Schrödinger Operators in L2(R ) with Pointwise Localized Potential

Journal of Mathematical Analysis and Applications 235, 180᎐191 Ž1999. Article ID jmaa.1999.6390, available online at http:rrwww.idealibrary.com on

Schrodinger Operators in L2 Ž ⺢. with ¨ Pointwise Localized Potential Ronan Pouliquen Departement de Mathematiques, Uni¨ ersite´ de Bretagne Occidentale, ´ ´ 29200 Brest, France Submitted by George A. Hagedorn Received March 5, 1998

1. INTRODUCTION Schrodinger operators with Dirac’s comb type potential have been ¨ largely studied by Albeverio et al. in their monograph w1x on the subject. In particular, they construct the self-adjoint realization of the formal operator y⌬ q ⌺k j ␦ a j in L2 Ž⺢., where  a j 4j g ⺪ is a real sequence satisfying infŽ a jq1 y a j . G d ) 0. A study of self-adjointness under general conditions has also been done by Mikhailets in w5x Žsee also w6x.. In these two papers, the author connects this problem to the self-adjointness of an infinite Jacobian matrix as an operator in l 2 Ž⺪.. He also relates the semi-boundedness of the operator to the boundedness of  k j 4j g ⺪ and Ž a jq1 y a j .y1 4j g ⺪ . Otherwise, a self-adjoint extension in L2 Ž⺢. of the formal operator y⌬ q W q ⌺k j ␦ a j with W g L1loc Ž⺢. is also presented by Gesztesy and Kirsch in w3x, with no conditions required on the increasing sequence  a j 4j g ⺪ . However, the weights  k j 4j g ⺪ are determined Žsee Eq. Ž3.8. in w3x. and the authors do not present any results of lower semiboundedness in their paper. Here, we give a rigorous rewriting of the operators y⌬ q ⌺k j ␦ a j and present a direct proof of self-adjointness and lower semi-boundedness under conditions sharper than those used in w1x, for the sequence  a j 4j g ⺪ , which allow a jq1 y a j to tend to zero. Our condition is of the type

lim jª⬁

< kn < j

a n jq 1 y a n j 180

0022-247Xr99 $30.00 Copyright 䊚 1999 by Academic Press All rights of reproduction in any form reserved.

- q⬁,

SCHRODINGER OPERATORS IN ¨

L2 Ž⺢.

181

where  k n j 4j g ⺪ is the negative subsequence of  k j 4j g ⺪ and  a n j 4j g ⺪ is the corresponding subsequence of  a j 4j g ⺪ . Moreover, we give, explicitly, the domain of self-adjointness. We do not need any hypothesis on the positive part of  k j 4j g ⺪ other than the boundedness of the whole sequence. The conditions required on this subsequence are, in fact, part of the domain itself. Explicitly, under the above conditions, we provide a self-adjoint realization of the operator y⌬ q ⌺k j ␦ a j on

½

D s H 1 Ž ⺢. l H 2 Ž ⍀ . l u

Ý k p < uŽ k p . < 2 - ⬁ j

j

jg⺪

5

y l  ur Ž ᭙ j g ⺪ . u⬘ Ž aq j . y u⬘ Ž a j . s k j u Ž a j . 4 ,

where ⍀ s D j g ⺪ x a j , a jq1w.

¨ 2. SCHRODINGER OPERATORS WITH POINTWISE LOCALIZED POTENTIAL IN ⺢ Consider two real sequences  a j 4j g ⺪ and  k j 4j g ⺪ . Assume that  a j 4j g ⺪ is strictly increasing and satisfies lim jªy⬁ a j s y⬁, lim jªq⬁ a j s q⬁, and that  k j 4j g ⺪ is bounded. We fix the following notations:

Ž ᭙ j g ⺪.

⍀ j s x a j , a jq1 w , ⍀ s

D ⍀j. jg⺪

Define the Hermitian form B in H 1 Ž⺢. for compact supported functions in x. B Ž u, ¨ . s ² u⬘, ¨ ⬘:L2 Ž⺢ . q

Ý k j uŽ aj . ¨ Ž aj . . jg⺪

In order to specify the domain of definition of B, we need the following result whose proof is in w7x: LEMMA 1. Let the sequence  a j 4j g ⺪ and  k j 4j g ⺪ satisfy lim jªy⬁ a j s y⬁, lim jªq⬁ a j s q⬁,  k j 4j g ⺪ g l⬁Ž⺪., and limjª⬁Ž k jrŽ a jq1 y a j .. - q⬁. Then, we can find C ) 0 such that, for any ␮ ) 0, the following holds in H 1 Ž⺢.:

Ý k j < uŽ aj . < 2 F jg⺪

1

␮

5 u⬘ 5 2L2 Ž⺢ . q C Ž 1 q ␮ . 5 u 5 2L2 Ž⺢ . .

Ž 1.

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Construction of the Operator. We complete the hypotheses on the sequences  a j 4j g ⺪ and  k j 4j g ⺪ : Denote by  k n j 4j g ⺪ the subsequence of negative terms of  k j 4j g ⺪ and assume that < k n j <4j g ⺪ and  a n j 4j g ⺪ satisfy the hypotheses of Lemma 1. Hence the sum Ý j g ⺪ k n j < uŽ a n j .< 2 converges. Note that we do not formulate any hypotheses other than boundedness on the positive subsequence  k p j 4j g ⺪ of  k j 4j g ⺪ . The Hermitian form B is defined on

½

DomŽ B . s H 1 Ž ⺢ . l u

Ý k p < uŽ ap . < 2 - ⬁ j

j

jg⺪

5

.

We have: THEOREM 2. Let the sequences  a j 4j g ⺪ and  k j 4j g ⺪ satisfy lim jªy⬁ a j s y⬁, lim jªq⬁ a j s q⬁,  k j 4j g ⺪ g l⬁Ž⺪., and limjª⬁ Ž< k n j
½

Ý k p < uŽ ap . < 2 - ⬁

D s H 1 Ž ⺢. l H 2 Ž ⍀ . l u

j

j

jg⺪

5

y l  ur Ž ᭙ j g ⺪ . u⬘ Ž aq j . y u⬘ Ž a j . s k j u Ž a j . 4

and the operator A on D by Au s y

Ý Ej Ž R j Ž u⬙ . . , jg⺪

where the operator R j , j g ⺪, restricts a function to the open set ⍀ j , while Ej , j g ⺪, extends a function of L2 Ž ⍀ j . by 0, outside ⍀ j . Then, the operator A is lower semi-bounded, is self-adjoint with domain D, and is the unique operator satisfying DomŽ A . ; DomŽ B .

Ž ᭙ u g DomŽ A . . Ž ᭙ ¨ g DomŽ B . .

² Au, ¨ :L2 s B Ž u, ¨ . .

Proof. Equality ² Au, ¨ :L2 s B Ž u, ¨ . on D = H 2 Ž⺢., and hence the symmetry of A on D, results from an integration by parts, using a family of truncation functions in the variable x, with a uniformly bounded derivative. Let u 0 g DomŽ A*.. We prove that u 0 g DomŽ A.. First, applying <² u 0 , A¨ :< F C1 5 ¨ 5 L2

Ž 2.

L2 Ž⺢.

SCHRODINGER OPERATORS IN ¨

183

to test functions ¨ g D Ž ⍀ . ; D, shows that Au 0 g L2 Ž⺢.. Therefore, Au 0 s ␻ g L2 Ž ⺢ . , which is rewritten as

Ž ᭙␭ ) 0 .

Au 0 q ␭2 u 0 s ␻␭ g L2 Ž ⺢ . .

Ž 3.

The parameter ␭ will be given a fixed value later on. For any x in ⍀ j , the square integrable solutions of Eq. Ž3. are given by u0 Ž x . s

1

x

H 2␭ a

ey␭ Ž syx .␻␭ Ž s . ds y

jq1

1

x

y ␭Ž xys.

␻␭ Ž s . ds

He 2␭ a j

y␭ Ž xya j .

q ␣␭ , j e

y␭ Ž a jq 1 yx .

q ␤␭ , j e

Ž 4.

.

Denote by u 0,c j the integral part Ž‘‘c’’ means ‘‘convolution’’. and by u 0,h j the rest of the right member of Ž4., which is the solution of the homogeneous equation associated with Ž3.. These are the restrictions on each ⍀ j of two functions denoted respectively by u 0c and u 0h. We are led to prove that u 0c and u 0h belong to H 2 Ž ⍀ .. Using Young’s inequality 5 f ) g 5 L2 F 5 f 5 L1 5 g 5 L2 we prove that for any j in ⺪, 5 u 0c , j 5 L2 Ž ⍀ j . F c du 0, j

dx

L ⍀ j. 2Ž

c d 2 u 0, j

dx 2

L2 Ž ⍀ j .

F

1

␭2 1

␭

5 Rj Ž ␻ . 5 L2 Ž ⍀ j . ,

5 Rj Ž ␻ . 5 L2 Ž ⍀ j . ,

Ž 5.

F 2 5 Rj Ž ␻ . 5 L2 Ž ⍀ j . .

Ž 6.

On the other hand, we can also write h 5 2 5 u 0, j L Ž⍀ j. h du 0, j

dx

L2 Ž ⍀ j .

h d 2 u 0, j

dx 2

L2 Ž ⍀ j .

F

Ž ␣␭2, j q ␤␭2, j .

1r2

␭1r2

F ␭ Ž ␣␭2, j q ␤␭2, j . F ␭3 Ž ␣␭2, j q ␤␭2, j .

, 1r2

,

1r2

Ž 7. .

Ž 8.

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Therefore, we have u 0c Ž x . g H 2 Ž ⍀ . with 5 u 0c Ž x .5 H 2 Ž ⍀ . F C2 5 ␻ 5 L2 Ž⺢ . and for any ␸ in D Ž⺢. Ž ␸ Ž x . ⭈ u 0h Ž x .. < ⍀ g H 2 Ž ⍀ .. However, as we do not know whether Ý j g ⺪ Ž a␭2, j q ␤␭2, j . converges or not, it is not possible to establish any global property for u 0h Ž x . Ži.e., a property on the whole ⺢.. The following boundary conditions hold for any j in ⺪:

⭸ u0

y u 0 Ž aq j . s u0 Ž a j . ;

⭸x

Ž aqj . y

⭸ u0 ⭸x

Ž ayj . s k j u 0 Ž a j . .

Let ␸ j g D be such that ␸ j Ž a j . s 0 and suppŽ ␸ j . ;x a jy1 , a jq1 w. Integrating ² u 0 , A ␸ j :L2 by parts in Ž2., we can write y < ␸ jX Ž a j . Ž u 0 Ž aq < 5 5 2 j . y u 0 Ž a j . . F C 3, j ␸ j L ,

Ž 9.

with C3 , j s 5 uY0 < xa jy 1 , a jq 1w 5 L2 Žx a jy 1 , a jq 1w. q C1 - q⬁. Let ␸ g D Ž⺢. be such that ␸ Ž0. s 0, ␸ ⬘Ž0. s 1. For any j g ⺪, define ␸ j, ␧ in ⺢ by

Ž ᭙␧ gx 0, ␦ j w . Ž ᭙ x g ⺢.

␸ j, ␧ Ž x . s ␸

ž

x y aj

␧

/

.

Taking the constant ␦ j - 1 ensures that suppŽ ␸ j, ␧ . ;x a jy1 , a jq1 w, so that ␸ j, ␧ belongs to D. Using the inequality Ž9. with the family ␸ j, ␧ , we obtain y < 3r2 < u 0 Ž aq C3, j 5 ␸ 5 L2 . j . y u0 Ž a j . F ␧

Finally, letting ␧ tend to 0 proves that u 0 is continuous at the point a j . We shall now prove the boundary conditions X y uX0 Ž aq j . y u0 Ž a j . s k j u0 Ž a j . .

Ž ᭙ j g ⺪.

Let j g ⺪ and ␺ j g D be such that ␺ j Ž a j . / 0 and suppŽ ␺ j . ;x a jy1 , a jq1w. Proceeding as for the inequality Ž9. and using the continuity of u 0 , we obtain X y < ␺ j Ž a j . ⭈ Ž uX0 Ž aq < 5 5 2 j . y u 0 Ž a j . y k j u 0 Ž a j . . F C 3, j ␺ j L .

Ž 10 .

Let ␸ g D Ž⺢., ␸ Ž x . s 1, in a neighbourhood of the origin. Defining the family ␺ j, ␧ in ⺢ by

␺ j, ␧ Ž x . s ␸

ž

x y aj

␧

/Ž

1 q k j ⭈ Ž x y aj . ⭈ H Ž x y aj . . ,

SCHRODINGER OPERATORS IN ¨

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185

the conditions on the parameter are the same as before to ensure that each ␺ j, ␧ belongs to D. Applying inequality Ž10. to this family of functions, we obtain X y 1r2 5 5 2 < uX0 Ž aq < ␸ L Ž⺢ . , j . y u 0 Ž a j . y k j u 0 Ž a j . F C 3, j ⭈ K j ⭈ ␧

where, for any j in ⺪, K j s 1 q < k j < Ž a jq1 y a j . . As ␧ tends to 0, we reach X y uX0 Ž aq j . y u0 Ž a j . s k j u0 Ž a j . .

We now show that ␣␭, j and ␤␭, j satisfy

Ý ␣␭2, j q ␤␭2, j - ⬁. jg⺪

Replacing u 0 by its restrictions in the preceding boundary conditions, we obtain y u 0, j Ž aq j . s u 0, jy1 Ž a j .

du 0, j dx

du 0, jy1

Ž aqj . y

dx

Ž ayj . s k j Pj u 0, j

¦ ¥ Ž a . .§

Ž 11 .

j

Denote e␭ , j s ey␭ Ž a jq 1ya j . A␭ , j s B␭ , j s

aj

Ha

ey␭ Ž a jys.␻ jy1 Ž s . ds

jy1

a jq1

Ha

ey␭ Ž sya j .␻ j Ž s . ds.

j

The equations Ž11. can be rewritten as

M␭ ⭈

⭈⭈⭈ ␤␭ , j

1

⭈⭈⭈ Ž k jr2 ␭ . B␭ , j y A␭ , j

2␭

Ž k jq1r2 ␭ . A␭ , jq1 y B␭ , jq1

0  ␣␭ , j ⭈⭈⭈

s

⭈⭈⭈

0

,

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where M␭ is the following matrix: ⭈⭈⭈ M␭ s



ž

1q

O kj

2␭

kj

/

2␭

ye␭ , j

e␭ , jy1

kj

ye␭ , jy1

2␭

ž

e␭ , j

1q

2␭ ⭈⭈⭈

O

We can write M␭ s R␭ q

1 2␭

kj

/

0

.

S␭ , with R␭ and S␭ defined by

⭈⭈⭈ R␭ s



O

0

ye␭ , jy1

0

0

0 1

0 ye␭ , jq1

1 0

0 0

0

O ⭈⭈⭈

⭈⭈⭈ S␭ s



k j e␭ , j

kj

k jq1

k jq1 e␭ , j ⭈⭈⭈

0

.

The matrix R␭ is explicitly invertible: ⭈⭈⭈ Ry1 ␭ s



0 e␭ , jy1

O

0 0

0 1

1 0

0

e␭ , jq1

0

0

O ⭈⭈⭈

0

.

Recall that if we denote by 5 5 L p the subordinate matrix norms to the vector norms 5 5 p , we have 5 A 5 L ⬁ s sup

Ý < ai j < ,

5 A 5 L 1 s sup

ig⺪ jg⺪

Ý < ai j < ,

and

jg⺪ ig⺪

5 A5 L 2 F 5 A5 L 1 5 A5 L ⬁ .

'

5 L 2 F 2, and 5 S␭ 5 L 2 F 2 5 k j 5 l ⬁ . We can easily prove that 5 R␭ 5 L 2 F 2, 5 Ry1 ␭ ⬁ 5 5 If ␭ ) 2 k j l , then M␭ is invertible and we have 5 M␭y1 5 L 2 F 5 Ry1 5 L2 ␭

1 1y

1 2␭

5 Ry1 5 L 2 5 S␭ 5 L 2 ␭

F

2␭

␭ y 25 kj5 l ⬁

.

L2 Ž⺢.

SCHRODINGER OPERATORS IN ¨

187

Denoting the right member of this inequality by cŽ ␭. and fixing a suitable value ␭0 ) 2 k 5 k j 5 l ⬁ for ␭, we obtain ⭈⭈⭈ ␤␭ 0 , j

 0 ␣␭0 , j ⭈⭈⭈

⭈⭈⭈

c Ž ␭0 .

F



ž / 2 ␭0

2

Ž k jr2 ␭0 . B␭ , j y A␭ , j Ž k jq1r2 ␭0 . A␭ , jq1 y B␭ , jq1 0

0

0

0

⭈⭈⭈

0

.

Ž 12 .

2

For any j in ⺪, simple computations yield 1

A␭0 , j F

ž

'2 ␭ Ha 0

1r2

aj

␻␭20 , jy1 Ž s . ds

/

jy1

,

and the corresponding inequality for B␭0 , j . From Ž12., we deduce that

Ý

␣␭20 , j

q

␤␭20 , j

F

jg⺪

F

c 2 Ž ␭0 .

Ž 2 ␭0 .

ž ž /ž ž / Ý

3

Ž 2 ␭0 .

jg⺪

5 k j 5 2l ⬁

c 2 Ž ␭0 .

Ž 2 ␭0 .

< kj<2

3

Ž 2 ␭0 .

2

q1

aj

Ha

jy1

q 1 5 ␻␭ 0 5 2L2 Ž⺢ . ,

2

␻␭20 , jy1 Ž s . ds

// Ž 13 .

which gives the desired result. The function u 0 satisfies d 2 u 0rdx 2 < ⍀ g L2 Ž ⍀ ., du 0rdx < ⍀ g L2 Ž⺢.. The first property is a consequence of inequalities Ž5., Ž7., and Ž13.. For the second one, we first use Ž6., Ž8., and Ž13. in order to obtain du 0 dx

< ⍀ g L2 Ž ⍀ . .

Then for any ␸ in D Ž⺢ n ., we have du 0

¦ ; dx

,␸

D⬘ D

¦

s y u0 ,

d␸ dx

;

.

L2

Integrating ² u 0 , ddx␸ :L2 by parts on each I j and using the continuity of u 0 at the a j ’s, we obtain du 0

du 0

¦ ; ¦ dx

,␸

D⬘ D

s

dx

<⍀, ␸<⍀

;

L2

F 5 u 0 < ⍀ 5 H 1 Ž ⍀ . 5 ␸ 5 L2 . The conclusion follows. Hence, u 0 g H 1 Ž⺢..

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Let us prove that the sum Ý j g ⺪ k p j < u 0 Ž a p j .< 2 is finite. Let Ž᭙ l g ⺞. ␸ l g D Žx y l y 1, l q 1w. be such that 0 F ␸ l Ž x . F 1, in suppŽ ␸ l ., ␸ l < xy l , l ws 1 and ␸ Xl being uniformly bounded in l . We note u 0, l Ž x . s u 0 Ž x . ␸ l Ž x .. An integration by parts yields, for any l in ⺞, ² Au 0 , u 0, l :L2 Ž⺢ . s

du 0 du 0, , dx dx

¦

l

;

q

L2 Ž ⺢ .

Ý k j u 0 Ž a j . u 0, l Ž a j . . jg⺪

Lemma 1 gives

Ý < k n < < u 0 Ž an . < 2 F C4 5 u 0 5 2H Ž⺢ . . 1

j

j

jg⺪

Therefore,

Ý

k p j ␸ l Ž a p j . < u 0 Ž a p j . < 2 F C 5 u 0 5 2H 1 Ž⺢ . q ² Au 0 , u 0 , l :L2 Ž⺢ .

/

jg⺪

du 0 du 0, , dx dx

¦

y

l

;

.

L2 Ž ⺢ .

We conclude the proof using the monotone convergence theorem for the discrete sum and Lebesgue’s theorem for the terms in between brackets. Finally, we have shown that u 0 g D. Thus Ž A, D, L2 Ž⺢.. is a self-adjoint operator. Its uniqueness follows from a classical result about self-adjoint operators associated with quadratic forms Žsee w8, Theorem VIII.15x.. From inequality Ž1., we can find a constant C ) 0 such that, in DomŽ B ., the following holds:

Ý < k n < < u Ž an . < 2 F 5 ⵜu 5 2L

2

j

j

q 2C 5 u 5 2L2 .

Ž 14 .

jg⺪

Hence B is lower semi-bounded by y2C in DomŽ B .. There remains to prove that ŽDomŽ B ., 5 ⭈ 5 B . is closed Žwith 5 ⭈ 5 B s Ž B Ž⭈, ⭈ . q Ž2C q 1.5 ⭈ 5 2L2 .1r2 .. The norms 5 ⭈ 5 B and 5 ⭈ 5 H 1 Ž⺢ n . are not equivalent, as no hypothesis was made on the positive subsequence  k p j 4j g ⺪ of  k j 4j g ⺪ , with respect to 1 k jrŽ a jq1 y a j .. However, using inequality Ž1. with ␮ s 1 q 2C we prove that 5 ⭈ 5 2B G min

ž

1

,

4␲ 2

2 2C q 1

/

5 ⭈ 5 2H 1 Ž⺢ . .

SCHRODINGER OPERATORS IN ¨

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189

Let  u m 4m g ⺞ be a Cauchy sequence in DomŽ B . for 5 5 B . It converges to a function u in Ž H 1 Ž⺢., 5 5 H 1 . and we have, for any j in ⺪, lim < u m Ž a j . y u Ž a j . < s 0.

Ž 15 .

mªq⬁

Besides, using Ž14., we have

Ý < k n < < u Ž an . < 2 - q⬁. j

j

jg⺪

On the other hand, the sequence Žindexed by m.  k p j < u mŽ a p j .< 2 4j g ⺪ is a Cauchy sequence in l 1 Ž⺪., since

Ý k p < um Ž ap . < 2 F 5 um 5 B . j

j

jg⺪

From Eq. Ž15. we deduce that the limit sequence is, in fact, equal to  k p < uŽ a p .< 2 4j g ⺪ and that the latter belongs to l 1 Ž⺪.. Consequently, a j j Cauchy sequence in ŽDomŽ B ., 5 5 B . is convergent in this set, which concludes the proof. Remarks. This method applies in the same way to a finite sequence of points  a j 4j g ⺪ . In this case the only requirement is that the sequence of weights  k j 4j g ⺪ should be bounded. A similar result can be constructed in ⺢ q with a potential localized on parallel hyperplanes. Its proof follows the same principle modulo a few technical arrangements: The adequate operator is A q u s yÝ j g ⺪ EjŽ RjŽ ⌬ u.., where Rj and Ej relatively are the restriction and extension operators to the open set ⍀ q, j s ⺢ qy 1 = x a j , a jq1w. Denote ⍀ q s Dj g ⺪ ⍀ q, j . The operator A q is to be studied in the set

½

D s H 1Ž⺢q . l H 2 Ž ⍀q . l u

½

l u Ž ᭙ j g ⺪.

⭸u ⭸ Njq

q

Ý k p 5 ␯p u 5 2L Ž⺢ 2

j

j

qy 1

.

-⬁

jg⺪

⭸u ⭸ Njy

s yk j ␯ j u

5

5

with ␯ j the trace operator on the hyperplane  x q s a j 4 and the normal derivatives

⭸u ⭸ Njq

s y␯ j

ž

⭸u ⭸ xq

<⍀j ,

/

⭸u ⭸ Njy

s ␯j

ž

⭸u ⭸ xq

< ⍀ jy1 .

/

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A partial Fourier transformation in the first q y 1 variables, denoted by ‘‘n’’, is applied to the equation A q u 0 s ␻ g L2 Ž⺢ q . in order to obtain the equivalent of Eq. Ž3. in ⺢ q : 2

2

Au ˆ0 q Ž 2␲ .  ␰ ⬘ 4 uˆ0 s ␻ˆ␭ g L2 Ž ⺢ q . , with

Ž᭙u g D.

Asy

Ý Ej jg⺪

⭸ 2u

ž ž // Rj

⭸

 ␰ ⬘ 4 s Ž ␭2 q < ␰ < 2 .

and

x q2

1r2

.

The solution of this equation is as follows: u ˆ0 Ž ␰ ⬘, x q . s

1

xq

H 4␲ ² ␰ ⬘: a y

1

ey2 ␲ ² ␰ ⬘:Ž syx q.␻ ˆ Ž ␰ ⬘, s . ds

jq1

xq

H 4␲ ² ␰ ⬘: a

ey2 ␲ ² ␰ ⬘:Ž x qys.␻ ˆ Ž ␰ ⬘, s . ds

j

q ␣ j Ž ␰ ⬘ . ey2 ␲ ² ␰ ⬘:Ž x qya j. q ␤ j Ž ␰ ⬘ . ey2 ␲ ² ␰ ⬘:Ž a jq1yx q. . Fix O , a bounded open set in ⺢ qy 1, and ␺ , a function in D Ž O .. The function of one variable ² u ˆ0 , ␺ :L2 Ž⺢ qy 1 . has the same regularity and satisfies the same trace equalities as in the monodimensional case. Hence, we prove local regularity for u 0 . Global regularity is obtained by showing that Ý j g ⺪ ␣ j Ž ␰ ⬘. 2 q ␤ j Ž ␰ ⬘. 2 satisfies some integration properties. The method is the same as in the monodimensional case Žinversion of an infinite system in l 2 Ž⺪.. and leads to the inequality

Ý ␣ j Ž ␰ ⬘. jg⺪

2

2

q ␤j Ž ␰ ⬘. F

c 2 Ž ␭.

Ž 4␲  ␰ ⬘ 4 .

3

ž

5 k j 5 2l ⬁

Ž 4␲ .

2

q 1 5 ␻␭ Ž ␰ ⬘, ⭈ . 5 2L2 Ž⺢ . ,

/

with c Ž ␭. s

2␲␭

␲␭ y 5 k j 5 l ⬁

.

There remains to prove the uniqueness of A q with Theorem VIII.15 of w8x, using a q-dimensional version of inequality Ž1. in order to conclude.

REFERENCES 1. S. Albeverio, F. Gesztesy, R. Hoegh-Krohn, and H. Holden, ‘‘Solvable Models in Quantum Mechanics,’’ Springer-Verlag, Berlin, 1988.

SCHRODINGER OPERATORS IN ¨

L2 Ž⺢.

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