nx

6.5.

2.2. I

Note that the proof of (6.13), which is the first requirement of

Definition 2.1, is straightforward. Surprisingly, the proof of (6.15) and (6.15,),

>: '::_

:.:

which is the second requirement of Definition 2.1, and which was immediate in the situation of the previous two sections, requires here a considerably longer

:,a,.

technical discussion. Clearly our discussion is indebted to those given by Agmon

?,

and H6rmander [2] and by H6rmander [7] Ch. 14. However, our proof of the

='2

Tns Ln\.rrrrNc AssoRptrolr pnrncrplB

bB

lirniting absorption principle for H6 in weighted

spaces seems to require fewer technical considerations than the proof given by the latter author for Besov spaces' apparently because we are able to apply the classical privaloff_Korn

theorem at several crucial points. Before we proceed to a discussion of limiting absorption for frs perturbed by a short-range potential, it is necessary to obtain a few extra facts. In particular, as

in the previous sections, we need a sufficient condition, better than that given

by Lemma 3.3, under which Assurnption 3.2 holds.

6.6. Let, p6 be simply cha.tacteristic anil p. €R\l(pe). Tlen tlrere exjsf 6, | ) O so that for every € ? {€ , .Po(€) : pt}, every surface lr : {€ , Po(€) : )} for ) e (p- 6, p*6) ias a representation in the balt B,(4) PnoposrrroN

with

center

q

and radius

€r where h is

C* in lv

the constant

: :

.:

irr

C

all

:

r as ft(€r,...,€r_r, €r+r,..., €o,)),

of its variables

(hl < C,

€'

:

(6.25)

and

(€r,

... , {r-r, €r+r,. .., €,),

(6.26)

depending only on ps.

This result is an immediate consequence of the assumption that ps is simpry characteristic and the implicit function theorem (see [Z] pC. 18).

6.7. Let i a e Cylnn), e < po, p €R,\^(.Po) and suppose that Ao(p)I: Ao(p)c:0. Ther there exist a6 > 0 and apositive constant c. depending only on 6, e and ps so that for _ l,\ pl < 6, LEMMA

:,,

l(Ao (^) Q (D)

L

,l],

Q

@) c)l

s cl.\ -

Proot.

r"lllI

Using the last proposition, every surface be represented, without loss of generality, by

l, .,':.

::

€,

:

h(€',

)),

({', €") e

l':' :.|:

where lr satisfies (6.10) and (6.10,).

lb,llgli,.

l1 ng,(4), lf _pl <

B,(tt), l) -

pl <

6,

(6.27)

6, may

M.

54

Let 6,p

Ir(f

* '

A.

BEN-ARTZI AND

0e Cf (.B"(o)) so that I6':1,

DEvINATZ

and for a"ny 4

€ Rn andl€(p-

6) set g;

\) :

@"

Po

{D

y

-

/ (C)=)

JI Rtr-

e

T

q)Q2 (D) Ao(^l

6"(,t:!,9-,'J€) lvPo(€)l

s)

f@i({a" (6.28)

q:!9.,7a

lvPo(€)l

where in the last integral

l,

+ ly rit€la(el(r \

€: (€',h(€',))).

1'

r,l21rtz

41,,

The Schwarz inequality yields (6.2e)

lrnu, c;I)l S un(/, f i\IL/z {14k, c;^)}t/',. By hypothesis,

:0.

I,(f , I ; p) :

In

k,

g;

:

P)

0 so

thai ?(g, t (€', p))

:

0(€', h(€', p)l

Thus using (6.10) we get

li(€', l,(6', )))l'

: l?G' ,h(€' ,r)) - i(€', h(€'

p))l' lr,(e',A)t aft

<

ctl -,,ltvP,(€)t-'

,

|

,/ l#l' ,rl, |It'(€"p)' I

where

C

depends only on 6 and P6, and

{:

(€',h(€"})).

Since

lQ({)| <

CIVP6(€)l (see (0.7)) if we use the above inequality in (6.28) we get

rl^?alt

I,(f ,f ;))< cl) - A

J lt**l

R.'

With a corresponding inequality for l(q'?(D)Ao())/, c)l

^

a'te -n)d€.

I

g, from (6.29) we have

: l.l r,u,si \dnl ti" I

I

< cl) - plll/llr' llgllr''

I

QZ(D)Ai,(.\) exi,sts in R,\l|(Po) in the weak topology, and is -Eldlder continuous in the norm topology of B(I",,(i). ?hus, there aree, 6 ) O so that tr f, c e I'" and Ao(lr)f : Ao(p)S -- O, then for

LEMMA

l)-pl

6.8. If s > 3f2, then

<6, l(Q2@)ao(^)/,s)l S

cl^- pl'+"lllllr,llsllr"'

(6'30)

TTTE LTT4TTING ABSoRPTIoN where

C

55

depends only on 6, s, Q and Ps.

ProoJ. If J:

PRINCIPLE

we diferentiate (6.28) with respect to

1, we find that for

Q2(D)A!'[)

ll - pl <

)

and invoke (6.10), (6.10') for

6 (6 given by Proposition 6.6) arid s > Bf2,

exists in the weak topology and is a bounded function of

norm topology of

)

in the

B(I",Ii).

If we differentiate (6.28) a second time and invoke (6.10), (6.10,) for

j :1,2

l)-pl < 6 and s > S12, Q2(D),4fi()) exists in the weak ropology and is a bounded function of ) in the norm topology of B(I",Ij).

we find that for

If we now interpolate we have the first statement of the lemma. The second

3.3. I

statement follows by Lemma For p € R\A(P9) and s

X:,": {f

e

L2'"

> 1/2 let us now use the notation

(k")

.

arl :0onIu,0<j<s-L/2\,

ai

(6.31)

: p} and the normal derivatives are taken in the trace lr. : {€ 'Po(€) sense. Clearly, Cf (R") o,foo, is dense in .f"or. Although we shall not do so, it where

can be easily shown that for Lf 2 < s

PRoPoSITIoN

6.9.

Let s t L,

1t

< Sf2, I!,r:

Lr,"(n

€ R\tt(Po) and e

<

) n ker,46(pr).

ps. Then there exist a,

6 > 0 and a posibive constant C, depending only on 6, s, p6 and e so that the

operator valued function

)*

Q2(D)Ay(A) € B(X:,,,,G:.*)-) satislles

llQ2@)Ao(^)1ilrg*,1ry,1.

Prool- Ftxt > 3/2

and for 0

s cl\ -


1,

plr+', lr - pl < t.

(6.32)

let [.fflr,, rf,r]e be the interpolated space.

As is shown in [rZ], Wf ,,,

r:.ul:

rl-o+e,,u.

(we should remark that Proposition 6.6 allows us to carry over the proof of this equality for the possibly non-compact manifolds considered here.) Also, it is well-known [12] that for any two spaces !, Z,

Iy, 216

: lz*,!*lto.

M. Bpl-Anrzr

56

If we apply this in our situation we W1,,, r:,ul,"

:

AND

A.

DsvrNArz

get

[Wf,)- ,(x1,,)"1'-t: [(rf,,,)., (xl.,,).1t.

Thus, using (6.27) anil (O.SO), we get (6.32) by

interpolation. I

We now give a sharpened version of Corollary 3.6.

6.10. .Lets ) 1, p €R\A(P9) andQ < Ps. Thena@)n"+!t) e B(f:.,,I'(R,")). In other words, THEoREM

no*

Proof

.

(p) e

B

(I!,,, r ;,

(6.33)

",").

Write

Q(D),{o()) ,., f -\-u--nI t lr-pl6

a@)n*u,): P.v.

d@(D)Eo(^l)

lr-,

'

:Tt*Tz,

where this is to be initially interpreted as acting from

X!,, to ,fj for s >

I

(see

(2.7), noting that .I"ou C ker,{s(p)).

Let

i,

Q

(0.s2) we get

e Cf , Ao(p) I for l) - pl < 6,

l@ (D) Ao(^)

f,

g) I

: 0. Using the Schwarz inequality

together with

s (& ( r) Q @) l, Q (D) ilr / 2 @o (\) c, d' / " < cl) - pl+ilnl"(Ao(^)s,dLtz.

Thus,

l(r,f ,s)ls cllill"

s cltilt,

I lr -r,l<

U-pl+(,{o(\)g,slttz4s 6

( ,

\'/"(

)''" (.a6(.\)e,s)d)) I[1^-i.1
s cililt, llcllo' (6.s4)

Tnn Lrurrrwc ABsoRprror pRrucrpLn it

OI

As for (T2f ,9), we observe that it, is equal to F(pt) in (6.14). From (6.1e) follows immediately that

l4!') (p)l < cilA(. To deal

witl .f{2)(p)

- ?)ill" lla(. - ,r)?llo.

(6.35)

we use the Schwarz inequality in (6.22) to obtain

t 's -

tpl4 (p)1,

,^-,l,rro,

:l^,n)dl t$(. (\ - Pn)'

illil!,

where we have set

: [ ,"."''''-1r)f.Qegt2_ 'Po(q)2lvPo'n(€)lfG+n)t"a".

/(),?)

'

"j., ?G+d:0

By our assumption

for € € lr,r.o. Hence, noting (6.21) and the argument following (O.ZZ) we see that f(,\, a) has a compact support (in )) independent of lal

r

81, and satisfies, by (o.sz) with a constant

c

independent

of n,

I(), a) : lI(^, n) _ r(p, n)l <

ct) - t",t'+o ll0Ag::a Po('r)

ll

it" * rrll', "ll,'

where'/ e Cfi(R") and is one on supp f'. Sir,." all derivatives of a(€)A(€ + dnfu)-L are uniformly bounded (in a) we get, for lnl 2 Rr, tet"t0")r

=

"il@#ff=rtll" tt^ _ z)all"

s

cili( - dfll,lli(

Thus from (6.35) and (6.36) we get, for

lF, (p)l <

(6 36)

-,r)ollo.

lrll> nr,

cllt ( _ dnl ( "il$

_,r)?llo.

(6.37)

By the definition (O.fO) for ,t'q, this inequality is clearly true for lnl

<

Rr.

lntegrating (6.37) with respect to 4 we get finally

lT,f

,g)l <

cllfli"llcllo.

The inequalities (o.ra) and (6.3s) constitute the contents of the

we are now in

a

(6.38)

theorem. r

position to state and prove the main theorem of this section.

M. BnN-AnrzI

58

AND

A.

DOVTNITZ

6.11. Assumethatforsomes > lf 2 thepotentialV € B(f ;r",,, f ") is compact and symmetric, Also assume thatV e B(Xito,", Iz"). Then H : Ho*V, restricted to Cff(R'), is essentially self-adjoint and the limits fi+(,\) THEoREM

exist in R,\A(Pg) except for a discrete set oo(H) of eigenvalues of 6,nite mul-

H

tiplicity. In particular,

has no singular continuous spectrum and the wave

operators

w+:

"

- rliL "it* "it*o

exist and arc complete in the sense that

Proof. From

Range

lZ1

: E(R\ap(I1))I'(R")'

Theorem 6.3,

.116

is of type (I,,Iito,",a,R\A(P6)) and clearly

the density hypothesis of Theorem 2.7 for any non-real z is satisfied for D,

Cf (R,').

:

Thus we have the self-adjointness assertion for 11.

The proof now proceeds completely parallel to the proofs of the correspond-

ing theorems of the last two sections. Our assumption means that range with respect

to I/e in the

get the estimate (S.A) for

sense of Definition 2.6. However,

6: -Vntjt)d, t > lf2,we

V is shortin order to

must apply a bootstrap

procedure by means of interpolation. Take 1f2

(

s1

( s; then

v nt0r) € B(f:,,p,.L"),

Vntkt) e B(f",,.I,)

clearly

and so a fortiori

where I"o.,, b given by (o.rr). Atso, by (6.33) and our

assumption on Y we have Iz.B$ (p) e B(I!,ay12,p, X"+r/z). As we have already noted, we have the interpolation equality

lr!,, r, r !,,ult

:

Xlr- et

",+o "",t,' proof proceeds now exactly as the proof of Theorem 5.8, and we Using this, the

shall not repeat the

details. I

REMARK 6. I 2. Various sufficient conditions may be given for which

7 satisfies

the compactness condition of the last theorem. Following Hiirmander [7] we may consider the subspace of Q

<

P6

for which 6G) I F"G)

{Qi , t < i < ^} be a basis for this

+

0 as l€l

- -.

Letting

class we can take

v(x, D) :ir,1,1q01r1, 1

(6.3e)

Tnn LrurrrNc Ansonprrou pRrncrpln where the

7i(r)

59

satisfy

c(l+ lrl)-1-"

lvi@)l<

(6.40)

To allow for local singularities one may consider the subspace of e < p6 for which l€llA(€)l < cnG), or for which l€f lq(€)l < cFoG).In the second

for example, we now take {Qi} as a basis for this space and consider the potential v(a,D) in the form of (6.89) where each zi(r) is locally square case,

summable and satisfies the condition .a.

(L

+ lal)2"Vi , )12 -

-L2

is compact.

As is well-known [13], a sufficient condition for this is given by f

1\p-[(t+ l'l)n' JI r€Rr

ly-cl<

.:

lv,@fW -,1-n+,dv: <

*,

(6.41)

1

t,a, .

where 0

< p < 4.

We refer

to Schechter [13] for conditions on the V1@) in

l,

the frrst case mentioned in the paragraph so that

ii.

H6rmander [7], p.2a6.

I r: ''{:,

m

{

v(r,D)

is compact. see also

For elliptic operators Ps of order m or operators of principal type of order 1, we note with H6rmander that we mav take

v(r,D):

f

v,@)D".

This falls under the first case mentioned in the last paragraph so that

w *,

v.(z).

some

n (a.rz) we sum only over lcl ( m- r, this falls under the second case of the last paragraph so that we can alrow (o.at) for each I/o. singularities are allowed for the

{i,

(6.42)

lol<^

rt

a:

'' t':

:

:l

: :.

'ii

t ;; ::,

7. SOME FURTHER PERTURBATIONS OF _A

t:

i

In this section we discuss

cases

in which the operator -tlq is obtained from

by the adilition of an operator depending on one coordinate only. However, unlike the case of the operator (4.1), the weight function involved here will

-A

depend on all coordinates. we shall not make any attempt

at achieving the

most general results for "entire classesn of operators. Instead we shall try to illustrate the applicability of the abstract approach and the fact that it does not require a very detailed study of the operators involved' We shall consider the following two classes of operators: (7'1) H: Ho*v(r,t),Ho: Ht@Iz*Ir@Hz in r2(Rl)a'2(l'), c R) where I/1 : -A in L2(R*), Hz is a self-adjoint operator in '2(f)(f whose spectrum does not cover the entire real axis, and [, 12 are the identity operators.

: Holv(r),1{o : -A-(sgn c1)lo1lp,o < B <2, in '02(R,*), r eR^' (7'2) In (z.z) we exclude the case f: 1, which corresponds to the operator (4'2)' In : both cases we take Y to be a real potentiat. The operator (7'1), with H2 -i& H

in

.L2[0,

T], is related to the time- dependent Stark Efrect with periodic behavior

in t, following a suitable transformation [8]. The operator Ho tn (7.2) has the form (7.1), with Ilz : {* - (sgn o)lclp. However, the spectrum of II2 covers the entire real axis. In dealing with these operators we shall need a theorem on the resolvent of a sum of tensor products. Our setup is as follows' T1,

T2

are

self-adjoint operators in Hilbert spaces

/;()) : dq;(\)ld), i:1,2

operators are denoted by

)/2' respectively' The

{.Er(.\)}' {Er())}' with deriva(when they exist)' The respective resolvent

respective spectral families are designated bv

tives

.V1,

Rt(zl, Rz(z). 60

* .:J:

THE LIMITING ABSoRPTIoN The Hilbert spaces ,I; C X;,

i:

L,2,

61

PRINCIPLE

are densely and continuously embed-

ded.

As is well-known, the operator ?s

Hilbert space

)/: lr

8.V2, where

:

Tt

11, 12

@

Iz

* It

& Tz is self- adjoint in the

arc the identity operators. We denote

Uv {.Eo(f)} its associated spectral family, with.46())

: dEo\)/d\

(when

it

exists).

M, N C R we set M- N - {m-n : m € M,n € .ff}.

Finally,'Lf

by

a(")

the spectrum of an operator ?.

THEoREM 7.1 (see [5], Th. 3.6). Let A is of type (Xt, 6

We denote

Ii,a,R\A)

cR

becompact and assume thatTl

(Defnition 2.1). Assume in addition that for every

> 0 tlre operator-valuedfunction

r{1(.1) satisfes,with aconstant C5 depending

only on 6,

ll,{1(.\)llr,,ri SCa, dis6 (),4) > 6, (b) lll1(^2) - ,{t(rt)llr., x; < C6l\1- lzl", disf ();, A) > 6, i : 1,2. Assume further that T2 is of type (Xz,Ii,a,U - Ll for some open set y C R, (a)

and.

Iet

f c {fL I

}/u) n

be a densely and conbinuously embedded

than those of

?.

,;

a; :a t?:

It

@

Xz, Xt @

(ha

Hilbert

Xz)

space (hence

its norn is sfronger

Iz).

X*,a,U), Furthermore, Iet K C U be a compaca interval and A6 an open neighborhood ofl such that K - lo C U - L, and let / e Cf'(R), 0 < d < l, where 6 : L on Ln(K -o(T2)) andsupp S c Ls. Then forleKwehave /o(r): JJ|ft(t-sQ,-v))A,(r -v\oanrfu)+ | 6(v)dEt(v)aA2(\-v). (7.3) Then Ts is of type (X,

In particular, tle limiis ,B#(f) : Iim"*e+(I6 - ) a ie)-l topology of B(I , X"), \ e U.

exist

in

the notm

We can now apply this theorem to the operators (7.1), (7.2). In what follows

we shall use the notation of the theorem whenever appropriate. In particular,

note that by Lemma 5.4 the operator posed on

fi, with A: {0}.

I/r : -A

satisfies the assumptions im-

62

M.

BEN-ARTZI AND

A.

DpvINerz

A) The operator (2.1).

no(H2): 0. Clearly H2 is of type (X2,X2,a,(f), above we can take ll, : {0} so that Ln(K-o(H2)) : 6, we may take { : O in (Z.B) to obtain

Let U C R, be open, U ){z

: L2(t). As observed

where

K C U.

Thus

,{o())

: Ior$-z)8d,82(u), ^€Kcu. il

For every s € R we let

(7.4)

,2''(R'), re''(Rtr)

be the weighted spaces as in section

L2'" g

corresponding norm,

5 and set

I" : ll/ll3 Let,

D(lHrlr/2)

:

12(f), with

IfR.I u + pl2)"lr(r,ll2atdt.

be the domain of

lHrlllz in )/2, equipped with the graph-norm.

We now define

Iito,": (x1'-"13";@ rr(r))n (r2,-"(rr.)

@

Dln2lr/2))

with corresponding norm

lllll'x;",": LpUul, 7 .2.

For s

of type (X,,Iito,",a,

ll/111"

+llea)u2fll1, + ll(r, @lr,f/z1yyz_".

> L/2 and some a > O depending on s, the operator Hs is I/) (Definition 2.1), so that, in particular, the limits

4o-())

: "\xr*(to

exisf

(2.5)

- ) aie)-l,

I eU,

in the norm topology of B(X", Xfio,) and are H6lder continuous.

Proof. Note that the integration in (?.a) extends actually over values of z such that,\ - v ) 6 ) 0 when I € 1( c U since r{r() z) : 0 for ) < rz. Thus, using

-

and,{t(}) : (Ef ()) - n;(\)/zr; in (z.a) we obtain the boundedness and H6lder continuity of A6(,\) from .f" into .Vr,-" I ,2(f). To handle the (5.16)

remaining part of the norm (7.b) we observe that, using (5.11), the function

K

))J R

or.(^ _ v) o lv1|/z4Ez(v) e B(r",

r_,)

....:

ar

._:

TTTE LTUTTTNG ABSoRPTIoN

PRINCIPLE

63

is Hiilder continuous. Finally, to verify the second part of Definition 2.1, we use instea.d of (7.a) the formula

fto+(.\)

:

[,fO il

-

in conjunction with {s.ro) and the

v)

@

dE2(v)

(see (a.22)),

same axguments as

in the first part of the

proof. I Rnuenx 7.3. Note that if

we assume that

IIe has a closure Fo in !_", *r"

could add a term llEofllr_" to (Z.S) (see Theorem 2.4). However, we shall not need this term.

Our limiting absorption principle for

.EI

is now as follows.

TunoRou 7.4' Let H be given by (z.t) and assume that the real potential V(r,t) is compact from Li'o,o into L1a6 for some e > 0. ?hen H is self-adjoint (with same domain * Ho)' Let R(z): (H z)-t, Imz I o. Then the limits (7.6)

in the norm topology of B(I",Ifto,), s > lf2, except possibly for a discrete set (in u) of eigenvalues, oo(H), of frnite multiplicity. Furthermore, A+()) are H6lder conrinuous in U\oo(I/). exist

Proof. Clearly, multiplication by (1 + lrl")"/, is bijective from .f" onto I._" and from ,ffi.,, onto Iio,"+". It follows that Ir I Iio,"o - I,o, so : (l+ e)/2, is short-range and symmetric in the sense of Definition 2,6. Thus the self-

I/ follows from Theorem 2.2. Next, let peKC(f,s >l,andassumethatAs(p)f:0where f eI".

adjointness of

Noting the similarity of equations (a.zs) and (2.4), we can prove as in Lemma 4.6 that, for some 6

> 0 and

)

e K.

(,{o(r)/, I) 3 c1s,"1t - pl'+ollfll?. Indeed, this inequality follows from the fact that

(7.7)

in (z.a) we have, for some if t € .If and v e o(H2). Then by Corolla.ry 5.2 we have, if g e L2" (]Rl, A{p - v)g : o,) € K, 4 > 0, ll - vll4

(,4.()

-

v)c,s) s

cn,"l^-

pl'*ollgll"",.,.

M. Bnn-Anrzr

64

AND

A.

DEVTNATZ

The proof of (7.7) is now almost identical to that of Lemma 4.6 (the only easy

that

change being

tr2(R)

space

Ifi

is unitarily equivalent now to multiplication by

) in an

with function values in L2(5"-r'r1.

The proof of the theorem will be complete, in view of Theorems g,S, g.z and Proposition 3.E,

if

we can verify (3.8) for our case, namely, that

-V Bt0t)f , f e I"o,

so

> tf 2, then

if / :

(?.?) holds rrue. But the proof of this fact

is an exact repetition of the proof of Theorem 5.E. Indeed, if we denote

I!: I"nker:{e(p), s> LfZ, then from (5.22) we have the interpolation formula

W!,, Taking now tf 2 <

sl <

r!,+,/r),

:

x!,*+r.

so we have by assumptio"VPd*04

e B(X:,, -f"o)

and

by (7.7), (5.f7) and Corollary 8.6, also V A"+U4 €

(I",+r/2, Xeo+t/z).It follows | € I"o+r/2, with ll/11""r.1 1z < Cxll|ll"r, so that we have (7.7) with s : so and the proof is complete. I

by interpolation that

To give a more concrete application of the last theorem, assume now that

| :

is a finite interval and H2

:

/Aq is a self-adjoint operator in I boundary conditions). Assume further that D(H2lr/r) c X,(f), the sobolev space of order r > 0. since )/"(r) is compactly embedded in r2(f), it follows that (0,

") (with suitable

QQ,

o(H2): {)r}[i--,

A

.16

(

]3-"1,

(7.8)

*oo are the only possible accumulation points of o(Hz). As for potential V(x,t), assume that, for some e ) 0, where

V(t,t) : (r + lol)-r-"V1(a,tl, \(a,t) e .[@(R'" x t). CoRoLLARY

7.5.

Under the forgoing assumpfions on H2

given by (7.1), and R(z)

: (n - z\-t,

Im z

andV,

the

(7.e)

H be 10. ftren the spectrum o(H) has Iet

no singularly continuous pa,rt, and fle set of eigenvalues oo{Hl accumulates at most at the (threshold""

{)r}.

Furf.hermore,

Tnn Lrurrrnc AnsoRpttor.r pRrNcrpl,E

6b

(a) The Iimits

,B+(r) exist

in

:

E() + ie),

,\p

the uniform operator topology of

),

(

oo(E) u

B(I",Ii,o,),

{}r}ii_s

(7.10)

> |f2, and are H6td.er

continuous.

(b) The wave-operators

w*: exist and are complete

in the

t

- ,IIL "it* ,-irl{o

sense

(7.11)

,

that

tl,-

Range I,Za

:

E(R\(ao(n) u {lo}ii_*))rr(R,).

Proot. It follows from our assumptions that riro,"

g (,v''o(R') s rr(t)) n (r2(R")

@

y'(r)),

(7.r2)

hence the compactness imposed on

z in the last theorem is satisfied here in view of (7.9) and the Rellich theorem. Also, by (7.E) we can take U: (_lo()1,1611). Finallg part (b) follows REMARK

as

in Theorem

6.11. r

7.6.

observe that this coronary extends corresponding results obtained by Iorio and Marchesin for H2 : -iA/At (see [E], Theorem 5.1 and Appendix). However, while the rate of decay imposed on I/ in (z.s) is the same as theirs, we were unable to alrow rocal singurarities for

I/

as

in [gl. of

course,

the inclusion (7.t2) allows some singularity of.v, but in order to relax further the assumptions on

z

one would have to take a croser look at the range of in individual cases, as is done indeed in [g] (see also Rema.rk 7.8).

fif,(l)

B) The Operator (7.2). The operator .tle has the structure of ?o in Theorem 7.1 where

Tr:-A'

c€Ro-I,

rr: -# -gsna)'lzll, However, in the present situation

o(T2): R and

(7'a)' Thus we must study the limiting

c€R. (2.3) cannot be reduced to

absorption properties of corresponding to Lemma 4.1 we have here.

12. rn fact,

66

M.

Lruue

7

for

a)

some

.7

.

BEN.ARTZI AND

For every s

O. Here

t"

A.

DEVINATZ

> L/z - ft, the operator T2 is of type (t",I! ,a,R),

and

its aorm

are given

by (a.A),

Proof. The idea of the proof is identical to that of Lemma

4,1, only that the

situation here is much simpler since we do not have to prove uniform estimates

lke (a.5). Thus, let u(c,.\) be real continuous on R. x R and such that

/ ,12 -\ \'\'"'t' \ - dr, -(ssnr).ltlp /lo(r,)):)u(u,)),

)eR.

I

(2.18)

Furthermore, as in (a.6) we may assume that the transformation

(7/)())

:

I f@)u(",^)0,, /ecfi(R),

* extends as a unitary map on se

L"(R). In analogy to

(4.10) we have here, for

/,

L2(P-l, n

fi@z(t)f

,s)

:

7

f

(^)'Ti(Xl,

for

a.e. I e n,.

let.K c R' be compact. It follows from Theorem \ e K, satisfies the estimates Now

(

cx$+

l--' lu(c,I)l< { I \ Cs e",

l4-*,

8.2

in

(7.14)

that u(r,,\),

[B]

.IC*(t+r)t-io r]o, lar l-lCae", c(0. lao(,,r)l

(7.15)

The proof proceeds now in exactly the same way as in the arguments leading to (4.13), (4.15) (notice again that we do not prove uniformity with respect to

I

in

infinite intervals). In particular, we have instead of (4.14),

17f(^+h)- 7l(^)lScx.lrl.ll/llr., so that by interpolation (see (a.15)) we have

^eK,,> for

every

0)

s/2-

18,

(2.16)

e K,

0 and

^ l7 f

(t + h) - 7l1)l

I

L/2- B/4+e/4<

cn,e . lnlffiL# y11r", s

<sl2-ln *r1r. r

(7.r7)

Trrr Lrurrwc CoRoLLARy and

f e I".

7.8.

Lets

ABsoRPTIoN

PRINCIPLE

) 1-

LF anaassume that A2(p)l Then for some 6 > O and aII ), e K.

:

O, where p e

(Az(I)f , f) < cx,,lA- pl'+6lllll?. Proof. Indentical to that of Corollary

4.3, in view of

67

K

(7.1E)

(7.L7). I

Lemmas 7.7 and 5.1 show that the conditions of rheorem 7.1 are satisfied if one takes

rt:L2'" (nl,-') , s>Lf2, Iz:I"(R), s>L/2-Bla. In particular, since we are not aiming at the sharpest possible result, it is obvious that we can take.I in Theorem Z.L as L2i(&.), s ) lf2,withnorm

llTlll:

.f*^(r + lrl2)"lf (r)l2do. CoRoLLARy

T

.9.

For every s

> If 2, the operator Hs given by (Z.2) is of type 0. In pafticular, the lfunits

(L''"(n), L2'-"(n*),a,R) for some a> .Rot(l): exist

in

"\p,%(r+;e),

I

e R,

the norm topology of B(L2,",

AIso,

L2,-"), and are Hillder contiruous. if Ao(p)I : 0 wlere p e K c (0,oo) and i e L2,",(Rn),s1 )

then, for some 6

>

1,

0,

(Ar(^)f ,.f) < cr<,,.1)

- pl'+6llfll?,.

(7.1e)

Proof. Except for the last statement, the coro[ary folrows from Theorem 7.1. To prove (7.f0) we note that since s1 > 1 we can use (7.18) ancl (5.1S) in (7.g). The proof of (Z.fS) is now identical to that of proposition

Finally, we turn to the operator

5.6. I

rr of (2.2). we assume that lz(r)

has the

form

v(r):

(r + 1r;;-*-. (vr(r)

+vr(,)), e ) o,

(7.2o)

€ r*(R") and.V2(t) is compacily supported and compact (as a multiplicative operator) from,V2(R') into .[2(R'). where V1(r)

Our limiting absorption result for

If

is now as follows.

; ,i.

M. BEN-ARTZI AND A.

68 THEOREM

7,1O. under

the above assumpbion

a

onv

the opetator

H givenby

Itsspectrumis absolutely coatinuous

(7.2)isessentiallyseF-adjoinbonCff(It'). except possibly for

DEVINATZ

discrete set oo(H)

of

eigenvalues

of finite multiplicity.

Furthermore,

(a) The limits,

fi*(.\)

:

+

ie), t e R\ao(II),

"IH.R(.\ exist

in the uniform operator

topology of

B(L2'"(Fcn),rt'-"(R')) , s > lf2,

and

are Hilder continuous'

(b) The

wave operators

W*:s-

lim

eit$e-it$o,

exist and are complebe.

Proof.

We note that by standard elliptic estimates D(Ho)

9.Vfl"(R').

'l'tso

clearly I/o is closable in r2'"(IL.) for every real s. Thus in view of Theorem 2.4 and. Corollary 7.9, we can take .ffi" in Definition 2.6 as D(Es)-tlz-e/2t the graph-normed domain of the closure Es in ;z'-t1z-c/2, It follows from our assumption on

I/

and the Rellich theorem that

I/

satisfies the assumptions of

Definition 2.6. Moreover, our assumption on I/ implies that so

I/ : Ifio'

72'rile,

that (3.a) follows from (z.ro). Thus our theorem follows from Theorem

3.7, and Proposition

3.5,

3.8. I

REMARK 7.11. Observe that our decay assumption (7.20) is certainly not the noptimal" one for the operator (7'2). In fact, it was shown in [a] that I/ satisfies a limiting absorption principle with a weight function that depends only

f1 and also "distinguishes" between the positive and negative sides of the o1-axis, in complete analogy with corollary 4.t2. Even within the context of the present section, it is possible to do better by taking a closer look at the

on

range of ,Ro+(l) (compa.re Lemma 7.2) and using interpolation techniques as in the proof of Theorem 7.4 and in various proofs in previous sections. At the same

time, the simplicity of the proof of Theorem ?.10 anil the obvious possibilities of generalizing this argument justify, in our opinion, the more restrictive hypothesis

(7.20). Compare also with the discussion following Lemma 3.3'

REFERENCES

[1] Agmon, S., Spectral properties of Schr6ilinger operators and scattering theory, Ann. Scuola Norm. Sup. Pisa (4) 2 (192b), 1b1-218. [2] Agmon, S. and H6rmander, L., Asymptotic properties of solutions of differential equations with simple characteristics, J. Analyse Math. 30 (1926), 1-3E.

[3] Ben-Artzi, M., An application of asymptotic techniques to certain problems of spectral and scattering theory of Stark-like Hamiltonians, Tbans. Amer. Math. soc. 278 (1983),817-839.

[4] Ben-Artzi, M., Unitary equivalences and scattering theory for Stark-like Hamiltonians, J. Math. Phys. 25 (1984), 951-964. [5] Ben-Artzi, M. and Devinatz, A., Resolvent estimates for a sum of tensor products with applications to the spectral theory of differential operarors, J. Analyse Math. 43 (1984), 215-250.

[6] Herbst, I.W., Unitary equivalence of Stark Hamiltonians, Math. Z.

155

(1977), 55-70.

nThe analysis of linear partial differential operators II", [7] H6rmancler, L.,

Ch. 14, Springer-Verlag,

1983.

[8] Iorio, R.J. and Marchesin, D., On the Schr6ilinger equation with timedependent electric fields, Proc. Royal Soc. Edinburgh, 96A (19g4), 117134.

[O] Kato, T., and Kuroda, S.T., The abstract theory of scattering, Rocky Moun-

tain J. Math.

I

(1971), t27-771.

[10] Kuroda, S.T., Scattering for difierential operators I, J. Math. Soc. Japan 25 (1973), 75-103.

[11] Kuroda, S.T., Scattering for differential operators II, J. Math. Soc. Japan 25 (1973), 222-234. oNon-homogeneous boundary value problems [12] Lions, J.L. and Magenes, E., and apptcations I,o Springer-Verlag, 1972,

sspectra of partial differential operators,, North-Holland, [13] Schechter, M., 1971.

69

*

M.

70

BEN-ARTZI AND

A.

:

DEVINATZ

4 [14] Schechter, M., A new criterion for scattering theory, Duke (Le77),863-872.

Math. J.,

44

[15] Schechter, M., Cornpleteness of wave operators in two Hilbert spaces, Ann. Inst. Henri Poincar6 XXX (1979), 109-127. Stone, M.H., ulinear transformations in Hilbert space and their applications

[t6l

to analysis," Amer. Math.

Soc., New York, 1932.

[1?] Yajima, K., Spectral and scattering theory for Schr5dinger operators with Stark-effect, J. Fac. Sci. Univ. Tokyo, Sec. 1A, 26 (1979), 377-389.

Matania Ben Artzi Department of Mathematics University of California

Allen Devinatz Department of Mathematics Northwestern University

Berkeley, CA 94720

Evanston, IL 60201

Permanent Address: Depa.rtment of Mathematics Technion-Israel Institute of Technology Haifa 32000, Israel

l.,i:

i

ttr