Lectures on Exponential Decay of Solutions of Second-Order Elliptic Equations: Bounds on Eigenfunctions of N-Body Schrodinger Operations. (MN-29) [Course Book ed.] 9781400853076

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LECTURES ON EXPONENTIAL DECAY OF SOLUTIONS OF SECOND-ORDER ELLIPTIC EQUATIONS: BOUNDS ON EIGENFUNCTIONS OF N-BODY

..

SCHRODINGER OPERATORS

by Shmuel Agmon

Mathematical Notes 29

Princeton University Press and University of Tokyo Press 1982

Copyr1ght

€D

1982 by Pr1nceton Un1vers1ty Press All R1ghts Reserved

Pub11shed 1n Japan exclus1vely by Un1vers1ty of Tokyo Press; 1n other parts of the world by Pr1nceton Un1vers1ty Press

Pr1nted 1n the Un1ted States of Amer1ca by Pr1nceton Un1vers1ty Press, 41 W1ll1am Street, Pr1nceton, New Jersey 08540

The Pr1nceton Mathemat1cal Notes are ed1ted by W11l1a~ Browder, Robert Langlands, John M1lnor, and E11as M. Ste1n

L1brary of Congress Catalog1ng 1n Pub11catlon Data w11l be found on the last pr1nted page of th1s book

Table of Contents Introduction

3 8

Chapter 0

Preliminaries

Chapter 1

The Main Theorem

11

Chapter 2

Geometric Spectral Analysis

32

Chapter 3

Self- Adjointness

41

L2 Exponential Decay. Applications to eigenfunctions of N-body Schrodinger operators

52

Chapter 5

Pointwise Exponential Bounds

83

Appendix 1

Approximation of Metrics and Completeness

99

Appendix 2

Proof of Lemma 1.2

102

Appendix 3

Proof of Lemma 2.2

104

Appendix 4

Proof of Lemma 5.7

110

Chapter 4

Bibliographical Comments

112

References

115

-8-

Introduction This volume presents an edited and slightly revised version of notes of lectures given at the University of Virginia in the fall of 1980. The subject of these lectures is the phenomenon of exponential decay of solutions of second order elliptic equations in unbounded domains. By way of introduction we discuss briefly the special problem of exponential decay of eigenfunctions of Schrodinger operators. a problem which motivated the present investigations. Consider the Schrodinger differential operator P = - A + V{x) on tin where V is a real function in Ll~c(tln). Assume also that V_ E Ll'oc{tl n ) for some p

> n/2 and that inff(PfI.fI): fI

E

CO' (tin)

•

II fill L"(tI") == 1) > -

co

where (PfI.rp)

== J(lVrpI2 + VlrpI2)dx. tift

Under these conditions P admits a unique self-adjoint realization in L2{tI") which we denote by H (see Chapter 3). The essential spectrum of H ( which is bounded from below) is denoted by u 8ss {H). We set E E

=+ co

= inf u 8ss {H).

if CTess(H) is empty.

There is a general decay phenomenon of eigenfunctions of H. Namely. for a general class of potentials V any eigenfunction of H with eigenvalue in the discrete spectrum decays exponentially.

This phenomenon was studied

extensively in the literature for the special class of eigenfunctions with eigenvalues situated below the bottom of the essential spectrum E. Thus it follows from the works of O'Conner [31J Combes and Thomas [7J and Simon [37J. under some restrictions on V. that if ,,(x) is an eigenfunction of H with eigenvalue p.

< E. then (1)

-4-

< (E-JL)U2 and

where a IS any number such that a type of estImate for precise If V(x)

-+

t,

CD. IS some constant ThIS

WhICh may be referred to as the 'tSotropic estimate, IS

0 as Ix

lImIt In all dIrectIons as

I -+ Ix I -+

However, If V(x) does not tend to the same

DO

DO

one would expect that the IsotropIc estimate

could be replaced by a more preCIse non-IsotropIC estimate of the form

11/I(x) I ~ for any l:

> 0 where

C"e-(l-r:)P(:Z:)

on

lin

p(x) IS some functIOn whICh tends to Infimty as

(2)

I x I -+

DO

and WhICh reflects the dIfferent behaVIOr of the potential V In dIfferent dIrections The study of estunates such as (2) for solutIOns of second order ellIptic equatIons In unbounded domaIns WIll be our maIn task In these lectures An Important techmcal POInt In such a study IS a good choIce of the functIon p In (2) whIch should desCrIbe as closely as pOSSIble the decaYIng pattern of a whole class of solutIOns of the equatIOn Pu mfimty

= JLu

In some neIghborhood of

What makes eIgenfunctions With eIgenvalues below the bottom of the essentIal spectrum decay exponentIally? The answer to thIS questIOn IS to be found In the observation that exponential decay propertIes of eIgenfunctIons and other SolutIons of the equatIon Pu

= P.U

are closely related to POSItiVIty

propertIes of the quadratIC form «P-JL)rp,rp) on certaIn subsets of test functions

To see thIS connectIon we Invoke a formula for the bottom of the

essentIal spectrum of H gIven by Persson [32] (see also Chapter 3 ) by whICh (3)

where the supremum IS taken over the famIly of compact subsets K C lin It follows from (3) that If JL

< E and l:

then there eXIsts a number R = RE

IS any posItIve number such that l:

< 1: -

JL

> 0 such that

«P-JL)rp,rp) ~ (E-JL-l:)

J

Irpl2dx

1:z:I>a

for all rp E Co(Oa) where Oa = (x E lin

Ix I > RI

(4)

Thus we see In partIcular

< 1: the quadratic form

«P-JL)rp,rp) IS strongly POSItIve over all

rp E CoCOa) for R SUffICiently large

We now claIm that the quadratic form

that for p.

lower bound (4) Impbes the exponential decay upper bound (l) for eigenfunc-

tions and other solutions of the equation Pu

=""u.

This follows from results

to be established in these lectures. The principle

which states that positivity of the quadratic form

((P-",,)rp,rp) on C;(OR) implies decay of solutions of Pu =

""U

can be extended

to yield non-isotropic exponential decay estimates. Thus suppose that (4) is replaced by a more general inequality of the form «P-",,)rp,rp) ~

J

A(x) Irp 12 dx

(5)

12:I>R

for all rp

E

CO'(OR) where A(x) is some positive continuous function on lin.

Suppose (for simplicity) that 0 c1

' C2

< c 1 :!:; A(x) :!:; c2( 1+ Ix I)N for some constants

and N. Let p(x) denote the geodesic distance from x to the origin in

the Riemannian metric ds 2

=A(x)(dxf

+ ... + dx~).

(6)

It will be shown in these lectures that if ,,(x) is a solution of which does not increase too fast in the sense that

JI,,1 2 e-2 (l--4)p(2:)dx
0, then in fact" decays exponentially in the following L2 sense:

!1,,1 2 e 2 (l-r:)p(2:)dx
O. Under a mild additional restriction on

V_we shall moreover

deduce from (7) the non-isotropic pointwise estimate (2) for x in OR' The last result is applicable Schrodinger operators.

with some modifications to

N-body

It yields non-isotropic exponential decay upper

bounds for eigenfunctions of various multiparticle quantum systems. The bounds are of the form (2) with p(x) defined as the geodesic distance in a Riemannian metric similar to (6) with a certain multiplier A{x) defined by the system ( we note that" depends only on x/ Ix I and that it is in general a discontinuous function of direction). The non-isotropic exponential decay upper bounds thus obtained are precise and cannot be improved in an essential way in case the eigenfunction" is the ground state (i.e. when II- is the

-8-

lowest eIgenvalue) of an N-body Schrodmger operator

ThIS follows from a

recent work by Carmona and SImon [6J on lower bounds for such eIgenfunctions These lectures are onented toward apphcatIOns to Schrodmger dIffer entlal equatlons on Rn The method used can however be adapted to Yield exponentIal decay results m other SItuatIOns WhIch are of mterest Thus one can obtam results on exponentIal decay of solutIOns of ell1ptlc equatIOns m an unbounded domam 0 With a non-compact boundary aD assummg that the SolutIons vamsh on aD outSIde some compact

Generahzmg m another dIrec-

tion one can study by the same method the rate of decay of solutIOns of elhptIC equatIons on non-compact RIemannian mamfolds The plan of these notes IS as follows In Chapter 0 we mtroduce variOUS functIOn spaces and recall some related techmcal results In Chapter 1 we present the basic weIghted L2 esllmates for solutIOns of second order elhpllc equatIOns Pu f III a domam 0 m Rn These esllmates are proved m

=

Theorem 1 5 under a A-pOSItiVIty assumptIon on the form Re(P~.~) and a growth restrictIon on 11. The theorem IS preceded by a dISCUSSIOn of some elementary properties of Rlemanman (and Fmsler) metnc functIOns whlch appear as weights m the mam estimates Chapter 2 prepares the ground for applIcatIOns of Theorem 1 5 The dISCUSSIOn m thIS chapter IS mollvated by the followmg questlOlil related to the "A-CondltIOn" GlVen an elhptic operator P on Rn find a functIOn A(x) whIch lS as large as pOSSIble such that for some neighborhood of mfimty OR (P~.~) ~ JA(x)I~12dx OR

for all rp e: CO(D R ) Lemma 28 gives an answer to thIS questIOn under the addillonal reqUirement that A IS as a homogeneous functIOn of degree 0 Chapter 3 deals WIth the self-adJomtness problem for ell1pllc operators on Rn The basIc results on the eXIstence of self-adJomt realIzatIOns of such operators and their spectral propertIes are given m Theorem 3 2 Chapter 4 contams the mam examples of L2 exponentIal decay results of these lectures TYPIcal examples are the non-IsotropIC exponentIal decay estImates for

-7-

for eigenfunctions of atomic type Schrodinger operators given in Theorem 4.12 and the non-isotropic exponential decay estimates for eigenfunctions of N-body Schrodinger operators given in Theorem 4.13. Chapter 5 deals with pointwise estimates. The aim in this chapter is to show that the various L2 exponential decay results established in Chapter 4 imply similar pointwise estimates (under somewhat stronger assumptions). Examples of such pointwise estimates for eigenfunctions of multiparticle Schrodinger operators are given in Theorem 5.2 and Theorem 5.3. The key result in Chapter 5 is Theorem 5.1 which gives pointwise bounds for weak solutions of second order elliptic equations with measurable coefficients and possibly complex valued lower order terms. The estimates are extensions of similar estimates established by Stampacchia which in turn generalize the well known pointwise bounds for solutions of second order elliptic equations in divergence form given by De Giorgi and Nash. The notes conclude with four appendices and with a section of bibliographical comments. We remark that some of the results on exponential decay of eigenfunctions of multiparticle Schrodinger operators as well as other results given in these lectures were announced by us in [1] and in [2]. The present volume gives complete details of these results together with some additional material. I am greatly indebted to Emanuela Caliceti, Richard Froese, Ira Herbst and James Howland for their help in preparing these lectures for publication. Not only did they take the notes of these lectures, in itself no easy task, but they have also improved the original presentation by supplementing the notes in several places with some of the missing details. It is with a great pleasure that I acknowledge this help and express my thanks for the work done. Last and not least special thanks are due to Danit Sharon for her very skillful computer editing on a DEC VAX 111780 under the UNIX (a trademark of Bell Laboratories) "troff" package. Shmuel Agmon

-8-

Chapter 0 Preliminaries In this chapter we introduce various function spaces which playa role in the sequel and state some of their basic properties. Let 0 be an open connected subset of lin. H(x;r) will denote an open ball in Iln centered at

x = (x 1

xn) and with radius r. Let

' ... ,

{j

i}i

denote .,.-. vXi

If F(O) is a space of functions on 0, then Fo(O) denotes members of F(O)

with compact support.

If for every open subset U of 0 we are given a space F( U) of functions on U, then

Ftoc (0)

denotes functions f such that for every x

neighborhood U of x such that f tU

E

0 there exists a

E

F( U).

C" (0) denotes the space of k times continuously differentiable functions.

The members of CoCO) (space of infinitely differentiable functions with compact support) are sometimes called test functions. LJ'(O) denotes the space of (equivalence classes of) functions which are measurable and have integrable p-th power with respect to the Lebesgue measure on O. L-(O) denotes the space of essentially bounded functions.

Hl(O) denotes the Sobolev space of functions U E L2(0) such that the distributional derivatives Biu are in L2(0) for i = 1 , ... , n. For 0 s 6 lows. For n

n = 2 and 6 that

~

< 1 we introduce the space of functions

Mcf(O) defined as fol-

3 (recall that n is the dimension of the underlying space) or for

"#- 0, the space Mcf(O) consists of the functions

u

£

LJc(O) such

lim

J

lu(x)1 Ix-xoI 2-n-4d:z: = 0

r -+0 B(:z:o;r) nO

uniformly in ZOEIl". In the case n =2 ,6=0, replace Iz-zoI2-n-4 with Ilog Ix -x0 II in the defining expression, and when n 1 replace Ix _x o 12-n-4

=

with 1.

M,(O) is an example of a Stummel space of functions [41). For most of the results given in these lectures we need to consider only the spaces Mo(O) and MO,lac(O) (the space M,(O) with 6> 0 will be needed however in the last chapter). In the following we shall simplify notation and write M{O) for the space Mo(O) and MIDc (0) for the space MO,tac (0). Note that by Holder's inequality MLoc (0) ::> ~c(O) for p and for n

> n/ 2 and n

O!!

2,

= 1 M1oc(0) =IJc(O).

Finally a real valued function on 0 is Lipschitz if for every exists a neighborhood U of zO and a number C

lu(x)-u{y) I ~ Clx-y

I

xO

E 0 there

> 0 such that

for all

x,y

E U.

We will have occasion to use the following two results: 0.1 Rademacher's Theorem: If

differential of

'II.

'II.

is a Lipschitz function then the

exists almost everywhere.

This theorem is proved in [27; Theorem 3.1.6, p. 65]. In the following A·'P is defined for VI E Co(Il") (or also for f1 E S(Il"» via the Fourier transform map:J: L2{1l")

-+

L 2(1l"), with:Ju = u, by

={1+ltI2)"/2~W on Il". (Thus in particular A = (l-fl)* where fl = f; iJ;,2 is the Laplacian.) :J{A·VI)(~)

t=l

Also, in the follOwing lemmas

II . II

denotes the norm in L2{1l") and func-

tions defined in 0 C Il" are assumed to have been extended as zero in Il"\. O.

-10-

0.2 Lemma: Let 9 E M4 .toc (0). Then for every I: > 0 and every compact subset K of 0 there exists a constant C(I:,K) such that for" = 1-(0/2) (0.1)

for all rp E CO'{O) with supprp c K. When 9 chosen independently of K.

E

M4 (0) the constant C(,;,K) can be

The proof of this Lemma can be found in [36, Theorem 7 3, p. 138]. When

o = 0 we obtain the following corollary 0.3 Lemma: Suppose 9 E Mloc (0) Then for any I: > 0 and every compact subset K of 0 there exist constants C1 (I:,K) and C2 (1:.K) such that

IIlg I*rpll

for all rp

E

S

I:llVrpl1 +

cO' (0) with supp If c K

Here

C 1 (1:,K)

II rp I!.

(0.2)

II Vu II denotes (/

t. ID~ u

0

JlR

B(x;l:I)

en for

such that {(x) = 0 if

1 and f~(x)dx = l. IlA

Define for 0

< l: < l:t {E =

£-n

~(=-).

( 1.11)

£

Then the following smoothed version of It is defined for x

As

£ --.

0 , hE'" It uniformly on compact subsets of

Now we must estimate K(x,VIt£(x)) for x

K(x ,Vlt (x)) :s; 1.

E

E:

n ':

n '. r

under the assumption

-17-

J

_ K.(x,1]) -

:s::

1 IIB(x)17ID '1.,on = IIB(x)-l~1I = M = [A(x )L~3 (x n'~J ]M, '03

where the fourth equalIty IS a consequence of Schwarz's inequalIty Therefore PA(x,y) =PK.(x,y) Smce K(x ,Vh (x»2 = A(x)-IIVAh(x) 12, (I) and (11) are direct consequences of Lemma 1 3

-19-

We pass to the main theorem of this chapter.

Let 0 be a connected open set in lin and let A(x,B) =- LBjati{x)Bt be the elliptic operator on 0 introduced in (1.1). Let 1.5 Theorem: t,;

q (x) be a complex valued function on 0 such that

(ii) q_

E.

M1oc(O) where q_{x) = max(O,-Req{x».

Suppose that there exists a positive continuous function A{x) on 0 such that

Ref(IVA~12 + q(x)I~12)dx ~ fA(x)I~12dx

o

for every rp

E.

0

(1.12)

CoCO). Let PA(x,y) be the geodesic distance in 0 between the

points x, y in the Riemannian metric

[~j]

= [a ij ]-l.

F'i.x a point yO

E.

0 and set PA(X)

= PA(X,yO).

Let h be a real

valued Lipschitz function on 0 such that

Suppose now that u differential equation

is a junction in H~c(O)

which satisfies the

Au+qu=f in the sense that qu

E. LI~c(O),

flu(x)

f

E. LI~c(O),

and (1.5) holds. Suppose also that

12 A(x)e-2 (l-.s)PA(z)dx

< co

(1.13)

o for some

~

> O.

Then the follOwing inequalities hold: (a) If 0 is complete in the metric PA ' then

flu(x) 12(A{x )-IVAh (x)

o

12

)e 2h (Z)dx

(1.14)

-20-

~ II! (x) 12 (A(X)-IV Ah(x) 12)-le 2h (Z)dx

o

'I.! 0 'tS not necessartly complete, let d > 0 and define > dj where p).(x,fool) 'tS defined by (19) 1hen

(b) in general, Od

= fx

E:

0 p).(x,fool)

Ilu (x) 12(A(X )-1 VAh(x) 12 )e 2h (z)dx Dd

~ II! (x) 12(A(x)-IVAh(x) 12 )-le 2h (z)dx

(1 15)

D

lu(x) 12A(x)e 2h (Z)dx

+ 2(1+;d) I d

D\Dd

Remark Suppose that fln\. 0 IS a compact set and that p). has the follOWlng properties

(1) For a fixed yO

E:

0

p).(x,yO)

-+

00

as

Ix 1-+

00,

x

E:

0

(n) p).(x ,y) admIts a contmuous extenslOn from 0 x 0 to () x () In thIS situatlOn It IS easy to see that p).(x ,fool) comcides Wlth the dIstance from x to the boundary of 0, so that m thIS case Od = fx

E:

0, p).(x,BO)

> dj

Smce the proof of thIS theorem IS long and the trend of the argument IS somewhat obscured by the details we first gIve a short discusslOn of the Ideas Involved

We

v

E:

begm

Cl(O) , '1/1

E:

by

Cl(O),

derIvmg

an

IdentIty

to

be

used

below

Let

'1/1 real Then a calculatlOn shows

VAv VA (,,21J) = IVA ("v)1 2 - Iv 12 1vA ,,1 2

-

['I/IvV A" VA'Il - ,,'IlVAV VA'I/I]

Smce the last term IS pure Imagmary,

(1 16) Now suppose u IS a solutlOn to the dlfferenlIal equatlOn Then we have

-21-

for every rp

E:

CoCO). Suppose we could replace rp with u'I/I 2 where '1/1 has com-

pact support. Then we would obtain, after taking the real part and using identity (1.16)

JUVA('I/Iu) 12 - lu 12 IVA '1/1 12 + Req lu 12'1/12]dx

(1.16)'

n

=Re Jn JU '1/12dx. Invoking the A-condition (1.12) with rp

='I/Iu we would get

j[Alu'l/l12 - lu 12 IVA '1/1 12]dx ~ ReJJu'I/I2dx. a a In fact as we shall later show this inequality is valid if '1/1 is a real Lipschitz function of compact support. It appears as (1.20) below. Suppose we ignore the requirement that '1/1 have compact support and use

'1/1 = ell,

in the above inequality. Then proceeding formally we obtain

Jlu 12 (A-IVA h 12)e 2h dx ~ Reflue 2h dx.

n

n

An application of Schwarz's inequality to the right side

ReJJfIe 2h dx = Reflu(A-IVAh 12)*(A-IVAh 12)~e2hdx

n 2 ~ !Jlu 1 (A-IVAh 12)e 2h dxl*!JII 12(A-IVAh 12)-le ilh dxl* D n D

produces, after dividing both sides of the resulting !flu 12(A-/VAh 12)e 2h l* and squaring both sides, n Jlu 12(A-IVAh 12)e 2h dx n ~

inequality

by

fll /2(A-IVAh /2)-le 2h dx n

which is precisely the result wanted when (0, PA) is complete. Note that we did not use the growth assumption on u with the illegal

= e h . If we now try to approximate ell, with legal choices for t, for example '1/1 = ehx with X a smoothed characteristic function of a compact set

choice

t

-22-

(whlCh wIll eventually become large). then quantItIes mvolvmg the derIvatIves of X must be estImated m the remote regIons of 0 where the growth of

U

wIll

be Important TIus IS preCIsely what happens m the proof of the theorem. to whIch we now turn

U E:

Proof By hypothesIs Ht!c(O) • qu E: 4~c(0) and

U

IS a

solutlOn of A (x .B)u + qu == J.

1

e .

(1 17) for rp E: CO' (0)

rp

E:

Hl(O)

n

By a

LO'(O)

densIty argument thIS equatlOn IS stIll true If

For suppose rp E: Hl(O)

of Lemma 13 (see (1 11)) then for 0

n

< e:S: £1

CO' (0) wIth support m a fixed compact set

LO'(O)

1f (" IS as m the proof

rp,,(x) == J(,,(x-y)rp(y)dy IS m n In additlOn rp" .... rp m Hl(O) as

0 whIle II\P"II L-:s: IIrpIlL- Usmg rpl: m (117) we see that the first and last terms converge as £.j. 0 whIle gomg to a sequence !£~ I . £• .j. O. for whIch

£ ->

rp",

-+

II' a e , the second term converges by Lebesgue's donunated convergence

theorem Let u" == u/ (1 +£ lu 12)

The distrIbutlOnal derIVatIves of u., can be calcu-

lated as If u were dIfferentIable m the ordmary sense

ThIS IS easIly seen by

first approXImatmg u by a smooth functIOn and gomg back to the defimtlOn of dlstnbutIOnal derIvatIve Hi~(O) as

£ .j.

A short calculatIOn also shows that u"

->

u m

0

If 1/1 IS a real LIpschItz functlOn of compact support an additlOnal approxI-

matIon argument can be used to show that

B,(1l e1/l2)

= (21/1B.t)1l" + 1/I2iJ.1l e

u z 1/l2 E: Hl(O) n

La (0)

and that

We thus set rp = 1l,,1/12 and substItute mto

(1 17) The result IS

fvAu VA(1l,,1/I2)dx + JqU1l,,1/I 2dx == JJ1le1/l 2dx D o n Next we replace u WIth u e + (u -u,,) and take the real part tIon assumes the form

Then the equa-

-23-

where

=Re JV A(U z -1.L )·VA(flr:1/I2)dx o

(Here Re q

+ J(q _ -q +)(u -1.Lr:)fle1/l 2dx. 0

= q + -q _ with q + = max(O, Re q ).) We now estimate I".

We first note

that (q_-q+)(u-1.L z )a z s; q_(u-1.L")a,, and then use Schwarz's inequality to obtain

I"s;

I

J

su.pp1

+! J

IVA(u-1.L z)1 2dx!*!

q_1/I2 lu-1.L.Ydx !*! J

supp1

Since

UFo

~

U

J

in HJ,c as

IVA(1/I2u,,)12dxl*

supp1

1/I2q_lu,,1 2dxl*.

supp1 l: -+

first term tends to zero as

0 and supp 1/1 is compact the first factor of the l: -+

O. The second factor stays bounded for the

same reasons. To estimate the factors of the second term we recall that q_

E

M,oc(O) and invoke lemma 0.3. This gives for any fixed d > 0

J supp 1

q_1/I2 Iu-1.L z I2 dx

S;

d J :EliJt (1/I(u-1.L,,)1 2dx supp 1 t

+ C(d,supp1/l) J 1/12 lu-1.L c 12dx , suppt

which implies that the first factor of the second term tends to zero as

l: -+

0

while a similar estimate shows that the second factor remains bounded. Therefore lim sup Ie S; O. z-+O

Now the identity (1.16) is used to rewrite equation (1.18) as follows:

JUVA(1/Iu,,) 12 + Re q IU z l21/12 - IU c 121VA112]dx D

=

ReJD JfI z1 2dx + Ie·

The condition (1.12) on heX) implies, by a density argument

(1.19)

/[ 1VA (¥'U.,) 12 + Re q lu" 12¥,2]dx ~ /J..I¥,u e I2dx.

o

0

This inequality is applied to equation (1.19): /[J..I¥'u,y - lu e I2 IVA¥'12]dx ~ RefJu,,¥,2dx + I.,.

o

Let

l; ....

0

0 to obtain the important inequality /[J..lu¥' 12 - lu 12IVA¥'12]dx ~ RefJu¥'2dx.

o

0

(1.20)

We now use our freedom in choosing ¥,. Recall from the remark preceding the proof we want "

to

approximate

ehC"l.

With this in mind let

,,(x) = egCzlx(x) where 9 and X are real Lipschitz functions, X has compact

support and 0 :s: X ~ 1. Suppose in addition that 1VAg 12

< J.. a.e. Then

IVA,,1 2 = e2g~IVAg 12 + 2e2gX VAX·VAg + e 2g IVAXI2. SO (1.20) implies fluxI2(A-IVAg 12)e 2g ~ Re/Jux 2e 2g dx + flu 12e 2g ( IVAXI2+2xIVAX·VAg \)dx.

o

0

0

The first term on the right is estimated as follows: RefJfI~e2gdx

o

=Re/JfI~e2g(J..-IVAg 12)*(J..-IVAg 12)-lI!dx 0

~ ffluxI2(A-IVAg 12)e2gdxJ*ffIJxI2(J..-lvAg 12)-le 2g dxJ*.

o

0

Therefore fluxl2(A-IVAg 12)e 2g dx ~

o

f/luxI2(J..-IVAg 12)e 2g dx J*fflJ xI 2(J..-IV Ag 12)]-le 2g dx 1*

o

0

+ flu 12e 2g (lVAxl2 + 2xI VAX·VAg I)dx. o

This inequality is of the form

where a band c are positive. We claim this implies

a

~

b

+ 2c.

-25-

Certainly If c

~

a thIS IS true On the other hand If a > c then a -c ,s; a*b* lInpiles a 2 -2ac +c 2 ,s; ab

so

c2 a + -,s; b + 2c a

whIch lInplles the result Therefore !luxI 2 V.. -lvA g

12)e

n

2g

r 1e 2g dx

dx,s; !lfxI 2(A-IVAg 12

n

+ 2!lu 12e 2g (lVAxI2 + 2xIVA XVA g n

I)dx

(1 21)

(Note that we do not exclude the tnV1al case when the first Integral on the nght sIde of (1 21) IS Infimte ) We now let funcllOns of the form

eg(:z:)x(x)

approxImate

eh.(:z:)

ThIS wIll be

done by constructing sequences XdJ and It,. whlCh sallsfy the hypotheses for X and g respecllvely, inserting these functIons In (1 21) and taking IlIDlts At

the same bme we WIll try to control the last term In (1 21)

Let

fKiI be a sequence of compact sets In 0 such that K,. c K; If 1, < J and

U (mtenor K,.)

=0

GIven d

> 0,

71d(t)

={ 1

define the funcbon 71d(t) on [O,co) as fol-

~=1

lows tid

If t E[O,d] tf t E (d,co),

and let

K, Xd." = 0 and mSlde K, Xd 1 = 1 except for pOints whIch d In the metnc PA to the extenor of K, On these pOints = p". 01

IJc(lt~». If u

E:

(as

usual

we

assume

that

u

E:

Hj~c(IR:)

and

L2(1l~). then

(1.31) for every I:

> O.

To prove this fix xl> N. Set 0

= (x

I:

> 0 and choose N > I: such that IRe V(x) I < 1:/2 for

: Xl> NI. For rp

E:

CoCO) we have:

JUVrpl2 + Re(V+x 1)lrpI2)dx ~ J(xl-t)lrpI2dx.

a

Thus in A(x)

n the

0

operator -b.+ V +x 1 satisfies the conditions of Theorem 1.5 with

== x1-1:/2 > O. Let PA(x.y) be the distance in 0 between the points

x = (Xl • ...• xn) and y = (Yl •...• Yn) in the Riemannian metric: d,s2 = (Xl-t)( dx f + ... + d.x~). Fix Y and set PA(x) = PA(x,y), Since It,

PA(x)

~ IJ(t-t)*cit I. f/,

it is clear that A(x)e-2(1-6)~(s) is a bounded function for any 0

E:

(0.1). Thus

it follows that u verifies the growth condition (1.13) in Theorem 1.5.

We can choose hex) =

~(Xl-I:)3/2 and note that

IVh{x) 12

= 1~Bl{Xl-Z;)3/212 =A{x) 3

so that h verifies the condition: 1Vh (x) 12

< A{x)

E.. 2

(1.32)

a.e. which was required in

Theorem 1.5. Applying the theorem. using (1.15) with d.

= 1. we obtain

-31-

Jlu(x) 12(A(X) - IVh(x) 12 )e 2h (.z)dx s 6 J 0,

lu(x) 12 A(x)e 2h (.z)dx (1.33)

D'O,

= fx : x

> 11. Now it is easily seen that 0 1 = (x : xl> Nd for some N1 > N which implies that the function

where 0 1

E:

0 , PA(X ,ao)

A(X) exp(2h(x)), which depends only on xl' is bounded in 0 \ 0 1 . Thus, using

(1.32) it follows from (1.33) that

which implies (1.31).

-32-

Chapter 2 Geometric Spectral Analysis Our goal is to apply Theorem 1.5 to measure the decay of solutions to

Au +qu =0. If q (x) = q 1 (X) - IJ. we thereby obtain estimates for eigenfunctions of A +q 1 with eigenvalue IJ.. To apply Theorem 1.5 the real part of the quadratic form associated with the operator A +q need be positive on CO' (0). Moreover, we must find a positive continuous function A(X) such that the quadratic form in question is strongly positive on CO'(O) in the sense that the ACondition (1.7) holds. (Since we are dealing with behavior of solutions "at infinity" the inequality (1.7) need to hold only for test functions supported in O,K for some compact set K.) In this chapter we introduce various quantities related to the spectrum of operators obtained by restricting A +q to various subsets of O. As we shall see these quantities will allow us under certain conditions to produce the desired A functions. To simplify the discussion we assume from now on, unless otherwise specified. that A (x ,B) is the elliptic operator (1.1) defined on 0 = Il". We shall assume that the coefficients aij (x) of A are continuous bounded functions on II". Again let q E 4.~c (Il") and q _ E MLoc (IR") and assume that q is real valued.

With these assumptions set P(x.B) = A(x,B)

+ q(x)

(2.1)

and define

(Pu,u)

=J( IVAu 12 + q lu 12)d:J:

(2.2)

Rn

for all u

E 14~ (Il")

for which the integral (2.2) makes sense. Our main aim in

this chapter is to find continuous functions A{X) on Il". not necessarily positive but "as large as possible", such that the following inequality holds

(Prp.rp) ~ JAlrp12d:J: Iln

From now on unless otherwise specified. the norm in L2(1l").

(2.3)

11·11

denotes the norm

-33-

2.1 Definition: Let P be the operator introduced above. For any y E lin

and R

> 0 define AR(y;P) = infl

(,f:irJ:

II' E Co (B(y ;R) . II'

¢

01

(2.4)

where B(y ;R) is the ball of radius R centered at y. Note that AR(y;P) can be identified with the lowest eigenvalue of the self-adjoint realization of P in L2(B(Y ;R» under zero Dirichlet boundary conditions. This property will not be used in the sequel. On the other hand we shall use the following properties of A R . 2.2 Lemma: AR(x ;P) is a continuous function of (x .R) on lin

X

11+.

]i'urthermore AR(x ;P) = AR(x;A +q) is also continuous in [a ij ] in the sense that if Am = -

n....

~ iJja:l..iJ, iJ=l

..

where [a:l..(x)] has all the properties of [a"(x)] for

m = 1.2 ..... and lim Ia ij (x) - a:t. (x) I = 0 uniformly on compact sets. then m~-

AR(x;A". + q)

-+

AR(x;A +q) uniformly for x in compact subsets.

A proof of Lemma 2.2 is given in Appendix 3. We now produce functions A(x) which verify (2.3). 2.3 Lemma: lJnder the above conditions on P for any Re

l:

> 0 there exists

> 0 such that (2.5)

for all II'

E

Co (lin) and for any R;a>:; Re.

Proof: Let {" be real valued function in Co (lin) such that {"(x) =0

t~::: For R

> 0 and y

E

lin let

1.

-34-

("R.y{X) ("R{X)

=("{ x7).

=("R.O{X)

Since the deriVatIves of (" are bounded

In

R"'. we have (26)

for some constant C

>0

depending only on (" and the upper bound for the

Temporarily assume that

a",j

a~3

Then f = Pf/I IS well defined and

E Cl{R"')

In

LJ (R"') If rp E CO' (R"') Clearly

Usmg IdentIty (1 16) thIS leads to

f(lVA «("R.yrp) 12 + q I("R.l/rpI2)dx

Rn

-

JIVA("R y I2IrpI2dx

(27)

Rn

The first term IS recogmzed and estImated as follows

:=:

~ A R/ 2 (y.P) Now suppose Ix -y

I ~ R/2

(2 8)

(P«("R.yrp).("R.yrp)

f

Irp("R.yI2dx

B(y R/2)

Clearly. B(y .R/ 2) c B(x .R/ 2+ 1x -y

I) c

B(x .R)

Therefore (29) The equatIOns (26) through (29) together WIth the fact that ("R(X-Y) IS supported m B(y .R/ 2) Imply

f

Rn

AR{x.P) Irp{x)

12 ("j(x-y)dx

- CR-2

J

B{y.R)

If/I(x) 12 dx

-35-

~ Re 1 J (x)~(x) d(x-y)dx. tI"

Integration with respect to y gives

R'" 1 AR(x;P)/9'(x)1 2dx - CR-2V1R"'- 119'(x)1 2dx II"

tI"

~ R'" Re

1J (x )~(x)dx tI"

where VI is the volume of the ball of radius 1 in IRn. Dividing by R'" and recalling the definition of J we get

(P9'.9') ~ 1 AR(x;P) 19'(x) 12dx - CR-2V1 119'(x) 12 dx tI"

tin

(2.10)

and the lemma for the case a'; E C1 follows by taking R" such that R:

=CV1

&-I.

Since C only depends on the upper bound for the a i ; (and on ('). we can approximate [a(i(x)] in the case when the a i ; are not necessarily Cl uniformly by a sequence of positive definite matrices (a,;{] with smooth entries which are uniformly bounded from above (see Lemma A1.1. Appendix 1 ). Then by Lemma 2.2 AR(x;Am +q)

-+

AR(x ;P) uniformly on compact sets so

that

Also «Am +q)9'.9')

-+

(Prp.9') as m

Since the aM. are uniformly bounded

-+ "".

from above we can choose the constant C such that for every m the inequality (2.10) holds with P

= Am +q.

Thus (2.10) is also valid for P

= A +q

and the

lemma follows. •

The following quantities will play an important role in what follows. 2.4 Definition:

A(P)

= inf{ (,~:irJ

;

rp

E

Co (tin) • rp F oj.

(2.11)

-36-

E{P) =

s'jP

inr{ ~f:irJ :

'II E CO'{R"\K) . 'II

~ oj

(2.12)

where the supremum is ta.ken over the fa.mily of compa.ct subsets K

C fI" .

A{P) and E{P) will be shown to be equal to the bottom of the spectrum

and essential spectrum respectively of the self-adjoint realization of P on

L2{fI") when E{P)

> -....

In general both may take on the value - ....

It is easy to see that A(P) = lim AR{x ;P) for any x R~·

E fI".

Note that since

AR{x;P) is a decreasing function of R this limit exists (although it might be -

0 0 ).

The relationship between E{P) and AR{x ;P) is the subject of the follow-

ing lemma. 2.5 Lemma: E{P) = lim lim inf AR{x ;P). R~ .. 1.. 1".

Proof: Let K be a compact subset of R" and R

Ix I is sufficiently large B{x ;R) inf

{(~:irJ

> o. Then for x such that

C IR"\ K. Therefore for such an x.

:

'II E CO'(R"\.K) . tp

~ oj ~

AR(x;P)

and hence inf

{{,f:i~

:

'II

e: CO'{R"\K) . 'II

~ o} ~ l~~!~f

AR{x ;P).

(2.13)

Since the right hand side of (2.13) does not depend on K and the left hand side does not depend on R it follows from (2.12) that E(P)

:S

lim lim inf AR{x ;P). I.. I"·

R-+·

By Lemma 2.3, given

l:

>0

there exists Re

(2.14)

> 0 such that (2.15)

for all R

~

Re and'll E CO' (fin). Now the definition of lim inf AR{x ;P) implies 1.. 1..•

-37-

that (2 16)

o

for all tp E: C (II"\B(O,Ro)) where Ro IS suffIcIently ldrge and (2 16) lmply

1f:llil ~ hm mf IItpll2 1",1-+-

Therefore, (215)

Ae(x ,P) - 2t:

for tp € CO'(lRn\B(O,Ro)) Therefore

I:(P)

~

hm mf AR(x ,P) - 2t:

(2 17)

'''' 1-+Now the only restnctlOn on R IS that R

I:(P)

~

~

R£ Therefore

hm hm mf AR(x ,P) - 2t: R-+-

(218)

I'" 1-+-

But t: IS arbItrary So (2 18) and (2 14) prove the lemma • We now mtroduce a functlOn K(C".l) = K(C".l,P) defined on the sphere

Ix I

8"'-1 C lin WhICh wIll allow US to estImate AR(x ,P) for large

depending only on xl

by functIOns

Ix I Roughly speakmg K(C".l) approXImates the lower

bound of the quadratIc form (Prp,tp) restncted to test functlOns tp WhICh are supported m the mtersectlOn of a "small" neIghborhood of mflmty and a "thm" open cone In the dlrectlOn of

2.6 Definition: Let 8"'-1 and N

C".l

= {C".l E: lin

1C".l1

= Ij

For

C".l

C 8"'-1 ,0

< t: < 'II"

> 0 define r~,N = (x E II"

I;e,N(C".l) = mf{ K(C".l)

In (2 19)

< , > denotes OX>

(~:ifJ

rp

E:

=K(C".l,P) = chm ......

Co(r:/')

,rp

¢

oJ

hm I:",N(C".l)

(2 19) (220) (221)

N ... oo

the usual mner product m Ill"

cone With angle of opemng t: mcreases as N ....

-38-

2.7 Lemma: K{",) is a lower semicontinuous junction of'" on ,sn-l. The following holds: min!K{",) : '" E ,sn-lj

Proof: Let

!"'i j

clear that

(2.22)

be a sequence ot points in ,sn-l such that ~im "'i ,~-

="'.

< K{",). From the definition of K{",) it follows > L. Since "'i -+ '" it is

Fix a number L such that L that there exist

=E{P).

(0.1T) and N> 0 such that E£·N{",)

l: E

r!{2.N c r:· Nfor all i

~ io for some io. Hence

K("'j) ~ E£/2.N("'i) ~ E"·N{,,,) > L for i

~

io which implies. since L is an arbitrary number

..

liIp. inf K("'j)

~

< K{",). that

K{",).

,~

This proves that K(",) is lower semicontinuous. Next. to prove (2.22). fix Ii E{P)

=K :!!::

> 0 and write

sup' infl(P9I.9I)/ 119111 2 : 91 E

compact

CO' (R'"

inf!(P9I.9I)/ 119111 2 : 91 E CO'(Il"'KO)

•

91"#

OJ

K) . 91 "#

OJ + Ii

(2.23)

for some compact set Ko. If Ro

> 0 is chosen so that Ko

C

B(O;Ro). it follows from (2.23) that

E(P):!!:: inf((P9I.9I)1 119111 2 : 91 E

cO'{r:'A)

.91"#

OJ + Ii

= E"·N(,,,) + 6 for any'" E ,sn-l. 0

< l: < 7f and N > Ro. which implies that

E(P):!!:: lim lim E",N(",) E-+-

for any'"

E

N ....

+ Ii = K(",) + f>

,sn-l. Hence (2.24)

On the other hand using Lemma 2.5 it follows that given Ii > 0 there is an Rl

> 0 and a sequence of points

that

xm E R".

m = 1.2 ..... with

IXm

1 ...

00

so

-39-

E(P)

=Ielim ....

~

for m

(xml

o 0 if m is sufficiently large. it follows from (2.25) that

Ix I)

for any

E

and N and thus l:(P) ~ lim lim I;£,N("'O)-li = K("'o)-Ii ~

. . .o

N .....

~ minIK(",) : '" E

sn-11-li.

which together with (2.24). proves (2.22) .• Finally we prove a lemma which will let us obtain non-constant "A functions for use in Theorem 1.5 where the functions "A(x) in question depend only on the direction of x. This lemma will be in particular useful in applications of Theorem 1.5 to eigenfunctions of multiparticle Schrodinger operators to be considered in Chapter 4.

2.8 Lemma: Let p = A+q satisfy the same conditions as before. Let g(",) be a continuous function on sn- 1 such that g(",) < K(",) K(",;P) for all '" E SS -1. Then there exists C > 0 such that

=

(2.26)

for all 'II

E

CO'(Od where Oc

= Ix : Ix I > CI·

Proof: Since K(",) - g ("') is lower semicontinuous and positive we can

> 0 such that g(",) + 26 < K(",) exists R > 0 such that choose Ii

for all '"

E

sn-1.

By Lemma 2.3 there

for every 'i

(0

CO'(Rn). Therefore it is enough to show A8 (x;P)-6

for \x \

> C. So fix r.lo

(0

sn-1 ,

0

(2.27)

\: \ )

< eo < 71"/2 and No>

0 so that

> g (r.lo) + 26

I;r:•. NO(r.lO) and let Uo be a neighborhood of r.lo in

g (r.l)

~ g(

(2.28)

sn- 1 satisfying

< g (r.lo) + 6

c "t2.No

r.,

rEo,No

c"o

(2.29) for r.l

(0

Uo.

(2.30)

Then (2.28) through (2.30) imply

I;"0/2,No(r.l) ~ I;r:o,No(r.lo) > g(r.lo) + 26 > g(r.l) + 6 for all r.l

(0

Uo. Now the compactness of

sn-1 and a

covering argument allows

us to conclude that

I;e.N(r.l) for every r.l

(0

sn-

1

and some e

> glr.l) + 6

> 0 , N > o. /x / > C

Finally let

C = max!N+R, R/sinel. Then, if

B(x;R) c

and therefore A8 (x;P)

=inf{ ~:irl

riff.., 1

:

'(I (0

~ inf{5J:!Ml,P , 2 : '(I / '(III

(0

CO'(B(x;R» CO'

,'(I

(r:J'I.., I) ,

=I;E,N(~) /xl

~g( 1:1) +6, which proves (2.27) and completes the proof .•

'(I

cF

oJ

cF

oJ

-41-

Chapter 3 SeU-Adjointness In this chapler we consider lhe realization of P = A +q as an operator on L2(Rn). We show that when ~(P) >- P together with a suitable domain define a self-adjoint operator H which is bounded from below. We also find that A(P) and ~(P) are equal to the infimum of the spectrum and essential spectrum of H respectively. The results of this chapter will allow us to apply exponential decay results to eigenfunctions of H. Throughout this chapter we assume that P satisfies the conditions imposed on P = A+q in Chapter 2. That is, A= -I;D,aiia.. where aii(x) are ;'.j

continuous real valued bounded functions on R" such that [ai;(x)] is positive for every x, and q (x) is a real valued function in L~c (R") such that q_ E Mloc(R").

We begin by proving a lermna which will be needed in our study of the self-adjoint operator H. 3.1 Lemma: Let P = A +q be an elliptic operator on R" veriJying the conditions described above. There exists a positive continuous Junction k (x) on R" such that

(3.1) Jor every fI E Co(R"). Proof: We claim that it is enough to prove the lermna assuming the additional hypothesis ai' E C-(R"). For it follows from the approximation result in Appendix 1 that there are real valued functions b i ' E COD(R"), 1 ~ i,j ~ n, such that [b i ; (x)] is a positive definite matrix satisfying (3.2) where for matrices A and B, A ~ B means that A -B is positive semidefinite. Since ai' E L-(R") it follows from (3.2) that bid E LOD(R") for 1 ~ i,j ~ n.

-42-

Now. suppose we have shown that there exists a positive continuous function k 1 (x) on fin such that

for all rp E Gci(fin ) where

-E

B(x.B)=-

iJjbij(x)iJi

•

P 1 =B+2q.

'J=l

Then we have from (3.2)

(3.4)

while again from (3.2)

J -21 k 1(x)(IVA rpI2 + Iq Ilrpl2)dx ~ J k 1{x)(IVB rpI2 + 21q Ilrpl2)dx R"

(3.5)

11."

so that using (3.3) through (3.5). (3.1) follows with k (x) assume in what follows that aij E C-(lIn).

=k 1(x)/ 4.

We thus

Since q _ E Mtoc (Rn) we can apply Lemma 0.3 to conclude that there exists an increasing sequence

fC m I . m == 0.1.2 ... '.

for each rp

+

setting

C

(y)

linear in Iy

Gci(B(O;m

E

= em

I.

if Iy

2».

I = m.

cm;;e 1.

such that

Define a continuous function c(y) of Iy I by

m

= 0,1.2, ...• and letting C (y)

be piecewise

Then c (y) ;;e 1 and

(3.6) for all rp

E

Gci(B{y;1».

Using (3.6) and the definition of q_ we get that for rp

E

Gci{B(y;1»

J (IVA rpl2 + Iq Ilrpl2)dx = (Prp,rp) + 2Jq_lrpl2dx II"

II"

~ (P",,,)+ +

t !U VA,,1 2 + Iq 11,,1 2)dx + 2c (y)!1,,1 2 dx. Rft

Rft

Subtracting t !(IVA,,1 2 + Iq 11,,1 2)dx from both sides and dividing by 2c(y), noting that c (y)

~

1. we obtain

(4c(y»-1 !(IVA,,1 2 Rft

for all "

E:

Let"

CO' (B(y ;

E:

(3.7). Let ("

+ Iq Ilrpl2)dx

s; (2c(y»-1(prp,rp)+

+ IIrpl12

1».

CO' (Rn). We now manufacture a suitable function rp for use in E:

CO'(Rn) be a fixed real valued function such that ("(x) = 0 if 1x 1 ~ 1 , ("(x)

!fGdx

Rft

Let ("1I(x) = ("(x-y).

Then ("111/1

E:

=1

if

1x 1st,

= 1.

CO'(B(y;l». Therefore, applying (3.7) to

" = ("II'"

(3.8) On the other hand application of the identity (1.16) (see (1.16)') gives

Setting P1/I = J and noting that IVA ("II (x) 12 s; D for some constant D not depending on :z: or y, it follows that

(P("~II),1/I("1I) ~ Ref Jj("~dx + D Rft

f

11/I1 2dx.

1",....",1 0 and all

x

E IIln.

Hence, if P1 (x ,y)

r

denotes the geodesic distance between x, y in the Riemannian metric ds we I which implies that Rn is complete in the metric Pl'

have: PI (x ,y) ~ d Ix -'II

Next we observe that without loss of generality we may assume that P verifies the condition (Prp,rp) ~

11'11112

(3.17)

for 'II € CO'(Il").

Indeed, Lemma 2.3 implies that for R larger than some R", (Prp,rp) ~ !(AR(x;P)-e) !rp(x) !2dx.

(3.18)

11ft

Since, by Lemma 2.5 lim R...

lim inf AR(x ;P) = 1:(P) 1%1".

>- ...

and lim inf AR(x ;P) is a non-increasing function of R, we have 1%1"· lim inf AR(X ;P) 1%1"·

~

1:(P)

for every R. Thus since AR(x ;P) is also continuous there exists a constant C such that Ao(x ;P) - e ~ C

for every x

€

R".

Together with (3.18) this implies that (Prp,rp) ~ C 11'11112

for all

'II

€

(3.19)

CO' (Il").

Now we add a constant 7 to P so that (3.19) holds for P+., with C

~

1 and

note that if we can prove the theorem for H7 = H+., then it is also true for H. We thus assume that (3.17) holds, observing that this implies that the ACondition of Theorem 1.5 is satisfied with A == 1. We now apply the theorem to the function u.

€

D(H) which we consider as a solution of the differential

equation Pu = J with J = Hu. We use the theorem with h == 0 (all other hypotheses are easily checked). Since as we have seen before 1\" is complete in the metric PI it follows from (1.14) that

IIu. II

~

II Hu II

for every

u.

€

D(H)

(3.20)

which says that H is injective. Our strategy now is to find a self-adjoint inverse for H. Then the selfadjointness of H will follow.

-47-

Define (3.21)

Then it follows from Lemma 3.1 and (3.17) that there exists a positive continuous function k (x) on Rn such that 2111f'11I2 ~ (Pf',f') + 1If'1I2 ~ Jk(x)(IVA f'1 2 + Iq 11f'12)dx

R"

for every f'

E

Cei (Rn). Adding tho s inequality to (3.17) we obtain

11If'11I2~ for every

~ E

111 f'1I 2 + -3

1... Jk(x)(IVA~12 + 3 IR"

Iq

11~12)dx

(3.22)

Cei(Rn). Let V be the completion of Cei(R n ) with respect to the

Hilbert space norm 111·111. Denote by (·")v the scalar product in the Hilbert space V. Note that in view of (3.22) we have V

In particular V

c lu : u

C

E

L2(Rn)

n H~c(Rn) ,

L2(Rn). Also by (3.17) /lu

+,(u) = (u,fhe Then

+,

Iq I~ E L'~c(Rn)!

(3.23)

~ /llu "' for u E V.

/I

for

u

E

V.

is a bounded linear functional on V since

Therefore by Riesz's representation theorem there exists a unique

v,

E

V

such that +,(u) for every u

J

E

E

L2(Rn) with

V.

=(u,fh.= (u,v,)v

Define T: L2(Rn) ... V

II! /I

C

L2(Rn) by T! = v,.

(3.24)

Then for

= 1

II T! /1 2 ~

111111112

=(v,.v,h = (v,,fh.

= (TJ ,f h. ~ which implies with u(T)

C

II TJ II

~

1 and

II T! " (T! ,f) L. ~ o. Thus

[0,1] and Ran T c V.

T is a self-adjoint operator

-48

Next, recalling (3.23) and the definition of TI it follows from the fact that

\II. liD that

Co (1\") is dense in V (in the norm

(rpJ

h. = (rp,TI)Y = I(VArp·VA TI + qrpTJ)dx II"

for every rp tion u = TI

Co(IR") and for any given! E L2(1\"). This shows that the func-

E

V verifies the equation Pu = I in the usual weak quadratic form sense. It thus follows that TI E D(R) and that RTI = I. Thus we have shown that R is a one to one map from D(R) onto L2(1I") such that R-l = T is E

self-adjoint. This proves that R is self-adjoint. Since a( T)

C

[O,lJ it follows

that a(R) C [l,GO). The above considerations show that D(R) = Ran T c V. Note that, in view of (3.23) this implies that

Iq l*u E Ll~C(Il") if u E D(R). We next observe that

D(R) is dense in the Hilbert space V. Indeed if this were not the case there would exist an element Vo E V, Vo -F 0, such that (vo, TJ)y = 0 for every !

E

L2(1I"). In view of (3.24) this would imply that (voJ

I

E

L2(1I") which contradicts Vo -F O.

h. =0

for every

Consider the operator JIIz. We claim that D(JIIz) = V. Indeed it follows from the spectral theorem that D(R'h) = closure of D(R) in the graph norm

( II u

112 + (Hu ,u»*

(Hu,u)'h

= III~ III

which

in

our

case

is

equivalent

(we are using here the relation (J ,TJ)'h

we have just seen that the closure of D(R) in the norm

to

the

= III TI III).

111·111

norm Since

is V the result

follows.

o

By the above remarks and the fact that C (II") is dense in the Hilbert

space V, we get inf u(R)

=

inf 'UED(Ef)

This proves (3.15).

(Hu,u) = inf lIIu 1112

lIu 112

'UEV

lIu 112

-49-

To prove (3.16) we use Lemma 2.3 again. It follows from the Lemma that given t:

> 0 there exists a

number R

>0

such that

CO' (11.71.). Since lim inf AR(x ;P) ~ l:(P), it follows that

for every fI

E:

for some a

>0

1201"·

AR(x;P) ~ l:(P)-}

for

Ix I ~ a

sufficiently large. Since as a function of x AR(x ;P) is locally

bounded, we have AR(x;P) ~ l:(P) - C

for

Ix I ~ a

for some constant C. Choose a non negative function X E: Cci(lI. n ) such that X(x) ~ C for Ix

I ~ a.

Set P x = P + X. Then it is clear from the above that (Pxf1,fI) ~ (l:(P)-t:)!lflI 2 dx

for every fI

E:

II.n

CO'(lI. n ).

Introduce now the multiplication operator X : U

...

XU which is a bounded

= H + X with D(Hll ) = D(H). Then Hx is self-adjoint and previous considerations applied to self-adjoint operator in L2(lln) and consider the operator Hx.

Hx show that (3.25)

Observe now that the operator X is H-compact. To show this it suffices to demonstrate

Y =

xTl!

E:

XJr1 = XT

that

L2(lI.n) :

II! II

~

Ii

is

a

compact

operator,

has compact closure.

i.e.

Since every

that U

E:

Y

belongs to Ht!c(ll n ) and has support in some fixed ball B containing supp X, it suffices to show that

lIu II + L" II iJiu II

~

c for every

U

E:

Y and some con-

i=1

stant c. For by Rellich's theorem [ 3; Theorem 3.B, p. 30] the closure of

lu e: Ht!c(Il") is compact in L2(lln).

: supp u

C

B,

lIu II + L"

'=1

lIiJiu

II

~ c!

-50-

Suppose u

=xTf

wIth

lIu /I =

IIf II s

/lxTf

1 Then

II s IIxll L-(JI") II Til II! /I s c1

We now compute /lB~ull

s II {B,X)Tf II + s c2 + c3fJ

/lX{B,Tf)/I

f: (B, Tf )2dz I~

B J=I :S;

Iq II Tf

c2 + c4fJ{IVATf 12 +

12)dzl~

B :S;

c 2 + c 5 111 Tf

S C2

III

(326)

+ C5

where we have used the lower bound [aij (x)] ~ 61 for some constant 6

> 0 for

B to denve the thIrd mequalIty, the mequality (3 22) (whIch by denSIty holds for f1 = Tf) to denve the fourth me quality and the relallons all x

E:

III Tf 1112 = (Tf,f h.:s; II Tllllf 112:s; 1 to derIve the fifth and SIxth mequalItIes m (3 26) It thus follows that X IS Hcompact Hence by Weyl's theorem [35, Theorem XIII 14, P 112] It follows that U ess

(327)

(H) = u ess (Hx)

Combmmg (3 27) WIth (3 25) we get mf u ess (H)

~

l:{F) -

l:

and smce e was arbItrary It follows that (328) Fmally we show the reverse mequality To this end we choose any POSitIve number I" such that I"

< mf uess{H)

Let E(I")

=E({I","'»

Jecllon of H which corresponds to the mterval (J.I.,co) rank proJecllon, so I-E(J.I.) =

N

E (.;',)V,

'=1

for ;"

E:

be the spectral pro-

Then I-E(I") IS a fimte

D{H)

Thus If rp IS a test

-51-

function with support outside B{O;R) we have, using Schwarz's inequality N

~

III (I-E{J.')) 91 III

:E

1{91,ti)l

IIlti III

~=1

N

J

~:El

""1 I'" I>R

Therefore given c

>0

Itil 2dx l* IIltilll 119111·

there is an Re so that (3.29)

for all 91 E CO'{lln'B{O;Re )). From the spectral theorem we have

III E(J.') 91 1112 = J Ad (91,E«-CO,A»91) II-

(3.30) (We are using throughout that

111'1/. III = (Hu ,'1/.)* for 'I/. E 11·11 ~ 111·111. we have

V

= D(JIh)).

Combin-

ing (3.29) and (3.30), recalling that

(P91,91)*=

11191111

~ IIIE(J.')91111

-11I(I-E(J.'»911I1

~ J.'* II E(J.') 91 II

- ell 91 II ~ J.'* II 91 II - J.'* II (I-E(J.')91 II - ell 91 II ~ (J.'lLc) II 91 II - J.'* III {I -E(J.'»91 III ~ (~-c(J.'*+l))IJ911I

(3.31)

Thus using (3.31) we find that

l:(P)

=K

sup

compact

infl ~ : 91 E CO' (R n , K) , 91

~ infl \~:irJ

1/9111

:

91

E

CO'(ll n , B(O;R,,) , 91

~ 01

~ OJ

~ (J.'*-c(J.'*+ 1»2

for c > 0 sufficiently small. Letting c -+ 0 we find l:(P) ~ J.'. Finally since J.' was an arbitrary positive number less than inf u ess (H) we get l:(P) ~ inf u ess (H). This together with (3.28) proves (3.16) and completes the proof of the theorem .•

-52-

Chapter 4

L2 Exponenttal Decay

Applications to eigenfunctions of N-body Schrodmger Operators In thIS chapter we apply Theorem 1 5 to prove L2 exponentIal decay results

In Theorem 4 1 we consIder the general class of ellIptIc operators

A +q studIed

lo

Chapter 2 and Chapter 3 wIthout takmg lOto account m a

detaIled way the vanatIOn of A(x .R) m dIstant regIOns of II'" Thus replaclOg A by a constant we prove the theorem usmg a metnc PA WIth A

= constant

In

Theorem 44 we consIder operators A +q WIth constant ai, but take Into consIderatIon the varIatIon of A(x .R) WIth dIrectIOn for large Ix

I

by makmg use

of the functIOn K(rJ) mtroduced lo Chapter 2 We thus obtam an L2 exponenllal decay result for solullons of Au+qu = zu at a neIghborhood of mfimty whlCh mvolve a metnc Pc WIth

C

discontmuous but stIll lower semlContmuous

The use of such metncs reqUIres an addItIOnal techmcal result gIven m Lemma 43 We then consIder decay problems for eIgenfunctIOns of SchrodlOger operators P =- A+ V for a general class of multipartlCle type potentIals V

For such operators the functIOn K(rJ) can be descnbed explI-

CItly and takes a relatIvely SImple form

Combmmg thIS lOformatIOn WIth

Theorem 4 4 we obtam In Theorem 49 a general exponentIal decay result We conclude thIS chapter WIth a dIScussIon of the N-body problem The exponenllal decay estImates for eIgenfunctIOns of the correspondmg Schrodmger operator are obtamed as a specIal case of Theorem 4 9 Throughout the chapter we use the results of Chapter 3 to apply the exponentIal decay theorems to L2 eIgenfunctIOns of the self-adJOlnt realIzation of P 4.1 Theorem: Let p = A+q = -}:;;}.ia~J(x)a~+q(x) be an elltpttc operator

'J on II'" sattsJy1.ng the hypotheses of Theorem 3;2 Let p(x .y) be the geodes'1.C dtstance from x to y ?n 11'" ?n the .RI.emann'l.an metT'1.C ds 2 = :ECl.£,(x)dx~dxJ i,.J

where [~.1(x)]

ts

the 'tnverse of the matrLX [ai,(x)]

Set p(x)

=p(x.xo) where

-53xO is

let

Z

some fixed point in 1'1"'. Let 0 Z = P. < ~ where 1:

E C with Re

=1'1"" K where K is some compact set and =1:{P) is defined by {2.12}.

Let 1(x) be a solution of the equation "E Hi~c(O).

ICVA"rVArp + q"rp)dx

o

for every rp

Pt = zt

in 0 in the sense that

q1 E 4!c(O). and

E

= Iz"rpdx 0

CoCO). Suppose that 11";(x)1 2 e-2Pp(:e)dx

o

< co

(4.1)

for some (J < (1:-p.)* • (J ~ O. Then

11";(x)12e 2ap (:Z:)dx < co

(4.2)

o

for any a < (1:-~)*. Proof: Given 0 < a < (1:-~)* choose e> 0 such that a < (1:-~-e)* and < (1:-~-£)*. It follows from the definition of 1: that there exists a number Ril > 0 such that setting Oil = fx : Ix I > RilL we have (J

(4.3) for all rp

E

Cci(O£). Without loss of generality we may assume that

all C

O. By

subtracting ~ from each side of (4.3) we obtain Re

I( IV D.

A rpl2

+ (q -z) Irp 12)dx ~ (1:-~-£) I I rp 12 dx. D.

This is the A-Condition (1.12) of Theorem 1.5 where the function q of the theorem is the present q-z and A(x) '" 1:-p.-&. We now check the remaining hypotheses. Clearly (q-z)

Let hex)

E

4!c(0£) and (q-z)_

E

M'oc(OIl) by our assumptions on q.

=a p(x). Then. using Theorem 1.4. IVAh(:r:) 12 = a 2 IVAP(x) 12 ~ a 2 < A

Take in Theorem 1.5

f ==

a.e.

(4.4)

0 and u = 1 which in view of our assumptions is a

-54-

solution of A (2: ,(1)1.£

+ (q -Z)1.£ = 0

in 0". Finally we need to show that

flt(2:)

12Ae-2 (1-4)P'(Z)d3;


0 where for some fixed 2: 1 E: 0" 1

PA(2:) = p>..(2:,2:1)

=inf fA*[~aij(l'(t))7i(t)7j(t)]~t 7

i.J

0

and the infimum is taken over all absolutely continuous paths 1': [0,1]

-+

0"

joining 2: 1 to 2:. Clearly A*p(2: ,2: 1) :s p,,(2: ,2: 1) so that by the triangle inequality A*p(Z) :Sp,,(z) + c where c=Al£p(2: 1,zO).

(l-cJ)p,.(2:)

~

We now choose ~=l-f3A-'h. Then 0 d J. From (4.4) it follows

that A-a 2 :s (A-IVAh

12)

so that recalling that h (:r:) = ap(z) we have

f

11'(:r:) 12

e 2ap (z)d3; :S

o..d

Cd (A-a 2)-1

f

o.,O..d

11'(:r:) 12

e 2ap (z)d3;

(4.6)

Since the a ij were assumed bounded we have, as before, that PA(:r:,y) ~ Const.lz-y I where 12:-Y

I is the Euclidean distance. This implies that

0,,' O",d is contained in a compact subset of 0. Thus, since l' E: Lk (0) the right side of (4.6) is finite. Finally the assumption (4.1) implies that

-55-

I,,(x) 12e 2ap (:a:)d.x
0, except Jor posstbLy fimteLy many x Set C (x) = hm cJ (x) and suppose that fCJ(x)j ,J

,......

C

(x)

'tS

locally bounded Then C (x)

'tS

lower sem'tConnnuous and (49)

untJormLy tn (x ,y) tn any compact subset oj II" x II" Proof We begm by showmg that If A IS a lower seffilcontmuous function then PA{X ,y) does not change when we change the value of A at fimtely many POInts

Clearly It IS enough to show this for one POInt

Suppose A(X)

=5:{x)

Then smce for any absolutely contmous path "/, ,,/{t) = 0 a e on .,-l(1x°J) (see Lemma 55) we have A{.,(t»I.,(t)1 2 =5:{7{t»17(t)/2 ae for

except at

xO

t

Thus It follows that

E

[0,1]

of generahty that cJ (x) ~ 6

PA

>0

= PA Therefore we can assume WIthout loss

everywhere on II"

Smce the cJ are non-decreasmg, the Pc, are a non-decreasmg sequence of continuous functlOns on II" x II" It suffICes, therefore, to prove pomtwlse convergence smce by Dim's theorem this ImplIes the uniform convergence of (49) on compact sets Clearly, bmpc,{x,y) sPc(x y)

J .....

(4 10)

We now prove the reverse me qualIty and thus establIsh the lemma

We

first prove this under the additIOnal assumptIOn that the cJ (x) are contmuous Without loss of generality we assume that the matrIX [~J] IS the IdentIty matrIx, for by makIng the non-smgular transformatIOn x ~ [~J]~ we can reduce the situatIOn to this case

Therefore we assume ds 2

=l:~Jdx~dxJ

IS

the EuclIdean Relmanman metrIC Let Li (7) denote the length of the path 7 In the metrIC dsc~ = cJ ds 2 and let £ ("I) denote the EuclIdean length m the metrIc ds 2 FIX two dlstmct pomts

-57-

x ,y e: Ill'. The definition of the Pel (x ,y) implies that we can find a sequence of absolutely continuous paths 7j : [0,1]

-+

Ill' joining y to x such that

Lj(7;) 0

(4.12) Therefore maxl17j(t)l : 0 ~ t ~

11 ~ Ix I + £(71)

(4.13)

s K(x,y) where K(x,y) does not depend on j. By (4.13) the sequence f7j bounded. It is also equicontinuous since

l

is uniformly

t.

17j(tl) -7j(t 2 )1

= 1!7;(t)dt I t1

s £ (7;) It l-tfl

s

6-~(Pc(X,Y)+1) It l -t 2 1

(4.14)

where the last inequality follows from (4.12). Therefore Ascoli's theorem applies. By passing to a subsequence we therefore can assume that there is a continuous path 7(t) such that 7j(t)

-+

7(t) uniformly on [0,1]' Since the

£ (7j) are bounded we can choose this subsequence so that £ (7j) number £. Taking the limit as j ....

CD

-+

£ for some

of the second line of (4.14) we obtain (4.15)

Hence 7: t .... 7(t), t e: [0,1], defines an absolutely continuous path joining y to x. Note also that in view of (4.15) we have

17(t)I:!l:£ a.e. in [0,1]' Now let i be fixed and j

~

(4.16)

i. Then 1

Lj (7j) = £ (7j)!Cj(7j(t»'Iz dt

o

1

~ £ (7j)! Ci (7j (t »'lzdt, o and therefore, lim Pc (x,y) ~ lim !nf Lj (7j)

i~-

j~-

I

1

~ lirp. inf e (7j)! Ci(7j (t »'lzdt ,....

0

1

=eJVm Ci(7j(t»'lzdt

(4.17)

0''''1

=e !Ci(7(t)Y*dt o

1

~

! Ci (7( t »'Iz 17( t) Idt , o

where the first inequality follows from (4.11), the third and fourth inequality from the uniform convergence of the functions 7j(t) on [O,lJ and the last inequality follows from (4.16). Finally we let i ...

CD

and apply the monotone con-

vergence theorem. Denoting by L(7) the length of the path 7 in the metric C

(x )ds 2 it follows from (4.17) that

lim Pc/x ,y) ~

j....

1

Jc (7(t»*17(t) Idt 0

= L(7) ~ Pc(x,y)

which completes the proof of the lemma when the Cj(x) are continuous.

We now remove the temporarily added continuity assumption on the cj(x). Thus suppose that Cj(x), j = 1.2, .. " is a locally bounded lower sem-

icontinuous function such that Cj (x)

~6

>0

on Iln. By a well known theorem

on semicontinuous functions (see [30; pp. 149-156]) there exists for every

-59-

fixed j

an increasing sequence of continuous functions CjA: (x) on 11ft,

k = 1,2, ... ,such that CjA:(x) t Cj(x) as k t

OD

for all x. Set

=max(c1j(x), C2j(X), ... , Cjj(x), B),

bj(x)

j = 1,2, .. '. Then f bj (x)l is an increasing sequence of continuous functions

such that

0< B ~ bj{x) Also bj(x)

~ Cij(X)

for i

~ j.

~ Cj(x).

(4.18)

Hence fixing i we have

lim bj(x) ~ ~im Cij(x) = Ci(x) , ... -

i~-

which in view of (4.18), since i is arbitrary, implies that lim bj(x) = ~im Cj(x) = c(x). i..... , ....-

(4.19)

We are now in a position to apply the special case of the lemma already proved to the sequence of distance functions PilI" It follows from (4.19) that ~impII/(x,y) =Pc(x,y) ,-. ..

for every (x ,y) e: 11ft

X 11ft.

Since PII/r ,y) ~ PCI (x ,y) ~ Pc (x ,y) this implies limpcl(x,y) =Pc{x,y)

j-. ..

which completes the proof. • We can now prove the following exponential decay theorem in which Lemma 2.8 is used to provide a non constant A function. 4.4 Theorem: Let P

=-

L"

a ~J., 8;. iJ j + q (x) be an elliptic operator on 11"

".1=1

such that [a tj ] is a positive definite constant matrix, q a real function in Lt~c{llft) with q_ e: Mloc {llft ). 8uppose that E{P) >- .... Let [) = 1l"\.K where K is some compact set and let z e: C with Re z = JL < E(P). 8uppose Py, zy, in [) in the sense that y, e: H~c([)' qy, e: Lt~c(O) and

=

/(VAY,'VArp

o

+ qy,rp)dx

= /zy,rpdx

Jor every rp e: CoCO). Pix N > E(P) - JL and set

n

-80-

C

where K«(,) ::: K«(,),P)

mm(K(x/ Ix I)-JL, N) If x 'F 0 (x)::: {0 If X ::: 0 'tS

defined by (221)

from 0 to x 'l.n the metnc

Let Pc(x) denote the d'tStance

ds c2 ::: c(x)I;a~,dx,dxJ

where [a.,]::: [a'3]-1

'I.n

fl'"

(Note

i,J

that c (x)

~

l:(P)-JL

> 0 for all x

'F 0 l.n mew of Lemma 2 7 ) Suppose

JI,,(x)1 2 e-2 (1-4)P.(z)dx

o for some 0

0 such that (4.23) for all rp

E:

can also assume that 00

C

= ~x : Ix I > Rol.

Choosing R o large enough we O. We are now in a position to apply Theorem 1.5 to

Cci(Oo) where 00

the function y in 0o, Indeed, the inequality (4.23) shows that the hypothesis for A(X) in Theorem 1.5 holds with A(X) = Cjo(x),

The function q (x )-p.

satisfies the hypothesis for q (x) in Theorem 1.5 and y is a solution of

At + (q-p.)y

=0

in 0o, We set h(x) = -(JPClo(x) where 0

< -(J < 1 then, using

Theorem 1.4, we have a.e.

IVAh(x) 12 = -(J2IVAPCJg(X) 12 :s: -(J2c;o(x)

< c;o(x)

which shows that -(JpCJg(x) satisfies the hypothesis for h in Theorem 1.5. It remains to show that for some 0 < lJ' < 1 (4.24) where PCl (x) = PCJg (x ,xo) denotes the distance in 00 from some xO E: 00 to x in o the metric dsc~o restricted to 00' Since PClo denotes the distance function in 11" with the same metric, we have

PCI O(x)

=PClo(x ,xO) ~ PC/o (x ,XO) ~ PClo(x,O) - PClo(XO,O)

-62-

where we have used (4.22) to derive the last inequality and where C does not depend on

%.

Thus

where C' does not depend on

%.

We choose 6' so that

(1-6')(1-(1:/2)) = 1-6 i.e.,

6' = (6-(1:/2)) 1 (1-(1:/2)). We can assume without loss of generality that

o < 6' < 1.

I:

is small enough so that

Then 2(1-6 ')PCJ (%) ~ 2(1-6)pc (x) - C' so (4.24) follows from the o

hypothesis (4.20).

Therefore we can apply Theorem 1.5 to conclude

f

11'(%) 12(c;0(% )-1 VAh (%) 12)e 2h(O:)d%

0 0.>1

~ 2(l+;d) d

for d

>0

and

Now for

00d ,

% E:

=

J

11'(x) 12c;o(%)e2h(O:)d:r

0 0 ,00.>1

~x E: 0 0 :

Pcio (%,aO o) > dJ.

0 Ci.(x)-IVAh(%) 12 ~ (l--.7 2)c;o(X) ~ (1--.72) inf Cj

(Col)

1.,1=1 •

=m

where m is a positive constant. Also, 2h(x) = 2'(JpCJo(x) ~ 2"(1-f)pc(x),

so that if we now choose iJ ;:::: (1-1:)/ (i-f) then 0

Therefore

< " < 1 and

-83-

J I,,(x) 12 e 2 (1-C)P,(:I:)d.1; 00.. ~ m- 1 1 I,,(x) 12 (c j .(x)-IVA h(x) 12 )e 2h (:I:)d.1; O.,d

J

~ 2(1+~d) md

1,,(x)1 2 cj.(x)e 2h (:I:)d.1;.

(4.25)

0.,0•.•

Since 0 0 \ Oo.ct is contained in a compact subset of 0 the condition 1{1 E LJ,c (0) implies the right side of (4.25) is finite. We also have

I,,(x) 12 e 2 (1-c)p,(:I:)dx

1 0'1.0•.•

~ C2

1 1'\V(x) 12e-2(1-6)p,(:I:)dx 0'00..

< "", where the first inequality follows from the continuity of Pc (x) on Rn and the fact that 0\ Oo.ct is contained in a compact subset of Rn and the second inequality follows from the hypothesis (4.20). Therefore

11'I{I(x) 12e 2 (l-I:)p,(:I:)d.1; < ""

°

which completes the proof. • Theorem 4.4 has the following corollary for L2 eigenfunctions. 4.5 Corollary: Let p =-

n

..

1; a ~1 iJ t iJ j + q (x) be an elliptic operator on Rn i,j=l

=

satisfying the hypotheses of Theorem 4.4. Suppose also that K(",) K(",;P) is a bounded function on sn- 1. Let H be the self-adjoint realization of P in L2(Rn) defined in Theorem 3.2. Let 'I{I be an eigenfunction of H with eigenvalue JL < inf uus (H). Then 11'I{I(x)1 2 e 2 (1-£)P,(z)d.1; Rfor any 4.4.

l:

> 0 where C (x)

= K(x /

1xl)

< ""

(4.26)

- JL and Pc (x) is defined as m. Theorem

We now want to apply these results to eigenfunctions of N-body quantum systems. Since the previous theorems gave exponential decay in terms of

K(",;P). our goal will be to describe this function when P satisfies certain properties which hold for quantum mechanical N-body Hamiltonians. We also change notation to conform with conventions in physics: q (x) is rechristened Vex)

and is

called the

potential and we let A =

tl.

..

~ a~1a;,iJi.

Then

i';=1

P =- A + V(x). Note that A is the Laplace-Beltrami operator associated with tl. the inner product given on Il.tl. by x·y = ~ ~jx;.y;.. where as usual ;'';=1

=

4.6 Lemma: Let A(P). I:(P). and K(",) K(",;P) be as defined in Definition 2.4 and Definition 2.6. Suppose that V(x+t",o) = Vex) for some "'0 E gn-1 C Il.tl. and all tEll.. Then

(i) A(P) = I:(P)

(ii) K(",o;P) = I:{P) = min{K(",) : '" Proof: Let rp

E

E

sn-11.

Co (Il.tl.) and define rpT(X) = rp(x +7"'0) for

7 E

Il.. Then

5.!:.!lifl _ (P rp rpT) IIrpl12 - IIrpTII 2 · T'

To prove (i) we need to show I:(P) from the defintions. Fix inf Pick rp

E

l:

~

A(P). since the opposite inequality follows > 0 and let K be a compact set such that

f~; rp E II rp II

CO'(Il.tl.'K)1

> I:-l:.

Co(Il.tl.) such that

f.E!li£l < A{P) + e.

IIrpll2

Since rp has compact support there exists

A{P) + l:

T

so that

> 5.!:.!lifl

IIrpll2

(PrpT.rpT)

II rpT 112

rpT E

Co{Il.tl.'K). Thus

-65-

which proves (i) since t:

> 0 was

Noting that for any fI E

arbitrary.

Cci (Il n ),

r:,~N (see Definition (2.6)) for any

f:

fiT is supported in the truncated cone

and N when

T is

large enough, the first

equality in (ii) follows from a similar argument. The second equality is equation (2.22) so the proof is complete .• In the N-body problem the potential Vex) on Il n is of the form V(x) =

e

L (=1

Vt (x) where each of the potentials

Vi (x)

depends only on some of .

the variables. We shall need the following lemma to show that if the class M& on the subspace on which it depends then definitions of the classes of functions MA(n) for 0 :s: 8

Vi

Vi

belongs to

E M&(lln). For the

< 1 we

refer to Chapter

O. We also recall that Mo{n) is another notation for the class of functions

M{n). 4.7 Lemma: Let Iln = Y E& Z where Y and Z are two complementary sub-

spaces of Rn. For a point x ERn we use the obvious notation x = (y ,z). Suppose V is a function on Rn depending only on y, i.e., V«y ,z)) = W(y)

Proof: We consider fln as an inner product space with norm

Ix I

and

associated measure dx. We denote by dy and dz the measures induced by dx on Y and Z respectively. We have dx = ady®dz where a is some positive constant. 2 - 8

Let m

=dim Y.

We give the proof under the assumption that

< m < n. The necessary modifications needed in the proof for m = 1

and m

=2 are left to the reader.

-66-

We shall estimate the integral appearing in the definition of M,,(Il"). (We note here that the definition of the class of functions M,,(I1") is independent of the inner product imposed on II"). For any

xO

=(yO,zO) E II" and r

> 0,

we

have

J

IV(x)llx-xOI 2---dd,xS:

l:e ....... olO,"'EBJ, then If x E C and

Ix I > L, we have III,. x I = Ixlln~( 1:1)1 ~ Ix l(III,."'ol- 0 where p(x) is the geodesic distance from x to 0 in the .Riemannian metric (4.32)

Remark: The function c (x) which appears in the definition of the Riemannian metric is defined on Rn , (01 where it is positive bounded and lower semicontinuous. We have neglected to define c (x) for x = 0 and to be

-7~

exact we should formally set c (0) =

some non-negative number.

From

remarks made before it is clear that p(x) does not depend on the special value chosen for c (0). We shall now consider some special multiparticle Schrodinger operators arising in physical problems. We consider an atomic type system of N + 1 particles with coordinates :z:i Ell", i

=0,1 , ... , N, and interacting potentials Vij(Y)

defined on

Ilv.

The

particle xo (the "nucleous") has infinite mass and is fixed at xo. We assume that the real valued functions Vij , 0

~

i

0 so that

II AQ("ED II L2(Z) < (j.

(4.55)

Using (4.52) through (4.55) it follows that

(PQ~"D'~EDhe(X)= «(""DQ~X'X("£Dh2(X)

=(Q~X'Xh2(Y) os; A( Q~)

which implies (in view of (4.53» that

- (XA"(""D,X(""")Lt(X) ('l."(""D'(""Dhe(z)

+ (j + II A"(""D II L2(Z) II ("ED II L"(Z) < A(Q~) + 2{j,

-76--

A(J5,.,) < A( Q~) + 215 Combmmg (456) with (451) we obtam (448) (smce 15

(456)

> 0 IS arbitrary) thus

completmg the proof • The precedmg discussion leads to the followmg charactenzatlOn of the functlOn K(Col,P) 4.11 Lemma: Let P be the atomtc type Schrodtnger operator defined by (4 33), Pacts tn the configuratwn space X = R"N For any unit-vector Col EX (in some inner product) define the subsystem Schrodtnger operator Q~ as the

restrtCtion oj the operator Q", (defined by (446» to the subspace X", {defined by (4 47» '!hen K{Col,P) = A(Q~)

(457)

Proof. To obtam (4 57) simply combme (437) with (4 41) and (448) • It was observed before that the atoIDlc type Schrodmger operator P verifies the conditions of Lemma 4 8

Hence It also venfies the condltlOns of

Theorem 3 2 Applymg the theorem we shall denote by H the self-adJomt reallzatlOn of Pm L2(X) If Col IS a umt-vector such that [(Col) IS not empty, we shall denote (temporanly) by H~ the self-ad]omt reahzatlOn of Q~ m L2(X.,) where Q~ and X", are defined as m Lemma 411

It follows from Theorem 32 when

apphed to H~ that A(Q~)

= mf u(H~),

whIch when combmed WIth (457) shows that K(Col,P) = mf u(H~)

If [(Col) IS not empty,

(458)

K(Col,P) = 0 If [(Col) IS empty

We shall now combme Corollary 45 WIth formula (458) to denve the exponentIal decay esllmates for elgenfuncllons of atomic type Schrodmger operators

For reasons of presentallon we shall make some changes m the

notatlOn used before

-77-

4 12 Theorem Let P be an atom7.c type Schrod7.nger operator gwen by

P

= EN (_(2~)-1 ~ +vO'L(x~)) + E

where z, E flv and

~

v~1 (x~-x1)

(459)

IS' n/2 (n

~

2) then q_

M{j(H) for some 0> O.

E

In view of our assumption on q it follows by Lemma 0.2 that there is a constant c 1 and a number"

E

(0.1) such that for all rp

E

Co (H): (5.1)

where A == (l-A)*. 5.1 Theorem: Suppose that

(i) ai.J(x) , 1 si,j sn, are reaL measurabLe junctions on B satisfying a:iJ (x) == a ii (x) and

c;11~12 S

t

aiJ(x)~i~i ~ c21~12

(5.2)

iJ=l

Jor all

~ ERn

and x E B . c 2 a positive constant.

(ii) q is a complex valued function in LL~c (B) with q _ satisJying (5.1) Jor all rp

E

Co(H) with some"

E

(0.1).

Suppose u is a weak solution oJ the equation n

Au+qu ""-

E

iJja(Jih u + qu == 0

(5.3)

;,J=l

in the sense that u

E

L2(B)

n

lft~c(B) . qu E Lk(H). andJor each rp E Co(B)

f(AAU·AArp

+ qurp)dx = 0

B

(5.4)

(we use the notation (l.3». Then u E Lt';c(B) and Jor each ball BIS = H(O;a) with 0 < a < 1, there is a constant D = D(c1.c2.a) (Where c 1 and c 2 are given in (5.1) and (5.2» such that ess sup Iu (x) I s D

• En..

II u II L2(B)·

(5.5)

-85-

Before we proceed with the somewhat long proof of Theorem 5.1 we shall apply it to derive exponential decay pointwise bounds for eigenfunctions of the atomic type Schrodinger operators considered in Theorem 4.12 and for eigenfunctions of the N-body Schrodinger operators considered in Theorem 4.13.

5.2 Theorem: Let

P =

f {-(2m;.)-1A, +

Voi{X i ))

I:

+

i=1

Vij{Xi-X j )

l:r£.i0 ,i =

t , ... , N.

Let H be the self-adjoint realization of Pin L2{X). Let V'{x) be an eigenfunction of H with eigenvalue p. < inf u ess (H). Then for every e > 0 there is a constant C~ such that a.e.

on X

where p(x) is the geodesic distance Junction from x to the origin defined in Theorem 4.12. 5.3 Theorem: Let

P =- Ax +

E

Vij lsi 0 define u,,=(luI2+t:2)~. We claim that uI! and iLlu" are in H&!c(O) with (5.9) (5.10)

To see this we will use the

J.g

product rule

lJi (/ g) = g iJ,/

+ / Big for

E Hi!c(O» and the chain rule (Lemma 5.5). The product rule is easily

proved by an approximation argument. To see (5.9) let Fl be in Cl(ll) with bounded derivative and F 1(x) = (x+t:2)~ for x ~

o. Then u"

= Fl. Iu 12 and

the chain rule can be applied. Similarly to compute the (distributional) derivatives of u;l we introduce a function F2 in Cl(ll) with bounded derivative such that F 2(x) Then

U;l

= x- 1 if x : 2 (c depending on

If n ,;, 2 then (5.33) holds if 2' stands for any number r

r), while for n

= 1 (5.33)

= = 3 for

holds with 2'

cases we define in the following 2' (5.33) holds for all n.

00.

It follows from (5.33) and the hypothesis

~/2)

lu I 2

0

E

In order not to consider special n

=1

or 2. With this definition

lu Ip 0/ 2

!.d,c(B)

E

H~c(B) that (5.34)

and thus the right side of (5.32) is finite if p - (Pol 2) s (2'1 2)(Pol 2), i.e., if p spo(2'+2)1 4.

J q _~ Iu P' dx. At this stage we impose B that it is of the form (" = a where ("1 is a non-negative

We now consider the expression on (" the restriction

function in Co(E). From Lemma 5.7 and (5.1) we have for some r

E

(1,2) (5.35)

for all tp e:

Co (E)

where c' is a constant. This easily extends to rp e: La (E)

with Vrp e: L-(E) and thus we can substitute rp 1 E Co(E), Vrp =

and

real

2r- 1 1" I(2/1')-I{sgn

valued.

= 11121'1' into (5.35) with

Using

Corollary

5.6

gives

")V,, so that Holder's inequality applied to (5.35)

results in (5.36) A straightforward limiting argument shows that (5.36) is valid for 1 e: HJ (B). We shall substitute 1

II q ~(' Iu

= (", /2 1u IPo/2.

(Note that ("'/2

=("[ e: Cd (E).)

Thus

I(2/1')(Pol 2) II L"(B) ~ c" ( II V«('1'/ 21 u Ip 0/ 2) II L2(B) II ("1' 121 U IPol1' IIl~iB?'''

so that J~q-Iu IPdx < ex>

2 + 1I(""2Iu IP0/ 11£'.

B

Using our freedom to vary (" it follows that (6, lu I) lu 1(P/2)-1 e: Lc,c{E). From (5.34) it follows that lu IP/2 e: Lc,,;(E) and thus Corollary 5.6 allows us to write (5.37) and conclude that Vlu IP/2 e: L'~c{E). Hence lu IP/2 e: H,,~c{E). By induction it follows that lu IP/2 E H';'c(B) for allp ~ 2. To find explicit bounds we go back to (5.31) which we rewrite in the form

-98-

4(P;1) P

J~2IVA lu IP/2 12dx ~ 2J~IVA lu I·VA~llu IP-1dx B

B

+ Jq-~Iu

Ipdx

(5.38)

B

where we have used (5.37) to write

IVA lu 11'/212 = (P2/ 4) lu 11'-2 IVA lu 112.

no longer make the assumption that ~ = ~f but continue to assume ~

E:

We

Co (B)

and non-negative. Using (5.32) with Po = P to bound the first term on the right side of (5.38) results in

This inequality is of the form 2

A2~ (~)AB + P C. p-1 4(P-1)

Since AB ~ A2/ 4 + B2 we have

A2[1 Since p

~

2 ,p / (P -1)

~

2

P ] ~ ~B2 + P C. 4(p -1) P -1 4(P -1) 2, and thus (5.39) implies

J~IVA I'-u. I p/2 12dx ~ 4Jlu II' IVA~12dx+pJq_~lu Ipdx B

B

(5.40)

B

and upon taking square roots

II ~IVA lu Ip/21I1L8(B) ~ 2111u Ip/2 1VA~IIIL'(B)

+vP II q ~~Iu Ip/2I1L2(B)· According to (5.1) and (5.25) there is a number" such that for any c

> 0 and rp E:

(5.41) E:

(0,1) and constants

C ,c 1

Co(B):

(5.42) Since (5.42) clearly extends to rp

E:

HJ (B) we substitue rp

= ~Iu 11'/2.

IVrpl ~ ci* IVArpl II ~I VA lu 11'/21 II LO(B) ~ 2111u IP/2 1VA~III L2(B)

the resulting inequality into (5.41) and using

Putting

results in

-97-

+vP cs(e II VA «("I u 1,,/2) IIL2(B) + (e-.J(1-.J) + e) II ("I u 1,,/2 II L2(B) I where we have abbreviated IIIVAv III L2(B) = IIVAv

(5.43)

II L 2(B) and

2

IVAv I = (lVAv 1 )*. Using II VA «("Iu 1,,/2) II L2(B) ~ II IVA ("II u 1,,/211 L2(B) + II (" I VA lu 1,,/21I1L2 (B) and choosing e such that csVP e

= 1/2, we find

II VA «("Iu 1,,/2) II L"(B) ~ 6111u 1,,/21 VA ("III L2(B) + 2c3P*(e-.J/(1-.J)+e) 1I("lu 1,,/2I1 L 2 (B) ~ (6+2c 3P*(e-.J(1-.J)+e) liliu 1,,/2( IVA(" I +(") II L2(B) ~ cJ>G IIlu 1"/ 2( IvA ("I +(") II L2(B)

where a

= [2(1---.7)]-1.

(5.44)

We put this in a form which is useful for deriving L-

estimates by using Sobolev's inequality, Eqn. (5.33): (5.45)

Let a

E

(D,l) and choose e > D so that with e, = e/i2, I-a =

Defining ~ = 1-

, em we note that

k

m=1

ing to

II-.

,

-

(~1':=1

-

I; e,. (=1

is a decreasing sequence converg-

Let

Since (5.45) clearly extends to ("

=f,

and since

IVA(".: I ~ {Canst. )(i+l)2

we

thus have (5.46) Bu." denoting the ball B(D;~).

We take P

=Pi = 2{2 ~12)':-1.

Rearranging

(5.46) we find (5.47) Because Pi increases exponentially with i it is easy to see that for some

-98-

constant D.

for all £ so that (5.47) yields the inequality

Ilu II L"'+1(B,,) ~ lIu II L"I+I(B",.,) ~ D Ilu II L2(B",) which holds for i = 1.2..... Letting i ess sup lu(x) I ze:B.

-+ co

it follows that

= "Fmllu II L",(8.) ~ D lIu IIL8(B) .... ~ II

This yields (5.5) and completes the proof of the theorem. •

Appendix 1

Approximallon of Metrics and Completeness

Lemma A1.1. Let

9~J(X),

1 :!>'i" ,s;n, be real val'l.U!d conttnuous func-

hans on an open subset 0 oJ FJ."" such that [9" (x)] = G(x) ts a posihve definite matrix Jar every x E 0 Then Jar every 6 > 0 there eXtsts a posihve defin'tte matnx [htJ (x)] = H(x) 'l.lJ'tth real htJ E C-(O) such that (1-6) G(x),s; H(x),s; (1+6) G(x)

Jar every x

(All)

E ()

We recall that for n xn real symmetric matrIces A and B,

,s; for every t

IIA II

E

FJ.""

A,s; B means

We also let

=sup( IAtl

t

E

FJ."" ,

It I ,s; 11

Proof of Lemma: Choose a partition of UnIty (91J ,

= 1,2.

J for

0 satis-

fymg

(11) For any compact set K. supp 91,

nK

IS empty for all but finItely

many,

Let (" E CO' (R"") WIth

J ("(x)dx = 1 and supp (" c

let (",,(x) = t:-'n("(x/ t:) Let mt

K,

::> supp 91,

B(0.1) and

K, be a compact subset of {) WIth

Let 7, = 2-i 6 mf SEliI

(II G(x)-1ID-1 and

choose 0

< t:J < 1/,

so

that (a) supp

«(""J -( 91i G»

c

K, (Here - denotes the convolutIOn operatIOn on

functIOns on FJ."")

for all x

E {)

Then

(AI2) Smce

G(x)~IIIG(x)-111-1

It follows from the definItIon of 7j

that

-100-

7,I$. 2-i 6G(x) for x

E:

~

Thus by (AI 2) and (a),

~,(x)G(x)-2-J6G(x),s; Re/-(~J G»(x)

Let

H(x) =

$.

~3 (x)G(x)+2-3 6G(x)

-

L

Re/-('I', G»(x)

(AI3)

(AI4)

i=1

We show that H IS well defined and sallsfies the desIred me qualitIes Suppose 0 o IS an open subset of n WIth a compact closure m 0 Then for large dISt(OO, supp '1',) ~ a

enough J

that If K IS a compact subset j

supp '1',

n

> 0 for some a of n WIth mt K

K IS empty For these J

dIst«mt K)c, 00)

>0

Thus

dIst(OO' supp (tel -('I', G»)

because from (11) It follow'. :::>

0o, then for large enough

dISt(OO' supp 'l'J) ~

because

I:i

< 1/ J , for

large

enough

J

>0

ThIS Implies that for x m 0 o all terms m the sum (AI 4) for J large enough vamsh Idenllcally Hence H IS well defined and belongs to C""(O)

Fmally, summmg the mequalitIes (AI 3) over J, usmg

1:'1', = 1, gIves the deSIred me qualitIes • Lemma A1.2

Let

n

be an open connected subset oj Rn and a" (x) con-

hnuous Junctwns on 0 such that [a..; (x)] PA be as gwen by (19) where A(x)

PA(X, (""!) Then (0, P.,.)

1.8

1.8

postitve defintte Jor every x

= SUP(PA(x ,O'K)

K compact J

a complete metrw space 'if! PA(x ,lcol)

Proof Suppose PA(x,(coJ)

Let

> 01.8 conitnuous and let

= co and

= ""

IxJJ IS a PA-Cauchy sequence Then,

smce

PA(XO'X,) IS a Cauchy sequence of numbers WhICh therefore converges and thus IS bounded

(x, J c K

Smce PA(x ,(""!)

= "" we

can find a compact set K such that

It IS easy to see that PA generates the usual topology on 0

(K, PAl IS a compact metnc space and hence complete verges

Thus

Therefore (x, I con-

-101-

Suppose, on the other hand, that (0, PAl IS complete Assume m addItIon that the matrIX A(x ) [a" (x)] has entrIes m C-(O)

Then the mfimum over

absolutely contmuous paths m the defimtIOn of P)' can be replaced by an mfimum over smooth paths and PA IS the metrIC treated m standard differentlal geometry text books [22, 13]

We sketch the rest of the proof

usmg Ideas from differentlal geometry

By the Hopf-Rmow theorem [18]

metrIC space completeness IS equIvalent to geodesIc completeness ImplIes that gIven x

E:

ThIS

0 and t m the tangent space Tz at x, there eXIsts a =x ,7(0) = t For x EO con-

geodesIc 7(t) defined for all t such that 7(0)

Tz -+ n WhICh maps t E Tz to 7(1) where 7 IS the geodesIc such that 7(0) = x and 7(0) GeodesIc completeness Imphes Expz IS

SIder the map Expz

={

defined on all of Tz where N 1,2,3,

Smce Expz IS contmuous the sets ExPz(B(O,N))

=

are compact and

In the case where the matnx A(x)[ a~J (x)] does not necessarIly have smooth entrIes we apply the approxImatIOn Lemma A1 1 to find a matrIX Hex) WIth smooth entrIes, such that for some 6 (1-6)A(x)[~J(x)]:s

Then If

E

(0,1)

H(x):s (1+6)A(X)[a..,(x)]

P denotes the dIstance functIOn

defined by H we have, for x ,y

E

n

So If 0 IS complete m the metrIC PA and (x)! IS a p-Cauchy sequence PA(Xj,Xm,):S (l-6)-1 P(Xj,x m )

Thus the sequence IS also p),-Cauchy and hence converges ThIS shows p-complete By the preVIOUS argument p(X,fOD!) = OD so that PA(x ,fOD!) ~ (1 +6)-1p(X ,f"'!)

=...

WhiCh completes the proof.

n IS

-102-

Proof of Lemma 1.2

Appendix 2

The fact that K. exists and satisfies (2) and (3) follows immediately from the definitions while (4) is a consequence of the fact that for fixed x • K'(x .. ) is the supremum of linear functions. To see continuity let G c 0 be compact and define. for x

E

G and .". { E

sn-1•

I ,,(x .{) :: / K(x •.,,) so that K.(x.{) = supU ,,(x.{) :." "'0 E

sn-

1

E

sn-1 1 .

Given x

so that K.(x .() = I ",,{x.6 Then for x'

K.(x·

E

E

G and

~ E

sn- 1

E

sn-

I TJo(x'

.r»

G and f

choose

1

.n ~ I "o(x' .f).

so that

=I 'f),{x .t) =ITJo(x' .0 + (f TJ.(x .t) -

K.(x .{)

~

K.(x· .f) + ,,£8"-1 sup I/TJ(x .{) - I TJ(x'.O I·

Reversing the roles of (x .t) and (x' .f) we thus have

Since J TJ{x .{) is uniformly continuous for .". (

E

sn- 1

and x

E G.

the con-

tinuity of K. follows. The fact that K •• = K follows from the bipolar theorem [33; Theorem V. 28]. We give a direct proof. Since the variable x is irrelevant we suppress it. Since K.(t)

= sup«(.1}>/ K(.,,) : ." "# 0\ we have

.,,>

K.WK(.,,) ~

we have = 1 and < 1 for all

tE

~

Since

= a -lfl (-)

C. So TO e: CO by (A2.1). Hence

K •• (71o) = suPt : T

E

cOl

= 1.

This contradicts (A2.2). Therefore K •• (71) the proof .•

for

~

K(71) for every 71 which completes

Appenduc3

Proof of Lemma 2 2

Let us prove that AR(x ,A +q) IS continUOUS In x and R we sImplIfy notatIOn and wrIte AR(x) for AR(x,A+q) functIOn of (x,R) In fln

X

FIXing P = A +q

We consIder AR(x) as a

IR+ wIth the usual EuclIdean metriC

We start by

showing that AR(x) IS an everywhere fimte upper semicontInUOUS functIon Let n be a bounded open set In fln

En

[aii (x)] C!: M for every x

Since q_ t n belongs to M(n) and Since

for some 6> 0, It follows from Lemma 03

1 6

(applIed WIth e = nunC '2' '2» that there eXIsts a constant Co' depending on n and A, such that Jq_I9I1 2dx D

for every 91

E

~

tJIV A 9I1 2dx + CoJI9I1 2dx 0

(A31)

0

CoCO) USing (A3 1) It follows that (P9I,91) = JOV A 9I1 2 + Q+I,,1 2 - Q_I,,12)dx

o

C!:

for every"

E

t /(1 VA o

Co(O)

91 12 + q+I9I12)dx - Co Jltpl2dx

ThIS inequalIty ImplIes In partIcular that AR(x)

whenever B(x,R) cO Thus, Since IS

(x' ,R,)

t

(1

E

->

be

(xO ,Ro) as J

a

C!:-

Co

IS an arbItrary bounded open set, AR(x)

everywhere fimte Now let (xO,Ro) be a fixed pOint In fln

«xi,R,)J ,J = 1,2, ,

(A32)

0

sequence

of

pOints

In

X

fl+ and let

fln

fl+

X

such

that

PIck a number e > 0 and choose a functIOn

-0 00

», such that IItll = II? II L" =

Co(B(xO,Ro

1, and such that

(P?, t) ~ ARo(XO) + e

WIth l' defined as zero

In

fln \. B(x O,Ro) , It IS clear that supp t c B(xi,RJ ) for

J large enough Hence, for large J we have A8j(xi) =Inf I(P9I,tp) :S

letting J

-+

CD

tpE Co(B(x',R,) ,

11,,11

=

11

(P?, 1/1) ~ ARo(xO) + e

m the last mequallty and then lettmg

l: ->

0, we get

-105-

(AS.3)

This proves that AR(x) is an upper sernicontinuous function.

Next we show that AR(x) is a lower sernicontinuous function which together with the preceding result will prove that AR(X) is continuous in

(x.R). To this end we introduce some classes of functions. Let 0 be a bounded open set in R". We denote by Ho1(0) the completion of CoCO) in the Hl(O) norm:

We also introduce the space of functions

Hjq .. (0)

=f'IL ; H.:o (0) • q ~ E L2(0) J

(A3.4)

where q + is the positive part of the function q in the lenuna. bert space under the norm

Hdq .. (0)

is a Hil-

111·111 D defined by (A3.5)

We shall need the fact that CoCO) is dense in Hjq .. (0). To prove this result suppose that'lL

E Ho~.

(0) is orthogonal to CoCO) in Hrk .. (0). that is

j(Vu'Vrp + (1+q+)urp)dx D

for every rp

(1 +q +)'IL

E

=0

CoCO). This means that -Au =- (l+q+)u weakly. Since

E LL~ (0)

lenuna that I'lL I

E

we are in a position to apply Lemma 5.4. It follows from the

Hi!c(O) and that -Alu I ~

-

Re«1+q+)u(sgn u»:S- 111.

I in

the sense that J(Vlul'Vrp+ lulrp)dx:SO

(A3.6)

D

for every rp

E

Co (0) • rp ~ O. Since 11.

E

Postponing the proof of this result. let CoCO) such that ~j

....

may

that

assume

Ho1(0) we also have that lu I E H.:o(O). be a sequence of real functions in

!~jl

111. I in H.:o(O). Choosing if necessary a subsequence we ~j (x) -+

I'lL (x) I

almost

everywhere.

Set

-lOS-

rpi

= ("J+)-2)* -

Then rpJ e: CoCO) . rpJ ~ 0 and It IS easIly

= 1.2.

)-1.)

checked that rpi .... 111.1 In H,,},(O) (here one uses the fact that B, lu I vamshes almost everywhere on the set So =

Ix

(A36) WIth 'I' = '1', and lettIng) ....

DO

u(x) = OJ, see Lemma 55)

12 +

It folows that f(IVu

o

ApplYIng

lu 12)dx ~ 0

whIch ImplIes that u = 0

Hiq. (0) we therefore need I e: Ho1(0) Suppose first that

To complete the proof that CO'(O) IS dense In only to show that If u

u e: CO' (0)

Set rpi (x)

E"

Ho1(0 then also lu

= (Iu 12+) -2)* -) -1 .) = 1.2.

It IS clear thal '1', (x) ... lu(x) I as)

-+

DO

Then '1'3 e: CO' (0) and

umformly In 0 We have (A37)

whIch shows that the sequence 1'1', I IS bounded In Ho1(0) Imply that there eXIsts a funcllon

u e: Ho1(0)

These propertIes

such that '1', .... ft weakly In

Ha1(0). and that ft(x) cOIncIdes WIth lu(x) I for almost all x ThIS proves thal

lu I e: Ho~(O)

USIng (A3 7) we also see that

IIlu I !lw(D) ~ lIm

i-+-

sup II '1', IIw(n) ~

lIu IIW(D)

Let u e: Ho1(O) and let !rp, I be a sequence of

Now consIder the general case

1101 (0) From the result Just proved It III rpJ III wen) ~ 11'1', II w(n) whIch Imphes bounded sequence In Ho1(0) SInce 1'1', I .... lu I In L2(0). Il fol-

functIOns In Co (0) such that 'P, .... u In follows that IrpJ I e: Ho~ (0) and that thalllrp, IllS a

lows as above that 111.1 e: Ho1(O) We proceed now WIth the proof that AR(X) IS a lower semlContmuous funclIon m x and R

o

We fix a pomt (xO,Ro) In R7\ x R+ and pIck a bounded open set

such that 0 :::> B(xo.Ro) By addmg to P a suffICIently large constant and

denotmg the resultmg operator agam by P, we may assume (m VIew of the estImate fa" (x)] ~ 61, WhICh holds for every x e: 0 and some d e: (0.1) and the consequent esllmate (A3 2» that (A38)

for every 'I' e: Cci{O)

Here III

1110 IS

the norm m Ho~.(O) mtroduced before

-107-

Also, smce the a ii (x) are bounded mOlt follows that (A39) for every IP E CO'(O) and for some constant c

(Pu,U)n for any functIOn

U E

Let us Write

=J(lVAtL 12 + q I'lL 12)dz

(A3 10)

n

Hl(O) such that q

¥u

L2(0)

E

(A38) and (A39) It follows (smce CO' (0) IS dense

lo

1n VIew of the bounds Ho~+ (0» that (Pu ,tL)~ IS

an eqUIvalent norm lo the HIlbert space Hot (0) We now consIder a sequence of polOts «xi ,R, H (x 3 ,R,) -+ (xO ,Ro)

as J

-+

GO

lo FIn X FI+

such that

where (xo ,Ro) IS the pomt we have fixed before

Recallmg that B(xO,R o) cO we shall assume WIth no loss of generahty that B(xi ,Ri ) c 0 for all J We now choose a sequence of functIons IPj E CO' (B(xi ,R,» such that (PfP, ,fP,) :s;

=1,2,

for J

A~(xj) +.1. J

and

We consIder the IP, as elements of

II fP,l!

=1

Hiq+ (0)

ffPJ I

(fP, belOg defined as

Smce fA~(xi)! IS a bounded sequence {by (A3 3», It fol-

zero in O'\B(xi,R,»

lows that the sequence of norms (PIP) ,IP,)it IS bounded that

(A3 11)

IS a bounded sequence m

WhICh converges weakly m subsequence agam by

fIP,I,

Hiq+ (0)

Hot (0)

Thus we conclude

We now pIck a subsequence fIP,~ I

Changmg notatIOn and denotmg thIS

we thus may assume that there eXIsts tL

E Ho~+(O)

such that

= m Hot (0) ,compactness theorem [3, Theorem 38] weak hm IPj

Applymg Relhch's

U

(A3 12) It follows from

(A3 12) that fP,

Smce

U

{PfP, 'ffi

-+

tL

strongly m L2(0)

IS a weak lImIt of lIP, J 10

)n, we conclude that

H;q+ (0),

(A313)

WIth the norm of IP, gIven by

-108-

(A3 14)

where the last me qualIty follows from (A3 11) Observe also that from (A3 11) and (A3 13) It follows that

II

11'11.

=0 a e

B(xO,R o) and note that'll.

= 1

Denote by

'11.0

the restnctlOn of

'11.

to

m O\. B(xO,R o) (m Vlew of (A3 13) smce 'P, IS

supported m B(xi,R,») Hence usmg (A3 14) we have (A3 15) We q!u.o

now

£

claIm

that

L2(B(xO,Ro)) that

'11.0 £

'11.0 £

Ho~(B(xO,Ro»

H~. (B(xO,Ro»

whlCh

ImplIes

(smce

Acceptmg thIS result for a

moment we proceed With the proof and show that AR(x) IS lower semlContmu-

»

ous at (xO,R o) Indeed, smce as we have shown CO'(B(xO,Ro 1S dense m H,/q. (B(xO,R o», It follows that there eXIsts a sequence of functlOns '"

£

» such that t, -+

CO'(B(xO,Ro

'11.0 10

Hot (B(xO,Ro»

II L8(B(zO,Ro» = II u II L8(0) = 1, and thus we lit, II L8(B(zO,Ro) = 1 Smce ARo(xO) S; (Pt"ti)B(zO,Ro)

11'11.0

We also have that

may

assume

that

for :J = 1,2, ,we obtam

upon passage to the lIffilt that ARo(xO) s; (PuO,UO)B(zO Ro)

whIch when combmed w1th (A3 15) shows that

AR (xO) s; hm mf AR (x') •

,_

t

ThIS proves that AR(X) IS a lower semlcontmuous function and thus estabhshes (smce AR(X) IS also upper seffilcontmuous) that AR(X) IS contmuous We stIll must show that affme T,x

t,

maps

'11.0 £

1', Ilfl -+Ilfl

H,ic,(B(xO,R o)

wh1ch

='I7 j (x-xo) + xi With 17, =R,I R o

= 'Pi

0

1',,:J = 1,2, ,where 'P,

mtroduced above Then of Ho1(0)

Smce

x'

-+

t,

£

£

map

B(xo,Ro ) onto B(x' ,R,) Cons1der the sequence of functlOns

CO'(B(x'

.R,»

CO'(B(xO,Ro»

Xo ,", -+ 1 and smce

Ho!,(o) It follows readIly that

(A3 12) It follows that '"

-+

t, -

To thIS end mtroduce the

1S the sequence of functIons

Cons1der '" and 'Pj as elements

frp, I

IS a bounded sequence

In

'P, -+ 0 weakly m Ho~(O), so that m Vlew of

u weakly m Ho1(O) Smce the

t,

are supported m

-109-

the ball B(xO;Ro) it follows further that when restricted to this ball tj .... Uo weakly in Hl(B(xO;Ro». This implies that Uo E: Hoo (B(xo;Ro as claimed. The main part of the lemma is thus proved.

»

We shall now prove the

Am (x .it)

statement that if

= -LBja;,{iti

•

iJ

= 1.2 ..... is a sequence of operators satisfying the same conditions as A and if a:,{ ~ a ij

m

as m

~...

uniformly on compact sets. then AR(x;Am +q)

~

AR(x;A +q) uni-

formly in x on compact sets. Thus let K be a compact set in R" and let R > 0 be fixed. Pick a bounded open set 0 in R" such that 0 :J B(x ;R) for every z

E:

K. By adding the same constant to the operator P and the operators

Pm =

Am + q

we may assume without loss of generality. in view of (A3.1) and

(A3.2). that

(Prp.rp) ~

IIrpll2

and Jq_lrpl2dx

S

o

for every rp E: CoCO). Since a:,{(z) ~ aij(x) as m ~

co

(Prp.rp)

(A3.16)

uniformly for x EO. and

since we also have [aij(x)] ~ 61 for every x in 0 for some constant 6> O. it follows that there exists a sequence of positive numbers le m !. with em .... 0 as m .... -. such that (A3.1?)

for all x rp

E:

E: [) •

m = 1.2, .... Combining (A3.16) and (A3.1?) we find that for

CoCO): (Pmrp.rp) ~ (1+e m )(Prp.rp) + em Jq_I9'1 2dx

(A3.18)

o

~

(1+2e m )(Prp.rp).

and similarly that (Pmrp.rp) ~ (1-2e m )(Prp.rp)

(A3.19)

From (A3.18) and (A3.19) it follows that (1-2e".)A R (x;P) ~ AR(X;P",,):s; (1+2e m )AR (z;P)

for

every

AR(x;Pm )

~

K and m = 1.2..... which implies AR(x;P) uniformly on K as claimed.

x

E:

in

particular

that

-110-

Appendix 4

Proof of Lemma 5.7

To begin with we prove that if " and 'I E Sell"), where decrease, then

-+

(0.1).

T

> 2(l+2(l--6)/n)-1 . T > 1.

Sell") is the Schwartz space of functions of rapid

1\·11 .. denotes the norm in L"(Il"».

(Here and in the following

Define B; S(Il")

E

JJ

" L 2(1l") = J=1 .ED

by (Brp)j

= iJjrp + n~.

A short calcu-

lation using the Plancherel theorem gives

IIBrpl\J1= IIArpll2' Note also that with (AUg)j

(M.l)

=AG.gj we have A-(1-.J)Brp = BA-(1-.J)rp

so that

Hence using (A4.l)

(M.2) If ,

E

S(Il") it is easy to see that

so that

IIA-(1-.J),

112 ~ r( 1-;' )-1

It is well known that etA,

= Kt.,

j

e-t

1Ie tA, Ibdt.

(M.3)

= (47Tt)-1'L/2 e-lzl"/4t.

Using

t-(1+1J) /2

o

with Kt(x)

Young's inequality we have

with 1 +

t=

p-1

+ r- 1 . A calculation shows

IIKt lip = c(p) t-n/2(1-p-l)

-111-

so that using (A4.3)

IIA-(I""")J 112:S r(

Ii )-IIIJ IIr jo e-tc

(p)

t - /2 {1-p-l)t-(I+'IJ)/2dt

(A4.4)

The power of t in the integrand is greater than -1 since r > 2(1+2(1--17)/n)-I. Thus

IIA-{l""")J 11 2 :s d(17,r)JIJ Ilr' hence from (A4.2)

IIA"qJ)~:S d(17,r)2

EII (BjqJ+*) II; vn

J=1

:S c(17,r)2(IIV~lIr + 1I~lIr)2 which proves (5.24). We now turn to the proof of (5.25). It is clear that if 17

E

(0,1) and x

there exists a constant c such that

x" :S Thus for a

> 0 (ax)":s ax + c

80

x":s

X

+ c.

that a 1 ...."x

+ a...."c.

Since A2 is a positive operator

A2" :S e2A2 + Thus for

~ E

e-2"(I""")c

CD' (Il n ), IIA"qJIIH:s e2I1V~IIH + e211~IIH + e-2"/(I""")c II~IIH·

Taking square roots gives

IIA"qJII 2 :s ellVqJII2 + cl(e....,,/(1""")+t:)llqJI12 for some constant c l ' This proves (5.25) and completes the proof.

C!:

0

-112-

Bibliographical Comments Chapter 1. A special case of The~rem 1.5 was proved by Lithner [25]. A version of Theorem 1.5 which holds for solutions of elliptic operators on Riemannian manifolds was described by us in [1]. Theorem 1.5 which states that solutions of certain second order elliptic equations in unbounded domains which do not grow too fast in fact decay rapidly belongs to a group of results sometimes referred to as "PhragmenLindelof type theorems". Results of this general type for solutions of possibly higher order elliptic equations were studied by various authors. In this connection see P.D. Lax (24) and Agmon and Nirenberg [4; p. 220]. Chapter 3. There is an extensive literature on the self-adjointness problem for the Schrodinger operator -A+q and for more general elE;>tic operators on Iltl. Among the many papers on the subject we mention here only the older papers by Stummel [41] and by Ikebe and Kato [19). and the more recent papers by Kato [20. 21]. A detailed bibliographical commentary on the subject can be found in Reed and Simon [34]. In his more recent work Kato studied the self-adjointness problem for -A+q under minimal assumptions on the potential q. Thus for instance in the paper [21] no assumptions are made on the positive part of q except that it be locally integrable. Theorem 3.2 on self-adjoint realizations of A +q which we prove in these lectures is of the same type. it is proved under "minimal" assumptions on q. Formula (3.16) for the bottom of the essential spectrum of the selfadjoint realization of P (under more restrictive assumptions on the operator) is due to Persson [32]. Chapters 4 and 5. There is an extensive literature on exponential decay of eigenfunctions of Schrodinger operators. We shall mention here a few of those papers which deal the subject of exponential decay of eigenfunctions of N-body Schrodinger operators. For additional references the reader should consult [6).

-u&We have already mentioned in the Introduction the papers by O'Conner [31], Combes and Thomas [7] and Simon [37] which prove the isotropic estimate (1) for eigenfunctions with eigenvalues below the essential spectrum. As a matter of fact these papers deal with eigenfunctions of N-body Schrodinger operators with center of mass removed. The Schrodinger operator in question is the same as the operator H studied in Theorem 4.13. It acts on L2(X) where XC 1l1lN is defined in (4.69). O'Conner's result is that if t(x) is an eigenfunction of H with eigenvalue p.

< 1: = inf u.ss (H), then

Jlt(x)l2 e 2QJ 2:Jdx

x

for any a

< ..Jr.-p..

Here

< ....

I x I denotes the norm of x in the inner product

(4.68). Combes and Thomas have simplified consideraby O'Conner's proof while Simon has shown that the L2 bound could be replaced by the pointwise bound: It(x) I ::s;;

ell. exp( -a Ix I) for any a < ..J1:-p., ell. some

constant.

The more refined non-isotropic upper bounds for eigenfunctions of Nparticle systems were discussed in the literature in a paper by Deift, Hunziker, Simon and Vock [10] and in papers by Ahlrichs, M. Hoffman-Ostenhof, T. Hoffman-Ostenhof and Morgan [17,15,14] (see [6] for additional bibliography). These papers give various concrete upper bounds for eigenfunctions of multip article Schrodinger operators (mostly of atomic type). We also mention here the paper by Mercuriev [26] where precise asymptotic results are given for eigenfunctions of three-particle Schrodinger operators with short-range potentials. The non-isotropic upper bounds for eigenfunctions of multiparticle Schrodinger operators which we prove in Theorem 5.2 and Theorem 5.3 were announced by us in

[1J

and [2]. The estimates given in these theorems turn

out to be "best possible" in case the eigenfunction t is the ground state (under somewhat stronger assumptions on the potentials). This follows from similar Lower bounds which can be established in this case. To be precise, suppose that the eigenfunction t considered in Theorem 5.2 or in Theorem 5.3 is the ground state which we may assume to be everywhere positive. Let p(x) be defined as in Theorem 5.2 or Theorem 5.3, respectively. Then the

-114-

following lower bound holds: 'jI(x)

for all x e: X for any given

t:

>0

~ cce-(l-C)P(.z)

and some constant

Cc

> O.

Combining this

lower bound with the upper bound established in Theorem 5.2 and Theorem 5.3, respectively, we obtain for the ground state the follOwing asymptotic relation: lim log 'f(x) =- 1. l.z I". p(x) The lower bound described above for the ground state of the N-body problem (case of Theorem 5.3) is due to Carmona and Simon [6]. For a special case of Theorem 5.2 (the ground state of two electron atom) the lower bound is due to T. Hotfman-Ostenhof [16] and to Carmona and Simon [6]. (For some special lower bounds see also Bardos and Merigot [5].) The technique used in the proof of Theorem 5.1 is essentially that of Moser [28]. The added complication in our case is due to the fact that we consider complex solutions u of Au +qu = 0 for quite general complex functions q. The special case of Theorem 5.1 when q is a real function in £P (B) with p

> n/ 2 follows from the results of Stampacchia [40; see Theorem 5.1

and Remark 5.2 on p.171].

-115-

References [1] S Agmon. On ezponBntiaJ. decay of solution.s of second order eUvphc equations 1.n

unbOUTLded domains. Proc. A PleLJel Conf . Uppsala. September 1979. pp 1-18 [2] S Agmon. How do eigenfu=tions decay? The case of N-body quantum systems. Proc VIth Int. Conf Math Phys BeriLn 1981. Lecture Notes m PhYSICS. Sprmger-Verlag. 1982 [3] S. Agmon. Lectures on EU~tic Brrundary Value Problem. Van Nostrand. Prmceton. 1965 [4] S Agmon and L Nirenberg. Properties of solutions of ordiTl.l1.ry differentwl equations in

Banach space. Comm Pure Appl Math 18 (1963).121-239 [5] C Bardos and M MerLgot. Asymptotic decay of the soluhon of a second order eUvphc equation 'in an unbOUTLded doma'in Applications to the spectral propert1.es of a Hamil-

tonian. Proc. Roy Soc Edmburgh 78 A (1977). 323-344 [6] R Carmona and B SLmon. Pointwise bounds on eigenfun.ct1.OTl.S and wave packets in N -body quantum systems. V: Lower bounds and path 'integrals. Comm Math Phys 80 (1981).59-98 [7] J M Combes and L Thomas. Asymptohc behavior of e'IfJenfu=1:um.s for m1Llhparticle SchrOd'inger operators. Comm Math Phys 34 (1973). 251-276 [8] R Courant and D Hilbert. Methods of Mathematical Physu:s. Vol II Intersclence. New York. 1962 [9] E De GLorgL. Su.Ua differenziabilua e l'analicita deUe estrllmali degli integrali multipli

regolari. Mem Accad SCI. Torino Cl SCL FIS Mat Nat. Ser 3. 3 (1957).25-43 [10] P Delft. W. Hunziker. B Simon and E Vock. PoiTLtw..se bounds on eigenfunctions and wave packets in N -body quantum systems, IV. Comm Math Phys. 84 (1978). 1-34 [11] P Fmsler. tiber KurtJen 1LTI.d FIiichen 'in AUgeme'inen Riiumen. Btrkhiiuser. Basel. 1951 [12] D Gllbarg and N S Trudmger.

EU~tic

Partial Dil!erential Equations of second Order.

Sprmger-Verlag. Berlm and New York. 1977.

-118-

[13] S Helgason. Differential Geometry and Symmetric Spaces. Acadermc Press. New York. 1962 [14] M HolImann-Ostenhot and T HolIman-Ostenhof. "Schro dinger inequa.lmes" a.nd a.sym:ptotic beha.viar of the electron density of a.toms a.nd molecules. Phys Rev 1762-1765

18 A (1977).

[15] M HolImann-Ostenhof. T. HolImann-Ostenhof. R Ahlrlchs and J. Morgan III. On the ell:ponentiIU Ia.UoJf

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electron dens1.ties. Ma.thema.twa.l Problems in

'lheoretica.l Physics. Lecture Notes m PhysIcs no 118 (1960).62-67. SprInger-Verlag [16] T HolImann-Ostenhof. A lower bound to the deca.y of ground sta.tes of two electron a.toms. Phys Letters 7? A (1960). 140-142 [17] T HolImann-Ostenhof. M HolIman -Ostenhot and R Ahlnchs. "Schrodmger mequalllles" and asymptotic behaVior of many-electron denSities. Phys Rev A 18 (1976).326-334 [16] H Hopf and W RJ.now. tiber den BegriJf der vollstiindtgen diJferentmlgeometrischen Jilache. Comment Math Helv 3 (1931).209-225 (19] T Ikebe and T Kato. Uniqueness of the self-a.dJD'i:n.t extension diJfrn-enlml opera.tors. Arch Rational Mech Anal 9 (1962). 77-92

01 sin.gu.la.r elhptic

[20] T. Kato. Schrodmger operators With smgular potentials. Israel J. Math 146

13 (1972). 135-

[21] T. Kato. A second look a.t the essentml selfa.djO'Lntness 0/ the SchrO· d'I:TI.ger opera.tars. Physica.l Rea.lity and Ma.thematica.l ])esCT'if'twn. DReidel Pubhshmg Co . 1974. pp 193-201 [22] S Kobayashi and K Norruzu. Foundatum.s York. 1963

0/ lJiI.ferentia.l Geometry. IntersClence. New

[23] 0 A Ladyzhenskaya and N N. Ural'tseva. Linear and qUllSilinear equations of elhptw type.Izd Nauka. Moscow. 1964. English translallon. Acadermc Press. New York. 1966 [24] P. D Lax. A Phragmen-lMI.delO·/ theorem in ha.rmonic analysis and its applicatum. to 80me questions in the theory of elliptw equations. Comm Pure Appl Math 361-369

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Agmon, Shmuel, 1922Lectures on exponentlal decay of solutlons of second order ellIptIc equatIons.

(Mathematlcal notes ; 29) Blbllography: p. Includes ,ndex. 1. Dlfferentlal equatlons, E11,pt,c--Numer,ca1 solutions. 2. SchrCi'dlnger operator. 3. Elgenfunctlons. I. Tltle. II. Serles: Mathematlcal notes (Prlnceton, N. J.) ; 29. QA377.A48 1983 515.3'53 82-14978 ISBN 0-691-08318-5

Shmuel Agmon 1S Professor of Mathemat1cs at The Hebrew Un1vers1ty of Jerusalem.