Schrödinger operators with form-bounded potentials in <span class="mathjax-formula">$L^p$</span>-spaces

A NNALES DE L ’I. H. P., SECTION A

M. A. P ERELMUTER

Schrödinger operators with form-bounded potentials in L

^p

-spaces

Annales de l’I. H. P., section A, tome 52, n

^o

2 (1990), p. 151-161

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151

Schrödinger operators with form-bounded potentials

in Lp-spaces

M. A. PERELMUTER

Urickij street, 11, apt. tS3, Kiev, 252035,

Ann. Inst. Henri Poincaré,

Vol. 52, ^n° 2, t990, Physique théorique

ABSTRACT. - Let

closure of

the operator (-

^{A + V +}

^C) f C~0 (Rl)

^is

generator

^of the contrac- tion

Co-semigroup

^{in For}

p = ~

^{we obtain a new theorem concern-}

ing

the essential

self-adjointness

^{of the}

Schrödinger operator.

^The

sharp

ness of the results is illustrated

by

^the

examples.

ture de est un

generateur

^{de semi-}

groupe

Co

de contraction dans Pour

p = 2

^{nous obtenons un}

Annales de l’Institut Henri Poincaré - Physique theorique - 0246-0211 1 Vol. 52/90/02/151/11/$3.10/ ^(è) Gauthier-Villars

152 M. A. PERELMUTER

nouveau théorème concernant

l’auto-adjonction

essentielle de

l’opérateur

de

Schrodinger.

^{Ces resultats sont illustres sur des}

_exemples.

~

INTRODUCTION

This article deals with the

Schrodinger operators -A+V acting

ⁱⁿ

Lp - LP

(IR’, ^cf x).

^{We want to}

investigate

^{the first}

spectral problem,

^{i. e.}

the

problem

^of

m-accretivity.

^Most of the results

concerning

^this

problem

contain the condition of ^- 0394-boundedness of the

operator

V _

==max{-V,0}.

^Our

^goal

^{is to}

investigate

the conditions of the

following type:

where F is some function _space.

The first result in this direction is due to Povzner

[1] ] [p=2, P=l, F=C(!R~)]. Recently

^{some results were} also obtained. In

particular

the essential

self-adjointness

^was

proved provided

^that

F = Lk,

My

purpose here is to extend the abovementioned results ^to the case of

LP-spaces

^{and to} choose the broader space F. Even for

p = 2

^{we obtain a}

new result which is very close to

optimal.

It should be noted that our method is an extension of the Semenov’s

one

[3].

Note that LP-A-boundedness

implies

the condition

(1) (see [4])

^{and in}

my

opinion

the condition

( 1 ) corresponds

^{to the}

point

^{of our}

problem.

We want also to

emphasize

^{that the exact}

eigenfunction

estimates for the

operator - A

^{+ V and the exact results}

concerning

^the

accretivity

^{of the}

Schrodinger operators

^{have been}

proved

^{under the same condition}

( 1 ) (see [5], [6]).

Annales de l’lnstitut Henri Poincaré - Physique théorique

SCHRODINGER OPERATORS 153

Some common

symbols

^are:

==

x) = the complex functions f’

^on

^fR’

^{such that}

f

p ^E

LP,

V (p e ^meas

(B)

⁼

Lebesgue

^{measure of B c}

R’ ;

Lp° ~° ^_--_ LP’ 00

d x)

^{= the}

complex functions f

^on

^~~

^{such that}

(note

^{that LP c LP’ 00}

and

^E

^{LP’ 00)} Lp,~comp

^{= the set} of functions in Lp,~ with

compact support. A= Y 2014_

^the

^{Laplacian acting}

^{on D’}

^(1R1).

j= 1

lx )

MAIN RESULTS

LEMMA 1. ^-

If V

^E

then ( ~ V ~p - £

^{= O 0.}

Proof. - &#x3E;t}.

^The

function (t)

^{is nonin-}

creasing and (0) = meas {supp V}

^{CIJ. The} definition of the space

implies

^that

~. ( t) _ C t - p, V t &#x3E; 0,

^so

DEFINITION. - Let 0 ~

V,

^{W E}

L1loc.

^{Then V ~}

PK03B2(-0394

⁺

^W)

^{if and}

^only

if

Vol. 52, n° 2-1990.

154 M A. PERELMU"fER

THEOREM 2. - tet V be a real valued measurable

function such

^that

Then the _range

of

^tk

operator (-0394+V+03BB) C~0

^{M dense M!}

LP for

any

X&#x3E;C(P).

Suppose

^{that the} range is not

dense,

^then

by

the Hahn-Banach

theorem~

there exists a

function 0 ~ u ~ Lp

^{such that}

Whithout loss we can assume u= Reu. _{r~ _}

~’ j~’.

^Let

~&#x3E;0 be a small

parameter

^{and Iet us}

and It follows that

n

^L’.

The

equality (3) gives

in the distributional sense; hence

(C£ 17l).

Moreover, implies

Let us consider the

operator 1~Z

^{with the domain}

where

(-A+V+)

^{is taken in the} distributional sense and

==

(-A+V+)~

H:

^{is the}

generator

^{of the}

strongly

continuous contraction

semigroup

== V~&#x3E;0 in LZ which has

properties

^{such as follows}

below

l’Institut Hener Poincaré - Physique théorique

SCHRÖDINGER ^OPERATORS 155

Let

03C6r=Ts(l r)u.

^h follows from

(5), (8), (4)

^and

C0-property

^that

According

^to

(I)

^{we can choose}

subsequence (which

^we shall denote

by

the same

symbol { ~ } )

^{such that}

Then

Indeed,

because

~(jc)-~(j~)

^{a. c. Hence in order to} conclude the

proof

^of

(3)

^it

is sufficient to prove the convergence of the norms

(a

^nice

proof

^{of this}

fact _{may be} found in

(8])

^but

and the convergence

U

^{cp, -}

it

^u is the consequence of

(~).

ln the same way ^we obtain

Let

~~=7~ ~ ~ By

^{weR known}

properties

^{of the}

operation ~ ~ (see,

e. g.~

[9D~

^{we have}

(1), (3), (4)

^{for the}

subsequence

^of

{ fP,. } ~ (7) implies i,2,...)

^{so that}

V+ p~~V+

qJ,.. This fact and the

equality

L"

A

(/~ ~ p~)=~ ~

^{show that}

Let us choose any

03C9~C~0 having

^{and let and}

It is _easy to see that

properties (!), (3), (4)

^{hoid for}

Vol. 52, n* ^2-1990.

156 M. A. PERELMUTER

Using

^the

standard diagonal

process ^{we can} choose the

subsequence

th. ⁼ s uch that t

Observe that so that

M,JM,)~’~ M,~’~e~(V)(~~e.

^g.,

^{[ 1 0],}

Chapter 2),

^and the direct calculation

yields

^the

identity

(4)-(6) imply

the finiteness of the

quantities

so that the

simple approximation arguments

^{show that}

(3)

^{is valid}

provided

that and

(9)-( 1 2 ) imply

Using (13), (14)

^and the condition V _ E + V

+ )

^{we have}

:ic° l’Institut Henri Physique theorique

157 SCHRODINGER OPERATORS

Note

Therefore,

According

^to

(!5), (!6).

~ ‘1 1 ...

From (14)

^we have that the

right

side of the last

inequality equals

So, using

^Holder

inequality

^{we obtain}

According

^{to the lemma I}

IIV - !1z

h:)

= f~’

(E - ~k~.

^{On the other hand}

Now:

assuming

^that

E!

^{0 in}

( 1 7)

^and.

using k ~ ~

^{I we obtain}

Then u - o as 03BB &#x3E; C

(03B2). []

~~marks. - l . The condition V _ E

F’I~~~( - ~

^{+ V}

^{+~ ~1}

is necessary iw the theorem 2 but it i.s not sufficient as it may ^{be seen in the}

example

- 0394 -

03BA 1 - 2 (see [ 11], C’hapter X).

^{The same}

example

^{shows the}

sharpness

~x~2

of the theorem 2.

Inded,

^{let 2} p ^m

l,

where

)((.)

^{is the indicator} of the unit ban in Then the range of the operator

(-A-V-~.)

^is dense in LP if and

only

^if

v 01. 5 =. 11 ?-~.11c1.

158 M. A PERELMUTER

3.

The analysis

^{of the}

proof

shows that the restriction V- may be

replaced by

the weaker condition

1]

^V_

]~_~= ^~(s).

4.. The main device in the

proof

of the theorem 2 is the

inequality

N. Th.

Varopoulos [12]

^has

proved

^{the abstract version of this}

inequality

for the

generators

^of submarkovian

semigroups.

^{Thus our} result may be

generalized

^{for these}

operators.

COROLLARY 3. - L~

Then ( - A + V) C~0

^is

essentially self-adjoint

^{operator in}

^{L 2.}

Corollary 3

^is

proved

^{under the} restriction Ve but

using

^the

results of C.

Simader [13]

^{or H.}

Brezis [14],

^{we have}

THEOREM 4. - Let

+

V) C~0

^is

essentially self-adjoint.

Remark. - In the case 1&#x3E; 5 theorem 4 is a

generalization

of the Kalf- Walter-Schmincke-Simon theorem

(see [ 11 ], §

^X.

4).

Example. -

^We consider the

N-particle

Hamiltonian

Annales de l’Institut Henri Poincaré - Physique théorique

159 SCHRÖDINGER OPERATORS

Hardy inequality jJ= 2 N 03B1 (l - 2)2.

^{one sees}

^easily

^that

V. ".

If 03B1 ~ (l-4) 2N

^it

^follows

from drorem 4 that H is essen-

tially self-adjoint 9cf. {15]. [16].

Note. that for thc fehtivisdc Hamiltonian

Lieb and

Thimng

^haw

proved

^that

with 03B2

^{= O}

(N ex)

and is unbounded from below tor This correla-

tion

between 03B2

^{and N} appears ^to be true for our nonrdativistic Hamil- tonian .as well so that the

dependence 03B1 = o(1/N)

^is

optimaL

To prove

m-accretivity

^{we need.}

THEOREM 5

1’1- -

^Let

is accretive operator ^{in Le.}

We will show that

where

~v, w]

is semi-inner

product ~(~~ ~ ~ ~ ~, ~ X.8).

^{In the case of LP}

spaces

so that

(18)

may be written in the

following

way

Vol. 52, n° 2-1990.

160 M. A. PERELMUTER

1.

The

sharpness

^{of the result is shown in}

[6] by

^the

example

2. T. Kato

~7~

raised the

problem

^of

quasi-accretivity

^{of the}

operator

-A+V in the

limiting

^case

p= 1

^{for the}

potentials

^{such that}

We

will. show that the answer is

negative,

^i. e., and if -A-V is

quasi-accretive

ⁱⁿ

L 1,

^{then V E LOO.}

Indeed,

the semi-inner

product

ⁱⁿ

^L

¹

is

[~ u~~

^{= / 1"}

~~;-) IJ will.

^The

quasi-accretivity implies

^that

B 117 I

Note

= 0, Therefore, (V,

^i.

3. If

V E Lp, ,~ &#x3E; l/2, l &#x3E;_ 3,

then - A - V is the

generator

^of the

positivity preserving Co-semigroup

ⁱⁿ

^{L which}

^{is not}

quasi-contractive (other examples may

^{be found in}

[18], [6]).

Theorems

2,

^{5 and}

Lumer-Phillips

^theorem

(see [ 11 ]., § X.8) imply

THEOREM 6. - Let

~.~here

k = (

^-~-

a) ~ -1

⁺

.~/ 1- . ~

Then the closure

of

the o p era-

tor

( - A

^{+ V + C}

(~)) r C~

^{is the} generator

of Co-semigroup of

contractions in LP.

Remarks. - 1. Theorem 6 is

generalization (when

supp V is

compact)

of the result stated in

[4].

^It is of interest to prove the LP-version of the result of C. Simader

[13]

^{and H.}

Brezis ^[14].

2.

Having completed

this paper the author ^was informed

by

^{Yu. A.}

Semenov about the

proof

of the theorem 6

provided

l’Institut henri Poincaré - Physique théorique

SCHRODINGER ^OPERATORS 161

REFERENCES

[1] ^{A. Ya.} POVZNER, ^{On the} Expansions ^of Arbitrary ^{Functions in Terms of} Eigenfunctions

of the

Operator -0394u+Cu,

^{Mat. Sbornik, Vol.} 32, No. 1, 1953, pp. 109-156 ; ^A.M.S.

Trans. Ser. 2, Vol. 60, 1967, pp. 1-49.

[2] ^{M. A.} PERELMUTER, Positivity Preserving Operators and One Criterion of Essential

Self-Adjointness, ^J. Math. Anal. Appl., ^Vol. 82, No. 2, 1981, pp. 406-419.

[3] ^{Yu. A. SEMENOV,} One Criterion of ^Essential Self-Adjointness of ^the Schrödinger Operator with Negative Potential, Kiev, preprint, ^1987.

[4] ^{V. F.} KOVALENKO, ^{M. A.} PERELMUTER, ^{Yu. A.} SEMENOV, Schrödinger Operators ^with

Ll/2w (Rl)-Potentials,

^{J. Math. Phys., Vol. 22, No. 5, 1981, pp. 1033-1044.}

[5] ^{Yu. A.} SEMENOV, ^{On the} Spectral Theory of Second-Order Elliptic Differential Ope-

rators, Math. U.S.S.R. Sbornik, ^Vol. 56, No. 1, 1987, pp. 221-247.

[6] V. F. KOVALENKO and Yu. A. SEMENOV, Lp-Contractivity of ^the Semigroup ^Generated by ^the Schrödinger Operator ^with Singular Negative Potential, Kiev, preprint, ^1985.

[7] ^T. KATO, Lp-theory ^of Schrödinger Operators ^{with a} Singular Potential, ⁱⁿ Aspects of Positivity ^{Funct. Anal. Proc.} Conf., Tübingen, ^{24-28 June 1985, Amsterdam e. a., 1986,}

pp. 41-48.

[8] ^{W. P. NOVINGER, Mean} Convergence ^{in Lp} Spaces, Proc. Am. Math. Soc., ^Vol. 34, No. 2, 1972, pp. 627-628.

[9] ^{R. A.} ADAMS, Sobolev Spaces, ^{New York e. a., Academic Press, 1975.}

[10] D. KINDERLEHRER and G. STAMPACCHIA, ^An Introduction to Variational Inequalities

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[11] ^{M. REED and B.} SIMON, ^Methods of Modern Mathematical Physics, ^New York, ^San Francisco, London, ^Academic Press, ^1978.

[12] ^{N. Th.} VAROPOULOS, Hardy-Littlewood Theory ^for Semi-Groups, J. Funct. Anal., Vol. 63, No. 2, 1985, pp. 240-260.

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(Manuscript received March 23, 1989.)

Vol. 52, n° 2-1990.