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)

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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

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^
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">-.. J ) :

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

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

forc) ' max(0, -))

so

,\
-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 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 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

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


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

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 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

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€,


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


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?).

,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


,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
: '::_

:.:

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
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


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\ -

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
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
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 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'

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[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