Distribution Theory for Discontinuous Test Functions and Differential Operators with Generalized Coefficients

Transcription

1 JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 1, ARTICLE NO 56 Distribution Theory for Discontinuous Test Functions an Differential Operators with Generalize Coefficients P Kurasov* Department of Mathematics, Ruhr Uniersity-Bochum, 4478 Bochum, Germany; Department of Mathematics, Lulea Uniersity, S Lulea, Sween; an Department of Mathematical an Computational Physics, St Petersburg Uniersity, St Petersburg, Russia Submitte by Joseph A Ball Receive September 14, 1994 Investigation of the ifferential operators with the generalize coefficients having singular support on a isjoint set of points requires the consieration of the istribution theory with the set of iscontinuous test functions Such a istribution theory for test functions having iscontinuity at one point is evelope A fourparameter family of Schroinger operators, forme by the operators with singular potential, singular metrics an singular gauge fiel, is consiere It is prove that this family of singular interactions escribes all possible selfajoint extensions of the secon erivative operator efine on the functions vanishing in a neighbourhoo of the point Approximation by operators with smooth coefficients is is - cusse 1996 Acaemic Press, Inc 1 INTRODUCTION Differential operators with coefficients equal to the generalize functions appear in ifferent problems of applie mathematics an mathematical physics These operators are closely relate to exactly solvable problems in quantum mechanics, atomic physics, an acoustics 3, 11 An important class of such operators is forme by the ifferential operators with the coefficients having singular support on a isjoint set of points Such operators will be name operators with singular interactions in the future Every such selfajoint operator can be escribe as an extension of the symmetric operator efine by the same ifferential expression on the set of functions with the support isjoint from the singular support of the coefficients All selfajoint extensions of these operators will be name selfajoint perturbations 97-47X96 $18 Copyright 1996 by Acaemic Press, Inc All rights of reprouction in any form reserve

2 98 P KURASOV The selfajoint perturbations of the ifferential operators can be stuie with the help of the von Neumann extensions theory for symmetric operators or the Krein theory The operator with singular interaction was introuce by Fermi The first mathematically correct investigation of the operator with singular interaction was carrie out by Berezin an Faeev 6 A Laplace operator in L ŽR 3 with singular interaction at one point has been consiere The most complete collection of solve problem involving singular interactions in imensions one, two, an three can be foun in the monograph by Albeverio et al 3 The monograph by Demkov an Ostrovskii 11 contains numerous applications of the escribe Hamiltonians to physics Generalizations of the singular interactions involving aitional space of interaction have been introuce by Pavlov 19 The singular interactions with support on the low-imensional manifols have been stuie in Differential operators in imension one are investigate in this paper The relations between the singular interactions an selfajoint perturbations for the first an secon erivative operators have been iscusse 35, 9, 1, 114, 1618, 1 The set of the selfajoint perturbations an singular interactions for the secon orer ifferential operator in imension one is much wier than in the three imensional spaceit is escribe by four real parameters The four parameter family of selfajoint perturbations has been stuie recently in 1,, 8, 9 The aim of this paper is to clarify the relations between the singular interactions an selfajoint perturbations for the secon erivative operator Four important questions arise in this context: 1 How oes one escribe the selfajoint perturbation corresponing to the singular interaction? Is it possible to escribe all selfajoint perturbations by the singular interactions? 3 How oes one escribe all singular interactions leaing to the selfajoint operators? 4 How oes one approximate the operators with singular interactions by operators with smooth coefficients? The answers to these questions require etaile analysis of the istribution theory 15 Consier for example the following first orer ifferential operator with the singular interaction M ix X X, where is the Dirac elta function, X R This ifferential expression is well-efine on the functions f W 1 Ž R because these functions are continuous at the origin But the range of an operator efine in such a way is not containe in L Ž R an consequently the operator is not selfajoint The selfajoint operator corresponing to this ifferential expression can be efine on

3 DISTRIBUTION THEORY 99 the iscontinuous functions only The prouct of the elta function an iscontinuous function is efine only in the framework of the istribution theory for the iscontinuous test functions Then the natural extension of the elta function to the set of iscontinuous test functions, Ž ŽŽ Ž, can be use Using this extension of the elta function, the operator MX is efine as the operator of the first erivative with the omain of functions, satisfying the bounary conition at the origin Ž ix Ž Ž ix Ž Žsee 9, Section 6 It is possible to consier the singular interactions with the coefficient X equal to using the projective space formalism Žsee for etails 7, 17 A similar problem appears uring the consieration of the following secon erivative operator with the singular interaction Ž1 X X 1 1 L X X, X, X R, Ž 1 1 x the Schroinger operator with the generalize potential The corresponing selfajoint operator cannot be efine on a space consisting of functions which are continuously ifferentiable at the origin Two examples consiere here show that the singular interactions for the first an secon erivative operators cannot be efine in the frame- Ž Ž work of stanar istribution theory except the operator x X 1 We present an prove here only the most important facts from the istribution theory for the iscontinuous test functions Žsee 7, 17 for etails The operators with the singular interactions will be efine on the basis of the evelope technique A positive answer to the secon question, originally pointe out by Seba 1, can be given only if a wier family of singular interactions is consiere The family Ž 1 is escribe by two real parameters, but the family of the corresponing selfajoint perturbations is escribe by the unitary matrix, which contains 4 real parameters 9, 1 Two other families of singular interactions have been stuie in 9, 1 But these families of singular interactions o not cover all selfajoint perturbations We are going to present here the four parameter family which escribes the whole class of selfajoint perturbations Žsee also 1, This family is forme by the operator with the singular potential Ž 1, singular metrics, an singular gauge fiel We exten the family of operators by consiering the parameters from the projective space P 4 It is prove that the set of all selfajoint perturbations of the secon erivative operator coincies with the family of operators with the singular interactions We give an answer to the thir question in the framework of the istribution theory evelope here It is prove that only the four parameter family of singular interactions consiere leas to selfajoint operators

4 3 P KURASOV We show that every secon orer ifferential operator with a singular interaction can be approximate by a certain sequence of ifferential operators with the interaction efine by continuous short range coefficients DISTRIBUTION THEORY FOR DISCONTINUOUS TEST FUNCTIONS 1 Test Functions an Distributions We introuce the set K of test functions with a possible iscontinuity at the origin DEFINITION 1 The set of test functions K is the set of all functions with compact support on the line Ž, having uniformly boune erivatives of any orer outsie the origin The support of these functions is not necessarily separate from the origin Functions from K can be iscontinuous at the origin, but the limits of the functions an all erivatives from the left an from the right of the point zero exist an are finite Convergence in this space is efine as follows: DEFINITION A sequence 4 n of functions in K is sai to converge to a function K if an only if Ž 1 There exists an interval outsie which all the functions n vanish; Ž Žk The sequence 4 n of erivatives of orer k converges uni- formly outsie the origin on this interval to Ž k for every k Distributions corresponing to these test functions can be efine in the stanar way: DEFINITION 3 A istribution f from K is a linear form on K such that for every compact set B R there exist constants C an n such that f Ž C sup, K, suppž B Ž x Ý ž / n x Stanar methos of the theory of istributions can be applie to stuying the set K We are going to compare these istributions with the istributions corresponing to the test space D C Ž R The set of istributions for the test functions D is usually enote by D The

5 DISTRIBUTION THEORY 31 ifference between the spaces K an D is local an relate to the special behaviour of the test functions from K near the origin Generalize Deriatie The erivative of any istribution in K will be efine using the formula, which is vali for any istribution efine by the function f C ŽR 4 DEFINITION 4 Let f K, be a test function from K, then the erivative Dx f of the istribution f is efine by the equation / Ž Dx fž fž, Ž 3 x where the erivative of the test function Ž x is calculate in the classical sense at every point x outsie the origin We note that accoring to our efinition the erivative of the test function Ž x oes not contain any elta-functional singularity at the origin, even if the test function is iscontinuous there This efinition of the erivative allows us to calculate the erivative of any istribution from K The efinition of the erivative involving the elta function woul restrict the class of ifferentiable istributions The erivative of a istribution in K oes not coincie with the erivative efine in the classical sense For example, the erivative of the constant istribution is not equal to zero We shall use in future the Ž1 notation x for the classical erivative an Dx - for the generalize erivative in K LEMMA 1 The eriatie of the constant istribution c is equal to the istribution c, where istribution is efine by the formula Ž Ž Ž Ž 4 Proof Let K, then Ž Dc x Ž cž x / ž H H x Dx / c x x x x cž Ž Ž

6 3 P KURASOV Higher erivatives of the constant istribution can be calculate in a similar way: n1 n1 n n1 x ž ž / ž / / x x Dc 1 c 5 All erivatives calculate here are istributions vanishing on the set of test functions infinitely ifferentiable at the origin 3 Delta Function an Its Deriaties We are going to iscuss the efinition of the elta function with support at the origin The elta function is efine usually as a functional on the set of C functions by the following formula: Ž Ž Ž 6 It is obvious that this linear functional can be extene to the set of all functions continuous at the origin using the same formula Ž 6 But this formula cannot be use for the elta function in K since the value of a test function from K at the origin is not efine The elta function is an even istribution We can use this property to calculate the elta function on the iscontinuous test functions If the istribution feven is even, then the following formula is vali for every test function : ž / ž / Ž x Ž x Ž x Ž x f Ž f f even even even evenž / Ž x Ž x f Every even function from K is continuous at the origin an formula Ž 6 can be use to calculate the value of the elta function We shall use the following efinition in the future: DEFINITION 5 The elta function in K with support at the origin is a linear functional on K efine by the formula: Ž Ž Ž The approximative elta-function sequence can be efine by any even function V x C, H VŽ x x1 The sequence of functionals corre-

7 DISTRIBUTION THEORY 33 sponing to the functions V x 1 V x converges to the eltafunction in the space K when, ž / 1 x lim V Ž lim H V Ž x x lim H VŽ x Ž x x H VŽ x Ž x VŽ x Ž x H Ž Ž The erivatives of the elta function can be easily calculate using Definition 4: n n Ž x Ž Ž x Ž n DxŽ Ž 1 Ž 7 The elta function an its corresponing erivatives so efine possess the same properties as the stanar elta function with respect to the inversion an scaling transformations The inversion an scaling transformations are efine on the test functions as follows Ž I Ž x Ž x ; Ž Sc Ž x Ž cx Similar transformations for the istributions are efine in the stanar way 1 Ž I f Ž f Ž I ; Ž Scf Ž f S1c Ž 8 c n ž / LEMMA The nth eriatie of the elta function is a homogeneous n istribution of orer n 1:S cd x 1c n1 Dx n The nth eriatie of the elta function is an een istribution if n is an een number an an o n istribution if n is an o number: I D Ž 1 n D n x Proof The proof can be carrie out by irect calculations We note that the elta function in D possesses the same properties with respect to the inversion an scaling transformations Moreover, the following lemma can be prove 17 LEMMA 3 Let the istribution f from K n n 1 be equal to D for the test functions from C R ; x x

8 34 P KURASOV be a homogeneous istribution; 3 be an een istribution if n is an een number an an o istribution if n is an o number; then this istribution coincies with the nth eriatie of the elta function on K: f Dx n 4 Generalize an Classical Deriaties We efine by K loc the set of all boune functions which are infinitely ifferentiable outsie the origin with possibly a jump iscontinuity at the origin We suppose that the limits of all erivatives from both sies of the origin are finite Distributions an have unique extension to this class of test functions Two ifferent erivatives are efine such functions: the erivative calculate as an orinary function at every point outsie the originthe classical erivative Ž x ; the erivative calculate as a istributionthe generalize erivative Dx Ž1 The ifference between these two erivatives is illustrate by the following LEMMA 4 The generalize eriatie Dx an the classical eriatie Ž x of an arbitrary function K are relate as loc Dx Ž Ž, Ž 9 x where is the elta function an is the eriatie of the unit istribution Proof The generalize erivative for any istribution K loc acting on an arbitrary test function K is equal to / H DxŽ ž Ž x Ž x x Ž x Ž x x x x x H Ž Ž H Ž x Ž x xž Ž x H Ž x Ž x x x Ž Ž Ž Ž Ž x

9 DISTRIBUTION THEORY 35 The last lemma shows another time that the erivative of the istribution in K oes not coincie with the erivative in D The ifference vanishes on the test functions from D Similar results can be prove for the secon erivative: LEMMA 5 The secon generalize eriatie Dx an the secon classical Ž eriatie x of arbitrary function K are relate as follows: loc Dx Dx Dx DxDx 1 x 5 Prouct of Distributions The prouct of two istributions can be efine if one of these istributions is a function from K loc DEFINITION 6 K loc The prouct of any istribution f K an any function is efine as fž fž fž, where K is an arbitrary test function This efinition is correct because the prouct of K loc an any test function from K is a function from K again We have in particular for the elta function an any K loc, K, Ž Ž Ž Ž Ž Ž Ž Ž Ž Ž Ž Ž Ž We use in the last formula a natural extension of the efinition of the istributions an to the set K loc The formula for the erivative of the prouct of two istributions involves their classical an generalize erivatives LEMMA 6 Let K loc, f K, then DxŽ f Dx ff Ž 1 x

10 36 P KURASOV Proof We have for any K / ž / ž / ž / ž / ž / DxŽ fž fž f f Ž f x x x x Dx fž f Ž Dx fž f Ž x x The prouct of any function K loc an the erivative of an arbitrary istribution f K can be calculate in accorance with the formula / Dx fdxž f ž f Ž 13 x The following formula can be erive using Eqs 11 an 13 for the elta function an arbitrary K : loc Dx Ž Ž DxŽ DxDxŽ Dx Ž SECOND-ORDER DIFFERENTIAL OPERATOR WITH SINGULAR INTERACTION 31 Selfajoint Perturbations We are going to stuy now the selfajoint perturbations of the secon erivative operator Dx in imension one A selfajoint perturbation at the origin for this operator is a selfajoint extension of the symmetric operator 4 L D, DomŽ L W Ž R, Ž Ž Ž 15 x Ž The ajoint operator L * D is efine on the omain Dom L * x W ŽR 44 LEMMA 7 Eery selfajoint extension of the operator L coincies with the operator L *, restricte to the set of functions, satisfying the bounary conitions at the origin of one of the types ž / ž / c ž / Ž Ž i a b Ž 1 J, Je Ž 16 Ž Ž

11 DISTRIBUTION THEORY 37 with the real parameters,,a,b,c,r such that a bc 1; h f Ž h 1 f Ž Ž, Ž 17 ½ h fž h 1 fž Ž 1 with the parameters h h, h1 from the projectie space P This lemma has been prove in 9, 1 Selfajoint operators, escribe by the bounary conitions of the first type, will be name connecte because these conitions connect the bounary values of the function on the left an right halflines Selfajoint operators of the secon type will be name separate These operators are equal to the orthogonal sum of the secon erivative operators efine on the halflines 3 Singular Interactions: Four Parameter Family We are going to stuy now the four parameter family of the secon erivative operators with singular interactions The selfajoint extensions, corresponing to the family, will be calculate See Section 36 for the physical interpretation of the parameters THEOREM 1 The secon orer ifferential operator with the singular interaction at the origin LX Dx Ž 1X4 idxž X3 ix4 Ž1 X1 Ž X ix3 Ž1, Ž 18 4 X X 1, X, X 3, X4 R, coincies with the secon eriatie operator D efine on the omain of functions W ŽR 4 x, satisfying the following bounary conition at the origin: 1 Ž Ž x Ž X X X X 4X Ž ix X X X Ž ix X X X 4X1 X X1X4X Ž ix X X X Ž ix X X X Ž, Ž 19 Ž x

12 38 P KURASOV if ix X X X ; Ž x 1 X Ž Ž ž /ž / X X 4 Ž Ž x if 4 X1X4X, X3, X4 ; X1 Ž Ž 3 x 4 Ž 1 Ž if X, X, X ; 3 4 Ž 4 X 1 Ž Ž Ž x 4 if X, X, X 3 4 Proof The omain of the operator LX coincies with the set of functions L Ž R, which are solutions of the equation LX f for some function f L Ž R We consier the last equation in the generalize sense with the set of the test functions D 1 Consiering this equation for the test functions with the support separate from the origin we euce that W ŽR 4 The functions from this Sobolev space are continuous outsie the origin an have continuous boune first erivative there The ifferential expression Ž 18 is efine on such functions The istributions an an their first erivatives can be efine on the functions from W ŽR 4 as follows: Ž Ž Ž Ž Ž1 Ž ; Ž ; Ž Ž Ž ; Ž1 Ž Ž Ž Ž 1 The set of test functions is chosen equal to D because it forms a ense subset of the omain of the secon erivative operator Dx Different choice of the test space woul lea to a nonselfajoint operator or an operator with the point interaction even if all coefficients X, X, X, X are equal to zero 1 3 4

13 DISTRIBUTION THEORY 39 Formulas Ž 11, Ž 14 efine the prouct of the elta function or its erivative an any function from W ŽR 4 The istribution LX K, W ŽR4 can have singular support only at the origin The singular term is equal to the linear combination of the istributions an an their first erivatives The istributions an Ž1 vanish on the test functions from D Then the istribution LX is equivalent to some function from L Ž R if an only if the coefficients in front of the elta function an its erivative are equal to zero We get the following linear system: : Ž1 Ž X1Ž Ž XiX3 Ž1 Ž Ž1 : Ž Ž XiX3 Ž X4 Ž1 Ž X1 XiX3 X1 X ix3 ž XiX3 X4 X ix3 X4 / Ž Ž x Ž Ž x Ž 3 The rank of the matrix in the last equation is equal to an it efines the two imensional subspace in the four imensional space of the bounary values Ž Ž Ž, x Ž,Ž, Ž x Ž We write conitions Ž 3 in the form: X X ix Ž 3 4 ž / X ix X 1 X1 XiX3 1 Ž XiX3 X4 Ž 1 ž / Ž 4

14 31 P KURASOV The eterminant of the matrix in the left-han sie of the last equation is equal to 1 Ž ix X X X 4 If, then the matrix is invertible an these bounary conitions can be written in the form Ž 19 Consier the case It follows that the coefficient X3 is equal to zero The bounary conitions Ž 3 can be written as X X 1 1 Ž x X4 X4 Ž x X1 X1 Ž X X ž Ž / 1 1 The eterminant of the matrix in the left-han sie of the last equation is equal to X 4If X4, then the inverse matrix can be calculate an the bounary conitions have the form Ž Consier now the case, X4 It follows that X Bounary conitions, efine by X, X4 an X, X4 are equal to Ž 1 an Ž corresponingly All possible values of the coefficients X 1, X, X 3, X4 R have been consiere Moreover, the image of every function W ŽR 4 satisfying these bounary conitions is equivalent to a certain function from L Ž R on the set of test functions from D This completes the proof of the Theorem The bounary conitions 19 can be consiere for infinite values of the parameters X, X, X, X A goo parametrization for this case can be one by using the formalism of projective space We are going to parameterize all singular interactions by X P 4 We get the bounary conitions for all elements from the projective space with the nonzero component X with the help of the stanar embeing of the space R 4 4 into the space P : Ž X, X, X, X Ž 1, X, X, X, X The bounary conitions corresponing to the other elements from the projective space

15 DISTRIBUTION THEORY 311 will be efine using the homogenize analog of the linear system 3 X1 X X ix3 X1 X X ix3 ž X X ix X X X ix X / Ž Ž x Ž Ž x Ž 5 We shall use the following efinition in the remainer DEFINITION 7 The algebraic set W is the set of elements from the projective space P 4, satisfying the following three algebraic equations simultaneously: X ; Ž 6 X3; Ž 7 Ž XiX3 X1X4X Ž 8 THEOREM Eery element X from the projectie space P 4, which oes not belong to the algebraic set W, etermines a unique selfajoint extension L X of the operator L, escribe by the following bounary conitions: 1 Ž Ž x Ž X X X X X 4X X Ž X ix X X X Ž X ix X X X 4XX1 XX X1X4X Ž X ix X X X Ž X ix X X X Ž, Ž 9 Ž x

16 31 P KURASOV 4 if X G X ix X X X ; Ž x 1 XX Ž Ž 3 ž /ž / X X X 4 Ž Ž x if X G Ž X ix X X X, X, X, X ; X Ž X 1 Ž 3 x Ž 31 Ž if X G 3 X ix3 X1X4X, X X, X, X4, X ; 4 3 Ž 4 Ž 3 4X Ž X1Ž x if X G Ž X ix X X X, X X, X, X, X ; Ž Ž 5 Ž Ž x x if X G Ž X ix X X X, X Proof The rank of the matrix in the linear system Ž 5 is equal to 1 if an only if X an X ix3 X1X4X If the rank of the matrix is equal to, then the homogeneous linear system efines two ifferent bounary conitions as was shown uring the proof of the Theorem 1 Corresponing bounary conitions are the homogenize analogs of the bounary conitions Ž 19 Ž an cover the cases 14 of the present theorem Consier the case when the rank of the matrix is equal to 1, ie, when conitions Ž 6 an Ž 8 are satisfie The unique bounary conition efines a certain linear subset Q in the omain of the ajoint operator L * The operator L * restricte on this subset is not symmetric We are going to prove that if the aitional conition X3 is satisfie, then there exists only one selfajoint extension of the operator L with omain equal to a subset of this linear set Q

17 DISTRIBUTION THEORY 313 The unique bounary conition, efine by the system 5, is equal to Ž X ix Ž Ž Ž X Ž Ž Ž Ž Every separate selfajoint perturbation in this case shoul be escribe by the Dirichlet bounary conitions It is possible only if X4 The last equality together with the conitions Ž 6, Ž 8 leas to the equation X3 X, which has only trivial solution X3 Thus, no separate selfajoint perturbations correspon to such an element X Consier the connecte selfajoint perturbations Substitution of the bounary conition Ž 16 into the equation Ž 33 leas to the equation Ž X ix Ž a Ž b Ž X Ž cž Ž 3 4 Ž X ix e i Ž X e i Ž, 3 4 which shoul be satisfie for all values of an It follows that the real coefficients a, b, c, are solutions to the following linear system with the real coefficients: X X 4 X cos X3 sin a X3 b X sin X3 cos X X4 c X4 cos X X sin 3 4 Coefficients a, b, c, can be calculate if X : 3 X X4 a sin cos ; b sin ; X X 3 3 X X3 X c sin ; cos sin X X X Then a bc 1 sin an it is equal to one if an only if The matrix J 1 ž 1/ accomplishes the proof of the theorem etermines the unique selfajoint operator It

18 314 P KURASOV THEOREM 3 Eery element X W etermines the families of selfajoint extensions of the operator L, escribe by the following bounary conitions: 1 if X1, X4 then Ž a Ž a1 Ž X 1 Ž X1 Ž x Ž a1 a x X X, or ar, Ž 34 X Ž X 1 Ž x ; Ž 35 X Ž X 1 Ž x if X, then 1 Ž Ž 1 b ž /, br, Ž 36 Ž 1 Ž x x or Ž x ; Ž 37 Ž x 3 if X, then 4 Ž Ž 1 ž /, cr, Ž 38 Ž c 1 Ž x x or Ž ½ Ž Ž 39

19 DISTRIBUTION THEORY 315 Proof We are going to consier the three ifferent cases separately 1 Suppose that X1, X4, X W It follows that X The linear system Ž 5 efines the unique conition / X1Ž Ž Ž Xž Ž Ž x x Every connecte selfajoint perturbation, corresponing to X, is efine i a b by the matrix J e ž / This matrix shoul be real Ž an the c coefficients shoul satisfy the following liner system: X1 c Ž a 1 ax cx X X 1 1 ½ bx1 X X X b Ž 1 The conition a bc 1 leas to the equation a Then the matrix J, corresponing to the element X, shoul be of the form Ž 34 Every such matrix efines the selfajoint perturbation Every separate selfajoint perturbation corresponing to the element X is efine by the bounary conitions Ž 35 Suppose that X W, X1, then X The unique bounary conition, efine by X, is equal to X1 Ž Ž x x This bounary conition leas to the matrices J of the following type: ž / 1 b J, b R 1 Corresponing separate selfajoint perturbations are efine by the Neumann bounary conitions 3 The case X W, X4 can be consiere in a similar way Theorem 3 is prove Thus the elements from W o not etermine the selfajoint perturbation uniquely Any operator from the corresponing family can be use to escribe the singular interaction Theorems an 3 cover all possible values of X from the projective space

20 316 P KURASOV LEMMA 8 The sets G 1, G, G 3, G 4, G 5, an W coer the projectie space P 4 We note that the element X cannot be uniquely efine by the omain of the operator For example, elements with X, X X1X4X3 correspon to the same selfajoint operator, efine by the bounary Ž Ž ž / ž / conitions 33 Classification of the Selfajoint Perturbations We are reay now to prove our main result THEOREM 4 The set of all selfajoint perturbations of the secon eriatie operator in L Ž R coincies with the family of operators with the singular interactions L, X P 4 4 X Proof Every operator LX is efine as the restriction of the secon erivative operator in W ŽR 4 on a certain linear set The bounary conitions efining the operators L are of the type Ž 16 or Ž 17 X It follows that every operator LX is a selfajoint extension of the operator L We have to prove only that every such extension can be escribe by certain singular interaction Consier first the arbitrary connecte perturbation, efine by a certain matrix J Ž 16 We are going to use the homogenize analog of the conitions Ž 4 If the element X efines the bounary conitions Ž 16, then the following equation is fulfille: ž 3 4 / X1 XXiX3 ž XXiX3 X4 / X1 X X ix3 i a b X X ix X c e ž / The last equation can be written as a 4 5 homogeneous linear system: e c e a1 e c ie c X 1 e e b e 1 ie i e a e a1 ie a i e c X e b e b ie b e 1 X i i i i i i i i X i i i i i i i i 3 X 4 Ž 4 Let us enote by, j, 1,, 3, 4, the eterminants of the 4 4 matrij

21 DISTRIBUTION THEORY 317 ces obtaine from the 4 5 matrix by erasing the jth column These eterminants are equal to i ie cos a ; 18ie i cž cos a ; 4ie i Ž a Ž cos a ; 38isin e i Ž cos a ; 48ie i bž cos a All the eterminants are equal to the prouct of the phase factor ie i an a certain real factor If, then the rank of the 4 5 matrix is equal to 4 The solution Ž i i i i i of the system 4 is equal to ie, ie 1, ie, ie 3, ie 4, i 4 ie 5 P This element oes not belong to W an it efines the selfajoint perturbation uniquely This perturbation necessarily coincies with the one efine by the matrix J 4 If, then the element, c, a cos, sin, b P is a solution of the linear system If sin, then this element oes not belong to W an consequently efines the matrix of bounary conitions J Consier the case, a The set of all such matrices is covere by the families a b Ž a1, a, br, b a b or 1 ½, cr 5 c 1 ž / These matrices can be escribe by the singular interactions X from the algebraic set W Both families are covere by the bounary conitions Ž 34, Ž 36, an Ž 38 It is prove that every connecte selfajoint perturbation is efine by a certain singular interaction Consier now the separate perturbations efine by the bounary conitions 17 Suppose that both zero components of the elements h

22 318 P KURASOV are not equal to zero, h, h The coorinates of the element X can be calculate h1 h1 XX4 h h ž / h1 h1 4XX3 ž h h / If h h as elements of P 1, then the elements X from G will efine the bounary conitions The coorinate X3 can be chosen equal to zero The first coorinate shoul be calculate from the conition 4 X X1X4 Ž X X 4X X 1 X 4 The case h h is escribe by the elements of W The bounary conitions Ž 35, Ž 37, Ž 39 cover all conitions of this type If h, then the element X Žh,4h,h,, 1 efines such separate bounary conitions If h, then the bounary conitions are efine by the element X Žh, 4h, h,, 1 The theorem is prove 34 Complete Description of the Singular Interactions We are going to prove here that only the consiere four parameter family of singular interactions can be escribe by the selfajoint operators in the framework of the evelope approach Every secon erivative operator with singular interaction having support at the origin has the following form: ž / ž / N N1 N n Žn n Žn n Žn x Ý x Ý 1 Ý n n n LD 1 a id a a This ifferential expression is efine on the functions from W ŽR 4 only if the coefficients a n, a1 n, a n, n, 3, 4,, are equal to zero Consier the formal operator L Dx Ž 1a a 1 Ž1 idxž a1 a1 1 Ž1 a a 1 Ž1 If W ŽR 4, then the singular part of the istribution L is equal to the linear combination of the istributions an an their first three erivatives This istribution is equivalent to a function from L Ž R on the set of test functions D only if the coefficients in front of the function

23 DISTRIBUTION THEORY 319 an its erivatives are equal to zero We get the following linear system: Ž1 a 1 Ž 1 1 aia1 a Ž1 Ž 1 1 ia1 a ia1 1 Ž a a 1 1 Ž This linear system efines a selfajoint operator only if its rank is equal to Thus the following conitions shoul be satisfie a 1, a ia 1 1 Ž 41 The bounary conitions efine by the linear system can be written as ž / ž / ž / Ž ia1 a 1 ia1 1 Ž Ž1 1 Ž1 a a Ž These bounary conitions efine a symmetric operator if an only if the coefficients an k satisfy the following homogeneous linear system: a a, a ia a, ia 1 ia These equations together with the equations 41 lea to the following conitions on the coefficients a, a, a R: a 1 ; a a Such coefficients escribe the four parmeter family of singular interactions consiere in Section 3 The following theorem is proven THEOREM 5 The set of selfajiont secon eriatie operators with singular interaction of finite strength coincies with the four parmeter family of operators L, X R 4 4 X 35 Approximation by Operators with Smooth Coefficients Every secon orer ifferential operator with singular interaction L X, 4 X X 1, X, X 3, X4 R, can be approximate by the secon orer ifferential operators with smooth coefficients The approximative opera-

24 3 P KURASOV tor can be chosen in the form L D 1XV Ž x id XV Ž x ix V Ž1 Ž x x 4 4 x XV 1 1 Ž x XV Ž1 Ž x ix3v3 Ž1 Ž x, where V Ž x i, i1,,, 4, are even continuously ifferentiable elta- functional sequences Ž may be ifferent constructe in Section 3: VŽ x i C R V Ž 1 VŽ x i The linear operators L are efine on the omain DomŽ L W ŽR 4 The sequence of the linear operators converges in the weak operator topology to the operator Ž1 Ž1 LX Dx Ž 1X4 idx X3 ix4 X1 Ž X ix3 X with the omain DomŽ L W ŽR 4 It is enough to show that Ž k Ž n Žk Žn D V x converges to D Ž x x x in K for all k, n N Let W ŽR 4, K, then Žk Žn kn Žk Ž n kn Žk Ž n Ž Dx V Ž Ž 1 Vž Ž / Ž 1 ž Ž / Ž Dx Žk Žn Ž Ž The operator L on the omain Dom L W ŽR 4 X X is not selfajoint This operator is an extension of the selfajoint operator L X Thus every operator with the singular interaction can be approximate by a sequence of selfajoint operators Ževery operator L,, is selfajoint on the omain W Ž R The convergence shoul be consiere in the sense of linear operators 36 Seeral Examples We are going to iscuss the interpretation of the parameters X 1, X, X 3, X 4, efining the four parameter family of singular interactions Three ifferent subfamilies of the operators, which appear in ifferent problems of mathematical physics, will be consiere Let us stuy first the two imensional subfamily of the Schroinger operators with the generalize potentials L D X X Ž1 Ž 4 X1X x 1 Every such operator coincies with the secon erivative operator efine on the omain of functions from W ŽR 4 satisfying the bounary

25 DISTRIBUTION THEORY 31 conitions X Ž Ž X Ž 43 Ž 4X1 X Ž x x 4X X The regularize Schroinger operator with the singular Gauge fiel Ž id X Ž X is the operator x 3 3 L D ix D Ž1 Ž 44 X3 x 3 x It is efine by the bounary conitions ix 3 Ž Ž ix 3 Ž 45 Ž ix3 Ž x x ix 3 x 4 x The Schroinger operator with the singular ensity D Ž 1 X D is the heuristic operator L D Ž 1X X D Ž1 Ž 46 X4 x 4 4 x It is equal to the secon erivative operator with the omain of functions from W ŽR 4 satisfying the bounary conitions ž / Ž Ž 1 X 4 Ž 47 Ž 1 Ž x x It follows that the coefficients X1 an X in the four parameter family of singular interactions Ž 18 can be interprete as the coefficients in front of the an potentials The coefficient X3 efines the strength of the Gauge fiel with singularity at the origin The coefficient X4 correspons to the singular ensity CONCLUSIONS The results of the present paper can be easily generalize to the case of a general secon orer ifferential operator with the singular support of

26 3 P KURASOV the coefficients on the isjoint set of points An infinite number of points can be investigate 3, The methos evelope have been applie alreay to ifferent problems in atomic an computational physics 1, 18 ACKNOWLEDGMENTS The author thanks Professor J Boman for his close collaboration an for suggesting proofs of several of the lemmas The author thanks Professor H Holen for a stimulating iscussion an Professors B Pavlov an S Albeverio for their continuous interest in the work The author is grateful to A Rivero for pointing out Ref 8 The author also thanks the referee for an extremely careful reaing of the manuscript an several remarks which helpe to improve it The author is grateful to Alexaner von Humbolt Founation for the financial support REFERENCES 1 S Albeverio, Z Brzezniak, an L Dabrowski, The heat equation with point interaction in p L -spaces, Integral Equations Operator Theory, 1 Ž 1995, S Albeverio, Z Brzezniak, an L Dabrowski, Funamental solution of the heat an Schroinger equations with point interaction, J Funct Anal 13 Ž 1995, 54 3 S Albeverio, F Gesztesy, R Hoegh-Krohn, an H Holen, Solvable Moels in Quantum Mechanics, Springer-Verlag, Berlin, S Albeverio, F Gesztesy, R Hoegh-Krohn, an W Kirsch, On point interactions in one imension, J Operator Theory 1 Ž 1984, S Albeverio, F Gesztesy, an H Holen, Comments on a recent note on the Schroinger equation with a -interaction, J Phys A: Math Gen 6 Ž 1993, F A Berezin an L D Faeev, Note on the Schroinger equation with the singular potential, Math USSR Dokl 137 Ž 1961, in Russian 7 J Boman an P Kurasov, Finite rank singular perturbations an istributions with iscontinuous test functions, in preparation 8 M Carreau, Four-parameter point-interaction in 1D quantum systems, J Phys A: Math Gen 6 Ž 1993, P R Chernoff an R J Hughes, A new class of point interactions in one imension, J Funct Anal 111 Ž 1993, Yu N Demkov, P B Kurasov, an V N Ostrovsky, Doubly perioical in time an energy exactly soluble system with two interacting systems of states, J Phys A: Math Gen 8 Ž 1995, Yu N Demkov an V N Ostrovskii, Zero-Range Potentials an Their Applications in Atomic Physics, Leningra Univ Press, Leningra, 1975 in Russian ; Plenum Press, New YorkLonon, 1988 English transl 1 F Gesztesy an H Holen, A new class of solvable moels in quantum mechanics escribing point interactions on the line, J Phys A: Math Gen Ž 1987, F Gesztesy an W Kirsch, One-imensional Schroinger operators with interactions singular on a iscrete set, J Reine Angew Math 36 Ž 1985, D J Griffiths, Bounary conitions at the erivative of a elta function, J Phys A: Math Gen 6 Ž 1993, L Hormaner, The Analysis of Linear Partial Differential Operators, I, Springer- Verlag, Berlin, 1983

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