Singularly perturbed self-adjoint operators in scales of Hilbert spaces

Finite rank perturbations of a semi-bounded self-adjoint operator A are studied in the scale of Hilbert spaces associated with A. A concept of quasi-boundary value space is used to describe self-adjoint operator realizations of regular and singular perturbations of A by the same formula. As an app...

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Datum:	2007
Hauptverfasser:	Albeverio, S., Kuzhel, S., Nizhnik, L.
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spelling	nasplib_isofts_kiev_ua-123456789-1641912025-02-09T10:55:22Z Singularly perturbed self-adjoint operators in scales of Hilbert spaces Сингулярно збурені самоспряжені оператори в шкалах гільбертових просторів Albeverio, S. Kuzhel, S. Nizhnik, L. Статті Finite rank perturbations of a semi-bounded self-adjoint operator A are studied in the scale of Hilbert spaces associated with A. A concept of quasi-boundary value space is used to describe self-adjoint operator realizations of regular and singular perturbations of A by the same formula. As an application the one-dimensional Schrodinger operator with generalized zero-range potential is considered in the Sobolev space Wp₂(R), p ∈ N. У шкалі гільбертових просторів, асоційованих з A, вивчаються скінченного рангу збурення напівобме-женого самоспряженого оператора A. Поняття квазіпростору граничних значень використовується для опису однією формулою самоспряжених операторних реалізацій як регулярних, так і сингулярних збурень оператора A. Як застосування, розглядається одновимірний оператор Шредінгера з узагальненим потенціалом нульового радіуса у просторі Соболева Wp₂(R),p∈N. The second (S.K.) and third (L.N.) authors thank DFG for the financial support of the projects 436 UKR 13/88/0-1 and 436 UKr 113/79, respectively, and the Institute fur Angewandte Mathematik der Universit at Bonn for the warm hospitality. 2007 Article Singularly perturbed self-adjoint operators in scales of Hilbert spaces / S. Albeverio, S. Kuzhel, L. Nizhnik // Український математичний журнал. — 2007. — Т. 59, № 6. — С. 723–743. — Бібліогр.: 28 назв. — англ. 1027-3190 https://nasplib.isofts.kiev.ua/handle/123456789/164191 517.42 en Український математичний журнал application/pdf Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	Статті Статті
spellingShingle	Статті Статті Albeverio, S. Kuzhel, S. Nizhnik, L. Singularly perturbed self-adjoint operators in scales of Hilbert spaces Український математичний журнал
description	Finite rank perturbations of a semi-bounded self-adjoint operator A are studied in the scale of Hilbert spaces associated with A. A concept of quasi-boundary value space is used to describe self-adjoint operator realizations of regular and singular perturbations of A by the same formula. As an application the one-dimensional Schrodinger operator with generalized zero-range potential is considered in the Sobolev space Wp₂(R), p ∈ N.
format	Article
author	Albeverio, S. Kuzhel, S. Nizhnik, L.
author_facet	Albeverio, S. Kuzhel, S. Nizhnik, L.
author_sort	Albeverio, S.
title	Singularly perturbed self-adjoint operators in scales of Hilbert spaces
title_short	Singularly perturbed self-adjoint operators in scales of Hilbert spaces
title_full	Singularly perturbed self-adjoint operators in scales of Hilbert spaces
title_fullStr	Singularly perturbed self-adjoint operators in scales of Hilbert spaces
title_full_unstemmed	Singularly perturbed self-adjoint operators in scales of Hilbert spaces
title_sort	singularly perturbed self-adjoint operators in scales of hilbert spaces
publisher	Інститут математики НАН України
publishDate	2007
topic_facet	Статті
url	https://nasplib.isofts.kiev.ua/handle/123456789/164191
citation_txt	Singularly perturbed self-adjoint operators in scales of Hilbert spaces / S. Albeverio, S. Kuzhel, L. Nizhnik // Український математичний журнал. — 2007. — Т. 59, № 6. — С. 723–743. — Бібліогр.: 28 назв. — англ.
series	Український математичний журнал
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first_indexed	2025-11-25T20:50:03Z
last_indexed	2025-11-25T20:50:03Z
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fulltext	Cej nomer Ωurnalu prysvqçu[t\sq pam’qti Marka Hryhorovyça Krejna (03.04.1907 – 17.10.1989) UDC 517.42 S. Albeverio (Inst. Angewandte Math., Univ. Bonn, Germany), S. Kuzhel, L. Nizhnik (Inst. Math. Nat. Acad. Sci. Ukraine, Kyiv) SINGULARLY PERTURBED SELF-ADJOINT OPERATORS IN SCALES OF HILBERT SPACES SYNHULQRNO ZBURENI SAMOSPRQÛENI OPERATORY V ÍKALAX HIL\BERTOVYX PROSTORIV Finite rank perturbations of a semi-bounded self-adjoint operator A are studied in the scale of Hilbert spaces associated with A. A concept of quasi-boundary value space is used to describe self-adjoint operator realiza- tions of regular and singular perturbations of A by the same formula. As an application the one-dimensional Schrödinger operator with generalized zero-range potential is considered in the Sobolev space W p 2 (R), p ∈ N. U ßkali hil\bertovyx prostoriv, asocijovanyx z A, vyvçagt\sq skinçennoho ranhu zburennq napivobme- Ωenoho samosprqΩenoho operatora A. Ponqttq kvaziprostoru hranyçnyx znaçen\ vykorystovu[t\sq dlq opysu odni[g formulog samosprqΩenyx operatornyx realizacij qk rehulqrnyx, tak i synhulqrnyx zburen\ operatora A. Qk zastosuvannq, rozhlqda[t\sq odnovymirnyj operator Íredinhera z uzahal\ne- nym potencialom nul\ovoho radiusa u prostori Soboleva W p 2 (R), p ∈ N. 1. Introduction. Let A be a semibounded self-adjoint operator acting in a separable Hilbert space H with inner product (· , ·) and let D(A), R(A), and kerA denote the domain, the range, and the null-space of A, respectively. Without loss of generality, we will assume that A ≥ I. Let Hs ⊂ H = H0 ⊂ H−s, s > 0, (1.1) be the standard scale of Hilbert spaces associated with A (A-scale) [1, 8]. Here, a Hilbert space Hs (s ∈ R) is considered as the completion of the set ∩n∈ND(An) with respect to the norm ‖u‖s = ‖As/2u‖, u ∈ ∩n∈ND(An). (1.2) By (1.2), the operator Ar/2 (r ∈ R) can continuously be extended to an isometric mapping Ar/2 of Hs onto Hs−r (we preserve the same notation Ar/2 for this continua- tion). In a natural way Hs and H−s are dual and the inner product in H can be extended to a pairing 〈u, ψ〉 = (As/2u,A−s/2ψ), u ∈ Hs, ψ ∈ H−s, (1.3) such that \|〈u, ψ〉\| ≤ ‖u‖s‖ψ‖−s. The present paper is an extended and modified variant of [4] and its aim consists in the development of a unified approach to the study of finite rank perturbations of a self-adjoint operator A in the scale of Hilbert spacesHs. We recall that a self-adjoint operator Ã �= A acting inH is called a finite rank pertur- bation of A if the difference (Ã− zI)−1 − (A− zI)−1 is a finite rank operator inH for at least one point z ∈ C \ R [16]. c© S. ALBEVERIO, S. KUZHEL, L. NIZHNIK, 2007 ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 6 723 724 S. ALBEVERIO, S. KUZHEL, L. NIZHNIK If Ã is a finite rank perturbation of A, then the corresponding symmetric operator1 Asym = A �D= Ã �D, D = { u ∈ D(A) ∩ D(Ã) \| Au = Ãu } (1.4) arises naturally. This operator has finite and equal deficiency numbers. It is important that the operator Asym can be recovered uniquely by its defect subspace N = H�R(Asym) and the initial operator A. Namely, Asym = A �D(Asym), D(Asym) = { u ∈ D(A) \| (Au, η) = 0 ∀η ∈ N } . (1.5) Moreover, the choice of an arbitrary finite dimensional subspace N of H as a defect subspace allows one to determine by (1.5) a closed symmetric operator Asym with finite and equal defect numbers. To underline this relation, we will use notation AN instead of Asym. Obviously, any self-adjoint extension Ã of AN is a finite rank perturbation of A. A finite rank perturbation Ã of A is called regular if D(A) = D(Ã). Otherwise (i.e., D(A) �= D(Ã)), the operator Ã is called singular. It is convenient to divide the class of singular perturbations into two subclasses. We will say that a singular perturbation Ã is purely singular if the symmetric operator Asym = = AN defined by (1.4) is densely defined (i.e., N ∩ D(A) = {0}) and mixed singular if AN is nondensely defined (i.e., N ∩ D(A) �= {0}). Important examples of finite rank perturbations of the Schrödinger operator are given by finitely many point interactions [1, 2]. The consideration of point interactions in L2(Rd) leads to purely singular perturbations and, in the case of Sobolev spaces W p 2 (Rd), p ∈ N, mixed singular perturbations arise [5, 26]. These applications can be served as a certain motivation of the abstract results carried out in the paper. It is well-known that finite rank regular perturbations of A can be described with the help of finite rank self-adjoint operators (potentials) acting in H. Typical examples of finite rank singular perturbations are provided by the general expression Ã = A + V, V = n∑ i,j=1 bij〈·, ψj〉ψi (R(V ) �⊂ H, bij ∈ C). (1.6) SinceR(V ) �⊂ H, the singular potential V is not an operator inH and it acts in the spaces of A-scale. Such types of expressions appear in many areas of mathematical physics (for an extensive list of references, see [1, 2]). In the present paper, we will study finite rank singular perturbations of A in the spaces of A-scale (1.1). The main attention will be focused on the description of self-adjoint extensions Ã of Asym in a form that is maximally adapted for the determination of Ã with the help of additive singular perturbations (1.6) and preserves physically meaningful relations to the parameters bij of the singular potential V = ∑n i,j=1 bij〈·, ψj〉ψi. In Section 2, such a problem is solved for the case of purely singular perturbations. Precisely, since the corresponding symmetric operator Asym = AN in (1.4) is densely defined, we can combine the Albeverio – Kurasov approach [2] with the boundary value spaces technique [15, 22]. The first of them allows us to involve the parameters bij of the singular potential in the determination of the corresponding self-adjoint operator re- alization of (1.6), the second provides convenient framework for the description of such operators. As a result, we get a simple description of self-adjoint realizations of purely 1The symbol A �X means the restriction of A onto the set X. ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 6 SINGULARLY PERTURBED SELF-ADJOINT OPERATORS IN SCALES OF HILBERT SPACES 725 singular perturbations (Theorem 2.2) and, moreover, we present a simple algorithm for solving an inverse problem, i.e., recovering the purely singular potential V in (1.6) by the given self-adjoint extension of AN defined in terms of boundary value spaces. Other approaches to the description of purely singular perturbations were recently suggested by Arlinskii and Tsekanovski [7] and Posilicano [27, 28]. The description of mixed singular perturbations of A is more complicated because the corresponding symmetric operator AN is nondensely defined and, hence, the adjoint of AN does not exist. To overcome this problem, a certain generalization of the concept of BVS is required. The key point here is the replacement of the adjoint operator A∗ N by a suitable object. In [13, 24], the operator AN and its ‘adjoint’ are understood as linear relations and the description of all self-adjoint relations that are extensions of the graph of AN was obtained. In [22], a pair of maximal dissipative extensions of AN and its adjoint (maximal accumulative extension) was used instead of A∗ N . This allows one to describe self-adjoint extensions directly as operators without using linear relations technique. The approaches mentioned above are general and they can be applied to an arbitrary nondensely defined symmetric operator. However, in the case where AN is determined as the restriction of an initial self-adjoint operator A, it is natural to use A for the description of extensions of AN (see [10, 11, 18]). In Section 3, developing the ideas proposed recently in [5, 26], we use A for the definition of a quasi-adjoint operator of AN . The concept of quasi-adjoint operators allows one to generalize the definition of boundary value spaces (BVS) to the case of nondensely defined operators AN and to preserve the simple formulas for the description of self-adjoint extensions of AN . One of the characteristic features of quasi-BVS extension theory that immediately follows from the definition of a quasi-BVS consists in the description of essentially2 self- adjoint extensions of AN . It should be noted that this property is very convenient for the description of self-adjoint differential expressions with complicated boundary conditions. Furthermore, it gives the possibility to describe finite rank regular and mixed singular perturbations of A in just the same way as purely singular perturbations. In Section 4, the results of quasi-BVS extension theory are applied to the study of finite rank singular perturbations of A in spaces of A-scale (1.1). In recent years, such kind of problems attracted a steady interest and they naturally arise in the theory of su- persingular perturbations [12, 21] and in the study of Schrödinger operators with point interactions in Sobolev spaces [5, 26]. 2. The case of purely singular perturbations. 2.1. Description. In what follows we assume that A ≥ I is a self-adjoint operator in H, N is a finite dimensional subspace ofH, and AN is a symmetric operator defined by the formula AN = A �D(AN ), D(AN ) = { u ∈ D(A) ∣∣ (Au, η) = 0 ∀η ∈ N } . (2.1) The operator AN is densely defined inH if and only if N ∩D(A) = {0}. In this case, D(A∗ N ) = D(A)+̇N and A∗ Nf = A∗ N (u + η) = Au ∀f = u + η ∈ D(A∗ N ) (u ∈ D(A), η ∈ N). (2.2) If AN is densely defined, then self-adjoint extensions of AN admit a convenient de- scription in terms of boundary value spaces (see [14] and references therein). 2I.e., those extensions that turn out to be self-adjoint after closure. ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 6 726 S. ALBEVERIO, S. KUZHEL, L. NIZHNIK Definition 2.1. A triple (N,Γ0,Γ1), where N is an auxiliary Hilbert space and Γ0, Γ1 are linear mappings of D(A∗ sym) into N, is called a boundary value space (BVS) of AN if the abstract Green identity (A∗ Nf, g)− (f,A∗ Ng) = (Γ1f,Γ0g)N − (Γ0f,Γ1g)N, f, g ∈ D(A∗ N ), (2.3) is satisfied and the map (Γ0,Γ1) : D(A∗ N )→ N⊕N is surjective. One of the simplest examples of BVS gives the triple3 (N,Γ0,Γ1), where N is taken from (2.1), (2.2) and Γ0(u + η) = PNAu, Γ1(u + η) = −η (∀u ∈ D(A) ∀η ∈ N), (2.4) where PN is the orthoprojector onto N inH. The following elementary result enables one to get infinitely many BVS of AN start- ing from the fixed one. Lemma 2.1. Let (N,Γ0,Γ1) be a BVS of AN and let R be an arbitrary self-adjoint operator acting in N. Then the triple (N,ΓR 0 ,Γ1), where ΓR 0 = Γ0 − RΓ1 is also a BVS of AN . The next theorem provides a description of all self-adjoint extensions of AN . Theorem 2.1 [17]. Let (N,Γ0,Γ1) be a BVS of AN . Then any self-adjoint extension Ã of AN coincides with restriction of A∗ N to D(Ã) = { f ∈ D(A∗ N ) \| (I − U)Γ0f = i(I + U)Γ1f } , (2.5) where U is a unitary operator in N. Moreover, the correspondence Ã↔ U is a bijection between the sets of all self-adjoint extensions of AN and all unitary operators in N. In cases where self-adjoint extensions are described by sufficiently complicated bound- ary conditions (see, e.g., [19]), the representation (2.5) is not always convenient because it contains the same factor U on the both sides. To overcome this inconvenience, we outline another approach that enables one to remove one of the factors in (2.5) but, simul- taneously, to preserve the description of all self-adjoint extensions of AN . The main idea here consists in the use of a family BVS (N,ΓR 0 ,Γ1) instead of a fixed BVS (see [23] for details). Let (N,ΓR 0 ,Γ1) be a family of BVS of AN defined in Lemma 2.1. For a fixed R, Theorem 2.1 implies that the expression AB,R := A∗ N �D(AB,R), D(AB,R) = { f ∈ D(A∗ N ) ∣∣ BΓR 0 f = Γ1f } , (2.6) where B is an arbitrary self-adjoint operator in N, determines a subset PR(AN ) of the set P(AN ) of all self-adjoint extensions of AN . More precisely, a self-adjoint extension Ã of AN belongs to PR(AN ) iff D(Ã) ∩ ker ΓR 0 = D(AN ). It is easy to verify, that the union ⋃ R PR(AN ) over all self-adjoint operators R in N coincides with P(AN ). Moreover, for a fixed Ã ∈ P(AN ), there exist infinitely many R such that Ã ∈ PR(AN ). Thus formula (2.6), where R and B play a role of parameters, gives the description of all self-adjoint extensions of AN . 3In fact, this BVS was already implicitly used in the classical works [9, 20]. ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 6 SINGULARLY PERTURBED SELF-ADJOINT OPERATORS IN SCALES OF HILBERT SPACES 727 2.2. Self-adjoint realizations. 2.2.1. Construction of self-adjoint realizations by ad- ditive purely singular perturbations. Let us consider the general expression (1.6), where ψj , 1 ≤ j ≤ n, form a linearly independent system in H−2 and the linear span X of {ψj}nj=1 satisfies the condition X ∩H = {0} (i.e., elements ψj areH-independent). Let {ej}n1 be the canonical basis of C n (i.e., ej = (0, . . . , 1, . . . 0), where 1 occurs on the jth place only). Putting Ψej := ψj , j = 1, . . . , n, we define an injective linear mapping Ψ : C n → H−2 such thatR(Ψ) = X . Let Ψ∗ : H2 → C n be the adjoint operator of Ψ (in the sense 〈u,Ψd〉 = (Ψ∗u, d)Cn ∀u ∈ H2 ∀d ∈ C n). It is easy to see that Ψ∗u =  〈u, ψ1〉 ... 〈u, ψn〉  ∀u ∈ H2. (2.7) Using (2.7), we rewrite the singular potential V = ∑n i,j=1 bij〈·, ψj〉ψi in (1.6) as fol- lows: n∑ i,j=1 bij〈·, ψj〉ψi = ΨBΨ∗, (2.8) where the matrix B = ‖bij‖ni,j=1 consists of the coefficients bij of the potential V. In what follows we assume that B is Hermitian, i.e., bij = bji. In order to give a meaning to Ã = A + V as a self-adjoint operator inH we consider a symmetric restriction Asym of A Asym := A �D(Asym), D(Asym) = D(A) ∩ kerΨ∗. (2.9) By virtue of (1.3) (for s = 2) and (2.7), the operator Asym is also defined by (2.1), where Asym = AN and N = A−1R(Ψ) = A−1X , i.e., N is a linear span of {A−1ψj}nj=1. Since N ∩ D(A) = {0}, the operator AN is densely defined inH. Any self-adjoint extension Ã of AN is a purely singular perturbation of A and, in general, it can be regarded as a realization of (1.6) in H. In this context, there arises the natural question of whether and how one could establish a physically meaningful correspondence between the parameter B of the potential V = ΨBΨ∗ and self-adjoint extensions of AN . To do this we combine the Albeverio – Kurasov approach [2] with the BVS technique. This approach consists in the construction of some regularization Areg := A+ + ΨBΨ∗ R = A+ + n∑ i,j=1 bij〈·, ψex j 〉ψi, (2.10) of (1.6) that is well defined as an operator from D(A∗ N ) to H−2. (Here, A+, Ψ∗ R, and 〈·, ψex j 〉 are extensions of A, Ψ∗, and 〈·, ψj > ontoD(A∗ N )). After that, the corresponding self-adjoint realization Ã of (1.6) is determined by the formula Ã = Areg �D(Ã), D(Ã) = { f ∈ D(A∗ sym) ∣∣ Aregf ∈ H } . (2.11) By (2.2), it is easy to see that for the definition of A+ in (2.10) one needs to determine the action of A+ on N. Assuming that A+ �N acts as the isometric mapping A in the A-scale, we get ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 6 728 S. ALBEVERIO, S. KUZHEL, L. NIZHNIK A+f = Au + Aη = A∗ Nf + Aη ∀f = u + η ∈ D(A∗ N ). (2.12) However, the principal point in the definition of Areg is the construction of Ψ∗ R or, equivalently, the definition of the functionals 〈·, ψj〉, j = 1, . . . , n, on D(A∗ N ). It is clear (see (2.2)) that 〈·, ψj〉 can be extended onto D(A∗ N ) if we know its values on N. Since N = A−1R(Ψ) and R(Ψ) coincides with the linear span of ψj , j = 1, . . . , n, the vectors ηj = A−1ψj , j = 1, . . . , n, form a basis of N. Using this fact and (2.2), we get that any f ∈ D(A∗ N ) can be represented as f = u + ∑n k=1 αkηk (u ∈ D(A), αk ∈ C). Thus the extended functional 〈·, ψex j 〉 is well-defined by the formula 〈f, ψex j 〉 = 〈u, ψj〉+ n∑ k=1 αkrjk ∀f ∈ D(A∗ N ) (2.13) if we know the entries rjk = 〈A−1ψk, ψj〉 = 〈ηk, ψj , 〉 of the regularization matrix R = ‖rjk‖nj,k=1. In this case, by virtue of (2.7) and (2.13), Ψ∗ Rf = Ψ∗ R ( u + n∑ k=1 αkηk ) = Ψ∗u + R  α1 ... αn  =  〈f, ψex 1 〉 ... 〈f, ψex n 〉  (2.14) for any f ∈ D(A∗ N ). If R(Ψ) ⊂ H−1, the entries rjk are uniquely defined and R is an Hermitian matrix. In the case whereR(Ψ) �⊂ H−1 the matrix R is not determined uniquely [2]. In what follows we assume that R is chosen as an Hermitian matrix. Lemma 2.2. The triple (Cn,ΓR 0 ,Γ1), where the linear operators ΓR i : D(A∗ N ) → → C n are defined by the formulas ΓR 0 f = Ψ∗ Rf, Γ1f = −Ψ−1(A+ −A∗ N )f = −Ψ−1Aη (2.15) (where f = u + η, u ∈ D(A), η ∈ N) is a BVS of AN . Proof. By (1.3), 〈u, ψj〉 = (Au, ηj). Taking into account this relation and (2.2), (2.7), (2.12) it is easy to verify that the mappings Γ0f = Ψ∗u, Γ1f = −Ψ−1Aη (2.16) satisfy the conditions of Definition 2.1. Hence, (Cn,Γ0,Γ1) is a BVS of AN . It follows from (2.13), (2.7), (2.14), (2.15), and (2.16) that ΓR 0 f = Γ0f −RΓ1f. By Lemma 2.1 this means that (Cn,ΓR 0 ,Γ1) is also a BVS of AN . Lemma 2.2 is proved. Theorem 2.2. Let Ã be a self-adjoint realization of (1.6) defined by (2.10), (2.11). Then Ã = AB,R = A∗ N �D(AB,R), D(AB,R) = {f ∈ D(A∗ N ) \| BΓR 0 f = Γ1f}, (2.17) ΓR 0 and Γ1 being defined by (2.15). Proof. Employing relations (2.10), (2.12), and (2.15), we get Aregf = A∗ Nf + Ψ [ BΓR 0 f − Γ1f ] . ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 6 SINGULARLY PERTURBED SELF-ADJOINT OPERATORS IN SCALES OF HILBERT SPACES 729 This equality and (2.11) mean that f ∈ D(Ã) if and only if BΓR 0 f − Γ1f = 0. Thus, the operator realization Ã of (1.6) coincides with the operator AB,R defined by (2.6). Since B is an Hermitian matrix, the operator AB,R is self-adjoint. Theorem 2.2 is proved. Summing the results above we can state that the choice of an extension Ψ∗ ex of Ψ∗ onto D(A∗ N ) plays a main role and precisely this enables one to choose (see (2.15)) a more suitable4 BVS (Cn,ΓR 0 ,Γ1) for the description of self-adjoint realizations of (1.6). 2.2.2. Recovering purely singular potentials by a given self-adjoint extension. Here we consider an inverse problem. Namely, for a given BVS (Cn,Γ0,Γ1) of AN such that ker Γ1 = D(A) and the corresponding self-adjoint extensions AB = A∗ N �D(AB), D(AB) = { f ∈ D(A∗ N ) ∣∣BΓ0f = Γ1f } , (2.18) where B is an Hermitian matrix, we recover an additive purely singular perturbation V = = ΨBΨ∗ such that the formal expression Ã = A + V possesses the self-adjoint realiza- tion AB. We start with the definition of Ψ. Since ker Γ1 = D(A), the restriction Γ1 �N de- termines a one-to-one correspondence between N and C n. Hence, (Γ1 �N )−1 exists and (Γ1 �N )−1 maps C n onto N. Putting (cf. (2.15)) Ψd := −A(Γ1 �N )−1d, where d ∈ C n, we determine an injective linear mapping of C n toH−2 such thatR(Ψ) ∩H = {0}. Set ψj = Ψej , where {ej}n1 is the canonical basis of C n. Putting f = u ∈ D(A), g = A−1ψj = A−1Ψej = −(Γ1 �N )−1ej in (2.3) and recalling the condition ker Γ1 = = D(A), we establish that 〈u, ψj〉 = (Au,A−1ψj) = −(Γ0u,Γ1A −1ψj)Cn = (Γ0u, ej)Cn . This formula enables one to determine an extension of 〈·, ψj〉 onto D(A∗ N ) with the help of the boundary operator Γ0. Namely, 〈f, ψex j 〉 := (Γ0f, ej)Cn . But then, reasoning by analogy with (2.14), we conclude that Γ0f = Ψ∗ Rf. Now, repeating arguments of Theorem 2.2, it is easy to see that the operator AB defined by (2.18) is a self-adjoint realization of the formal expression A+ + ΨBΨ∗ R. Example 2.1. General zero-range potential in R. A one-dimensional Schrödinger operator corresponding to a general zero-range po- tential at the point x = 0 can be given by the formal expression − d2 dx2 + b11〈·, δ〉δ + b12〈·, δ′〉δ + b21〈·, δ〉δ′ + b22〈·, δ′〉δ′, (2.19) where δ′ is the derivative of the Dirac δ-function (with support at 0) and the coefficients bij form an Hermitian matrix. Putting Ψ ( 1 0 ) = δ and Ψ ( 0 1 ) = δ′, we get Ψ∗u = ( 〈u, δ〉〈u, δ′〉 ) (u(x) ∈ W 2 2 (R)) and, hence, ΨBΨ∗ = b11〈·, δ〉δ + b12〈·, δ′〉δ + b21〈·, δ〉δ′ + b22〈·, δ′〉δ′. 4From the point of view of the simplest relations between coefficients of singular potentials and parameters of BVS. ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 6 730 S. ALBEVERIO, S. KUZHEL, L. NIZHNIK In the case under consideration, A = − d2 dx2 + I, D(A) = W 2 2 (R), where W 2 2 (R) is the Sobolev space; Asym = ( − d2 dx2 + I ) �{u(x)∈W 2 2 (R)\|u(0)=u′(0)=0} and Asym = AN , where a subspace N of L2(R) is the linear span of functions η1(x) = A−1δ = 1 2 e−\|x\|, η2(x) = A−1δ′(x) = − sign x 2 e−\|x\|. (2.20) Further A∗ Nf(x) = −f ′′(x) + f(x) (f(x) ∈ D(A∗ N ) = W 2 2 (R)+̇N = W 2 2 (R\{0})), where the symbol f ′′(x) means the second derivative (pointwise) of f(x) except the point x = 0. It follows from the description ofD(A∗ N ) that any function f ∈ D(A∗ N ) and its deriva- tive f ′ have right(left)-side limits at the point 0. Thus, the expressions gr = g(+0) + g(−0) 2 , gs = g(+0)− g(−0), (g = f or g = f ′) (2.21) are well-posed. To obtain a regularization of (2.19) it suffices to extend the distributions δ and δ′ onto D(A∗ N ). The most physically reasonable way, based on the extension of δ by the continuity and parity onto W 2 2 (R\{0} and preserving the initial homogeneity of δ′ with respect to scaling transformations [2], leads to the following extensions5: 〈f, δ〉 = fr, 〈f, δ′〉 = −f ′ r (f(x) ∈W 2 2 (R\{0}). These extensions can also be determined by the general formula (2.13), if we set R = ( 1/2 0 0 −1/2 ) . In this case, Ψ∗ Rf = ( fr −f ′ r ) and the corresponding boundary operators ΓR 0 and Γ1 in the BVS (C2,ΓR 0 ,Γ1) determined by (2.15) have the form ΓR 0 f(x) = ( fr −f ′ r ) , Γ1f(x) = ( f ′ s fs ) ∀f(x) ∈W 2 2 (R\{0}). (2.22) Here the operator ΓR 0 f turns out to be the mean value of f(x) and −f ′(x) at the origin and Γ1 characterizes the jumps of f(x) and its derivative at the origin. Taking into account the fact that the operator A+ = − d2 dx2 + I acts on f(x) ∈ ∈W 2 2 (R\{0}) by the rule A+f(x) = − d2 dx2 f(x)+f(x), where the action of− d2 dx2 f(x) is understood in the distributional sense, i.e., − d2 dx2 f(x) = −f ′′(x)− f ′ sδ(x)− fsδ ′(x) and employing Theorem 2.2 we obtain a description of self-adjoint realizations AB,R of (2.19) that are defined by the rule AB,Rf(x) = −f ′′(x), f(x) ∈ D(AB,R) = { f(x) ∈W 2 2 (R\{0}) ∣∣∣ ( b11 b12 b21 b22 ) ( fr −f ′ r ) = ( f ′ s fs )} . 5We omit index ex for such natural extensions. ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 6 SINGULARLY PERTURBED SELF-ADJOINT OPERATORS IN SCALES OF HILBERT SPACES 731 Example 2.2. Point interaction in R 3. Let us consider the self-adjoint operator A = −∆ + µ2I, D(A) = W 2 2 (R3) acting in L2(R3) and its restriction Asym = −∆ + µ2I �{u(x)∈W 2 2 (R3)\|u(0)=0} . It is easy to see that Asym = AN , where N is the linear span of e−µ\|x\| \| x \| , µ > 0. The triple (C,Γ0,Γ1), where Γ1f = lim \|x\|→0 \| x \| f(x), Γ0f = lim \|x\|→0 ( f(x)− (Γ1f) e−µ\|x\| \| x \| ) (2.23) (f(x) ∈ D(A∗ N ) = W 2 2 (R3)+̇N) forms a BVS of AN . Moreover ker Γ1 = D(A). It follows from (2.18) and (2.23) that the operators Ab(u(x) + bu(0) e−µ\|x\| \| x \| ) = (−∆ + µ2I)u(x) ∀u(x) ∈W 2 2 (R3) are self-adjoint extensions of AN . By virtue of the results of Subsection 2.2.2, the oper- ators Ab can be considered as self-adjoint realizations of the heuristic expression −∆ + + µ2 + b〈·, δex〉δ(x), where −∆ is understood in the distributional sense and the ex- tension δex(x) of δ(x) is determined in terms of the boundary operators Γi as follows: 〈f, δex〉 = Γ0f (f ∈W 2 2 (R3)+̇N). 3. The case of mixed singular perturbations. 3.1. The concept of quasi-BVS. In the case of mixed singular perturbations, the operator AN determined by (2.1) is non- densely defined and its adjoint operator A∗ N does not exist. Thus some modification of BVS is required to describe all self-adjoint extensions of AN . Let us suppose that there exists a real number m > 1 such that N ∩ D(Am) = {0}. Then, the direct sum Lm := D(Am)+̇N (3.1) is well defined and we can define on Lm a quasi-adjoint operator A(∗) N by the rule A (∗) N f = A (∗) N (u + η) = Au ∀f = u + η ∈ Lm (u ∈ D(Am), η ∈ N). (3.2) Formula (3.2) is an analog of (2.2) for the adjoint operator A∗ N and we can use A (∗) N as an analog of the adjoint one. It is easy to see that, in general, A(∗) N is not closable and it turns out to be closable only if AN is densely defined. The concept of quasi-adjoint operators allows one to modify Definition 2.1 and to extend it to the case of nondensely defined symmetric operators. Definition 3.1. A triple (N,Γ0,Γ1), where Γi are linear mappings of Lm in an auxiliary Hilbert space N, is called a quasi-BVS of AN if the abstract Green identity (A(∗) N f, g)− (f,A(∗) N g) = (Γ1f,Γ0g)N − (Γ0f,Γ1g)N ∀f, g ∈ Lm (3.3) is satisfied and the map (Γ0,Γ1) : Lm → N⊕N is surjective. Proposition 3.1 [23]. The following assertions are true: 1. If AN is densely defined, then an arbitrary BVS (N,Γ0,Γ1) of AN also is a quasi- BVS of AN . 2. If AN is nondensely defined, then the triple (N,ΓR 0 ,Γ1), where ΓR 0 (u + η) = PNAu + Rη, Γ1(u + η) = −η (u ∈ D(Am), η ∈ N, (3.4) is a quasi-BVS of AN for any choice of self-adjoint operator R in N. ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 6 732 S. ALBEVERIO, S. KUZHEL, L. NIZHNIK 3. Let (N,Γ0,Γ1) be a quasi-BVS of AN . Then the symmetric operator A′ N = A (∗) N �D(A′ N ), D(A′ N ) = ker Γ0 ∩ ker Γ1 (3.5) does not depend on the choice of quasi-BVS and its closure coincides with AN . Let (N,Γ0,Γ1) be a quasi-BVS of AN . An unitary operator U acting in N is called admissible with respect to (N,Γ0,Γ1) if the equation (I − U)Γ0f = i(I + U)Γ1f ∀f ∈ D(AN ) ∩ Lm (3.6) has only the trivial solution Γ0f = Γ1f = 0. If AN is densely defined, then D(AN ) ∩Lm = D(AN ) ∩D(Am) = D(A′ N ) and, by virtue of (3.5), any unitary operator U in N is admissible. Otherwise (AN is nondensely defined), D(AN ) ∩ Lm = D(A′ N )+̇F , (3.7) where dimF = dim(N ∩ D(A)). Vectors f ∈ F have the form f = u + η, where η is an arbitrary element of N ∩ D(A) and u is determined by η with the help of relation PNA(u + η) = 0 ( this determination is unique modulo D(A′ N ) ) . It follows from (3.5) and (3.7) that the condition of admissibility takes away the lineal F from the set of solutions of (3.6). Theorem 3.1 (cf. Theorem 2.1). Let (N,Γ0,Γ1) be a quasi-BVS of AN . Then any self-adjoint extension Ã of AN is the closure of the symmetric operator Ã′ = A (∗) N �D(Ã′), D(Ã′) = { f ∈ D(A(∗) N ) ∣∣ (I − U)Γ0f = i(I + U)Γ1f } , (3.8) where U is an admissible unitary operator with respect to (N,Γ0,Γ1). Moreover, the correspondence Ã↔ U is a bijection between the set of all self-adjoint extensions of AN and the set of all admissible unitary operators. Proof. Let U be an admissible operator and let Ã′ be the corresponding operator defined by (3.8). Since (Γ1f,Γ0g)N − (Γ0f,Γ1g)N = 1 2 ‖(Γ1 + iΓ0)f‖2N − 1 2 ‖(Γ1 − iΓ0)g‖2N, formula (3.3) implies that Ã′ is a symmetric extension of A′ N . Furthermore, there exists a linear subspaceM of Lm such that dimM = dimN = dimN and D(Ã′) = D(A′ N )+̇M. (3.9) It follows from the property of admissibility of U and (3.9) thatM∩D(AN ) = 0. The latter relation and assertion 3 of Proposition 3.1 mean that Ã′ is closable and its closure Ã is a symmetric operator defined by the formula Ã = A (∗) N �D(Ã), D(Ã) = D(AN )+̇M. (3.10) Since dimM = dimN, the defect numbers of Ã in the upper (lower) half plane are equal to 0 and hence, Ã is a self-adjoint extension of AN . Thus we show that the closure of Ã′ defined by (3.8) is a self-adjoint extension of AN . Conversely, let Ã be a self-adjoint extension of AN . It follows from Theorem 5.15 [22] (Chapter 1) that Ã is determined by (3.10), whereM ⊂ Lm and dimM = dimN. ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 6 SINGULARLY PERTURBED SELF-ADJOINT OPERATORS IN SCALES OF HILBERT SPACES 733 But then the symmetric operator Ã′ = Ã �D(Ã)∩Lm defined by (3.9) is an essentially self-adjoint restriction of Ã. The domain D(Ã′) = D(Ã) ∩ Lm admits the representation (3.8), where the admissibility of U follows from the relationM∩ D(AN ) = 0 and the unitarity of U follows from the property of Ã to be a self-adjoint operator. Theorem 3.1 is proved. Remark. If U is not admissible, then the domain D(Ã′) of a symmetric operator Ã′ defined by (3.8) has a nontrivial intersection with F and Ã′ is not closable. By analogy with the densely defined case we can describe self-adjoint extensions of AN as the closure of the symmetric operators A′ B = A (∗) N �D(A′ B), D(A′ B) = { f ∈ Lm \|BΓ0f = Γ1f } , (3.11) where (N,Γ0,Γ1) is a quasi-BVS and B is a self-adjoint operator in N. In such a setting, the operator B is called admissible with respect to (N,Γ0,Γ1) if the equation BΓ0f = = Γ1f (f ∈ D(AN ) ∩ Lm) has only the trivial solution Γ0f = Γ1f = 0. Proposition 3.2 [23]. If B is an admissible operator, then the closure of A′ B is a self-adjoint extension of AN . A self-adjoint extension Ã of AN can be represented as the closure of a symmetric operator A′ B defined by (3.11) if and only if D(Ã) ∩ ker Γ0 = D(A′ N ). Since (3.11) does not describe all self-adjoint extensions of AN , a situation where any operator B is admissible in (3.11) is possible. Proposition 3.3. If (N,Γ0,Γ1) is a quasi-BVS of AN such that ker Γ0 ⊃ D(AN )∩ ∩ Lm, then the closure of A′ B defined by (3.11) is a self-adjoint extension of AN for any self-adjoint operator B in N. Proof. If ker Γ0 ⊃ D(AN )∩Lm, then the equation BΓ0f = Γ1f (f ∈ D(AN )∩Lm) has only the trivial solution Γ0f = Γ1f = 0 and hence, any self-adjoint operator B is admissible with respect to (N,Γ0,Γ1). Proposition 3.3 is proved. Let us specify the obtained results and present more constructive condition of admis- sibility for the family of quasi-BVS (N,ΓR 0 ,Γ1) determined by (3.4). Proposition 3.4. 1. A self-adjoint operator B acting in N is admissible with re- spect to (N,ΓR 0 ,Γ1) if and only if the equation BPNAη = (I + BR)η ∀η ∈ N ∩ D(A) (3.12) has the unique solution η = 0. 2. Formula (3.11) (where Γ0 = ΓR 0 ) determines self-adjoint extensions of AN for any choice of B if and only if the operator R satisfies the relation PNAη = Rη for all η ∈ N ∩ D(A). Proof. Assertion 1 follows directly from (3.4) and the description of the elements of F ⊂ D(AN ) ∩ Lm. To establish assertion 2, it suffices to observe that ΓR 0 f = PNAu + + Rη = −PNAη + Rη for all elements f = u + η ∈ F . Thus, ker ΓR 0 ⊃ F ⇐⇒ PNAη = Rη for all η ∈ N ∩ D(A). Employing now Proposition 3.3, we complete the proof. Example 3.1. Let us consider a Schrödinger operator that is determined by analogy with (2.19), where δ′ is replaced by a function q ∈ L2(R): − d2 dx2 + b11〈·, δ〉δ + b12(·, q)δ + b21〈·, δ〉q + b22(·, q)q. (3.13) ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 6 734 S. ALBEVERIO, S. KUZHEL, L. NIZHNIK In our case, A = − d2 dx2 + I, D(A) = W 2 2 (R) and the defect subspace N ⊂ L2(R) is the linear span of the functions η1(x) = A−1δ = 1 2 e−\|x\|, η2(x) = A−1q(x). For the sake of simplicity, we assume that the function q(x) coincides with a funda- mental solution m2k(x), k ≥ 1, of the equation ( − d2 dx2 + I )k m2k(x) = δ. In this case, η1 = m2, η2 = m2k+2. Let us fix m = k + 1, then, according to (3.1), Lm = W 2k+2 2 (R)+̇N ⊂ ⊂ W 2k+2 2 (R\{0}). It is easy to see that an arbitrary function f ∈ Lm admits the rep- resentation f(x) = u(x)− f ′ sm2(x)− f [2k+1] s m2k+2, where u ∈ W 2k+2 2 (R) and f ′ s and f [2k+1] s mean the jumps of the functions f ′(x) and f [2k+1](x) at the point x = 0. Here, f [2k+1](x) := d dx ( − d2 dx2 + I )k f(x), x �= 0. By the direct verification, we get that the triple (C2,Γ0,Γ1), where Γ0f(x) = ( f(0) (f,m2) ) , Γ1f(x) = ( f ′ s f [2k+1] s ) ∀f(x) ∈ Lm is a quasi-BVS of AN . In our case, all conditions of Proposition 3.3 are satisfied and, hence, the restriction of A (∗) N (A(∗) N f(x) = −f ′′(x) + f(x), x �= 0) onto the collection of functions f ∈ Lm that are specified by the boundary conditions f ′ s = b11f(0) + b12(f,m2), f [2k+1] s = b21f(0) + b22(f,m2) is an essentially self-adjoint operator in L2(R). The closure of such an operator has the form Aq + I, where Aq is a self-adjoint realization of the heuristic expression (3.13). The operator Aq can be interpreted as the Schrödinger operator with nonlocal point interaction [6]. Its domainD(Aq) consists of all functions f ∈W 2 2 (R\{0}) that satisfy the boundary conditions fs = 0, f ′ s = b11f(0) + b12(f, q) and the action of Aqf is determined as follows: Aqf = −f ′′(x) + b21q(x)f(0) + b22(f, q)q(x), x �= 0. 3.2. Quasi-BVS and finite rank regular perturbations. Here we are going to show that the concept of quasi-BVS enables one to describe finite rank regular perturbations of A in just the same way as finite rank purely singular perturbations. To illustrate this point, we consider the following one-dimensional regular perturbation: Aα = A + α(·, ψ)ψ, ψ ∈ Hs \ Hs+ε (∀ε > 0). (3.14) The rank one operator α(·, ψ)ψ is a bounded operator in H and the operator Aα is self- adjoint on the domain D(A). On the other hand, we can consider Aα and A as two self-adjoint extensions of the symmetric nondensely defined operator (cf. (2.1)) AN = A �D(AN ), D(AN ) = {u ∈ D(A) \| (u, ψ) = (Au,A−1ψ) = 0}. (3.15) Here N is the linear span of η = A−1ψ (i.e., N =< η >) and η ∈ Hs+2 \ Hs+2+ε. ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 6 SINGULARLY PERTURBED SELF-ADJOINT OPERATORS IN SCALES OF HILBERT SPACES 735 Let us describe self-adjoint extensions of AN . To do this, we fix m > s + 2 and consider the direct sum Lm = D(Am) +̇ 〈η〉. In what follows, without loss of generality we assume that ‖η‖ = 1. Then, any ele- ment f ∈ Lm admits the presentation f = u + βη, where u ∈ D(Am) and β ∈ C and the operators ΓR 0 ,Γ1 defined by (3.4) have the form6 ΓR 0 (u + βη) = PNAu + rβη = [(Au, η) + rβ]η, Γ1(u + βη) = −βη, where the parameter r is an arbitrary real number. The triple (N,ΓR 0 ,Γ1) is a quasi-BVS of AN and Theorem 3.1 gives the description of all self-adjoint extensions of AN . In particular, formula (3.11) (where Γ0 = ΓR 0 ) shows that the closure of operators A′ bf = A′ b(u + βη) = Au, D(A′ b) = { f = u + βη ∣∣ b[(Au, η) + rβ] = −β } (3.16) are self-adjoint extensions of AN and they coincide with operators Aα (see (3.14)) if we put b = α 1 + α[(Aη, η)− r] . In particular, if r = (Aη, η), then b = α. 4. Finite rank singular perturbations of A in spaces of A-scale. Let p be a fixed integer (p ∈ N). SinceHp is a Hilbert space, all known results on finite rank perturbations of A can automatically be reformulated for its image A �D(Ap/2+1) acting inHp as a self- adjoint operator. However, the specific of Hp as a space of the A-scale (1.1) enables one to get a lot of new nontrivial results (see, e.g., [5, 26], where the spectral analysis of Schrödinger operators with point interactions in the Sobolev spaces W p 2 (Rd) was carried out). The aim of this section is to generalize the results of [5, 26] for the abstract case of a self-adjoint operator acting inHp. 4.1. Construction of BVS for powers of AN . Let N be a finite dimensional sub- space of H such that N ∩ D(A) = {0} and let AN be the corresponding symmetric densely defined operator constructed by N (see (2.1)). The following statement shows that an arbitrary power of AN is a symmetric restric- tion of the same power of A defined by the special choice of a defect subspace M̃ inH. Lemma 4.1. For any p ∈ N, Ap+1 N := (AN )p+1 is a symmetric densely defined operator inH and Ap+1 N = (Ap+1) M̃ , where M̃ = N+̇A−1N+̇ . . . +̇A−pN and (Ap+1) M̃ = Ap+1 �D((Ap+1) M̃ ), D((Ap+1) M̃ ) = { u ∈ D(Ap+1) ∣∣ (Ap+1u,m) = 0 ∀m ∈ M̃ } . Proof. Since D(Ap+1) ∩ M̃ = {0}, the operator (Ap+1) M̃ is densely defined. To prove Ap+1 N = (Ap+1) M̃ it suffices to observe that D(Ap+1 N ) = D((Ap+1) M̃ ). Lemma 4.1 is proved. The next statement gives a convenient algorithm for the construction of BVS of Ap+1 N starting from a fixed BVS of AN . 6We use the notation r instead of R to emphasize that R is an operator multiplication by a real number r. ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 6 736 S. ALBEVERIO, S. KUZHEL, L. NIZHNIK Theorem 4.1. Let (N,Γ0,Γ1) be a BVS of AN and let p ∈ N. Then the triple (⊕Np+1, Γ̃0, Γ̃1), where ⊕Np+1 := N⊕N⊕ . . .⊕N︸︷︷︸ p+1 times and Γ̃0f =  Γ0f Γ0A ∗ Nf ... Γ0(A∗ N )pf , Γ̃1f =  Γ1(A∗ N )pf Γ1(A∗ N )p−1f ... Γ1f  ∀f ∈ D((A∗ N )p+1) (4.1) is a BVS of Ap+1 N . Proof. It follows from Lemma 4.1 that ((AN )p+1)∗ = (A∗ N )p+1. Hence, the oper- ators Γ̃i are well defined on D(((AN )p+1)∗) = D((A∗ N )p+1). Furthermore employing (2.3) and (4.1) we directly verify the following equality for any f, g ∈ D((A∗ N )p+1): ((A∗ N )p+1f, g)− (f, (A∗ N )p+1g) = ((A∗ N )p+1f, g)− ((A∗ N )pf,A∗ Ng)+ + ((A∗ N )pf,+A∗ Ng)− ((A∗ N )p−1f, (A∗ N )2g) + . . . . . . + (A∗ Nf, (A∗ N )pg)− (f, (A∗ N )p+1g) = = (Γ1(A∗ N )pf,Γ0g)N − (Γ0(A∗ N )pf,Γ1g)N + (Γ1(A∗ N )p−1f,Γ0(A∗ N )2g)N− − (Γ0(A∗ N )p−1f,Γ1(A∗ N )2g)N + . . . (Γ1f,Γ0(A∗ N )pg)N − (Γ0f,Γ1(A∗ N )pg)N = = (Γ̃1f, Γ̃0g)⊕Np+1 − (Γ̃0f, Γ̃1g)⊕Np+1 . To prove that (Γ̃0, Γ̃1) maps D((A∗ N )p+1) onto (⊕Np+1)⊕ (⊕Np+1) some auxiliary preparations are required. At first, the property of (N,Γ0,Γ1) to be a BVS of AN and (2.2) yield D(Ap+1 N ) = ker Γ̃0 ∩ ker Γ̃1. (4.2) Further, since D(Ap N ) is dense in H and dimN < ∞, the relation PND(Ap N ) = N (PN is the orthoprojector onto N in H) holds for any p ∈ N. This equality enables one to verify (with the use of (2.1)) that A−1D(Ap N ) +D(AN ) ⊃ A−1N. But then recalling that D(A) = D(AN )+̇A−1N we get A−1D(Ap N ) +D(AN ) + N = D(A)+̇N = D(A∗ N ). (4.3) Let us prove the surjective property of the map (Γ̃0, Γ̃1) for p = 1. To do this we present an arbitrary vectors F̃0, F̃1 ∈ ⊕N2 = N ⊕ N as the vector columns F̃i = = (Fi0, Fi1)t (i = 0, 1 and t denotes the transposition). Then equations Γ̃if = F̃i (f ∈ D((A∗ N )2)) are equivalent to the following system of equations: Γif = Fi0, ΓiA ∗ Nf = Fi1, f ∈ D((A∗ N )2) i = 0, 1. (4.4) Since (N,Γ0,Γ1) is a BVS of AN , there exists g′ ∈ D(A∗ N ) such that Γig ′ = Fi0, i = 0, 1. (4.5) It is important that such g′ is not defined uniquely. Precisely, by virtue of (4.2), any g = g′ + u, where u ∈ D(AN ) satisfy (4.5). ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 6 SINGULARLY PERTURBED SELF-ADJOINT OPERATORS IN SCALES OF HILBERT SPACES 737 Let us consider the element f = A−1g+η = A−1g′ +A−1u+η, where u ∈ D(AN ) and η ∈ N are arbitrary elements. Clearly, f ∈ D((A∗ N )2) and, by (4.5), ΓiA ∗ Nf = Fi1, i = 0, 1. Taking into account the definition of f, we can rewrite the rest equations of (4.4) as follows: Γ0(A−1u + η) = F00 − Γ0A −1g′, Γ1(A−1u + η) = F10 − Γ1A −1g′, where u ∈ D(AN ) and η ∈ N play the role of ‘free’ variables. Employing now (4.3) for p = 1 and recalling the equalityD(AN ) = ker Γ0∩ker Γ1 we conclude that the latter two equations have a solution for a certain choice of vectors u = us and η = ηs. So, we prove that f = A−1g′ + A−1us + ηs is a solution of (4.4). Hence, (F̃0, F̃1) maps D((A∗ N )2) onto (⊕N2)⊕ (⊕N2). The general case p ∈ N is verified by the induction. Theorem 4.1 is proved. Example 4.1. Let H = L2(R), A = − d2 dx2 + I, D(A) = W 2 2 (R) and let AN and (C2,ΓR 0 ,Γ1) be a symmetric operator and its BVS, respectively, that are defined in Example 2.1 (see (2.22)). In this case, D(A∗ N ) = W 2 2 (R \ {0}) and A∗ Nf(x) = = −d2f(x)/dx2 + f(x) (f(x) ∈W 2 2 (R \ {0}), x �= 0). Let p ∈ N. Then Ap+1 N = ( − d2 dx2 + I )p+1 , D(Ap+1 N ) = { u(x) ∈W 2p+2 2 (R) \| u(0) = u′(0) = . . . = u(2p)(0) = u(2p+1)(0) = 0 } and (A∗ N )p+1f(x) = ( − d2 dx2 + I )p+1 f(x) (x �= 0) for all f(x) ∈W 2p+2 2 (R \ {0}). To simplify the notation we will use the following symbol for quasi-derivatives of f(x) ∈W 2p+2 2 (R \ {0}): f [2k](x) := ( − d2 dx2 + I )k f(x), f [2k+1](x) := d dx f [2k](x), k ∈ N ∪ 0. Thus (A∗ N )p+1f(x) = f [2p+2](x). According to Theorem 4.1 and (2.22), a triple (C2p+2, Γ̃0, Γ̃1), where Γ̃0f =  fr −f [1] r ... fr [2p] −f [2p+1] r  , Γ̃1f =  fs [2p+1] fs [2p] ... f [1] s fs  ( f(x) ∈W 2p+2 2 (R \ {0}) ) is a BVS of Ap+1 N . Here the indexes r and s mean, respectively, the mean value and the jump at x = 0 of the corresponding quasi-derivative f [τ ](x) (see (2.21)). The Green identity related to (C2p+2, Γ̃0, Γ̃1) has the form ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 6 738 S. ALBEVERIO, S. KUZHEL, L. NIZHNIK (f [2p+2], g)L2(R) − (f, g[2p+2])L2(R) = = 2p+1∑ τ=0 (−1)τf [τ ] r g [2p+1−τ ] s − 2p+1∑ τ=0 (−1)τf [2p+1−τ ] s g [τ ] r , where f and g are arbitrary functions from W 2p+2 2 (R \ {0}) [26]. 4.2. Construction of quasi-BVS for a symmetric operator AM in Hp. As was noted above, the self-adjoint operator Ap := A �D(Ap/2+1) acting in Hp can be considered as an image of the initial operator A0 := A in Hp. In this case D(Ap) = D(Ap/2+1). By analogy with (2.1), we fix a finite dimensional subspace M of Hp and determine a symmetric operator AM = Ap �D(AM ), D(AM ) = { u ∈ D(Ap) ∣∣ (Apu,m)p = 0 ∀m ∈M } (4.6) acting inHp. In this subsection, we will consider the case where M = p/2∑ k=0 +̇A−p+kN := A−pN+̇A−p+1N+̇ . . . +̇A− p 2 N. (4.7) Here p is assumed to be even and N is a finite dimensional subspace of H such that N ∩ D(A) = {0}. For such a choice of M the definition (4.6) of AM can be rewritten as follows: AM = = A �D(AM ), D(AM ) = { u ∈ D(Ap/2+1) ∣∣PNAu = PNA2u = . . . = PNAp/2+1u = 0 } , (4.8) where PN is the orthoprojector onto N inH or, that is equivalent, AM = AN �D(AM ), D(AM ) = D ( A p/2+1 N ) . (4.9) Thus the operator AM is closely related to AN defined by (2.1). It follows from (4.7) that M ∩ D(Ap/2+1) ⊃ A−pN �= {0}. Hence, AM is a non- densely defined symmetric operator in Hp and for it we can construct a quasi-BVS only. To do this, we chose m = (p + 1)/(p/2 + 1). Then D(Am p ) = D(Ap+1), the direct sum Lm = D(Am p )+̇M = D(Ap+1)+̇M is well posed and we can define the action of A(∗) M f on any element f = u + m ∈ Lm by the formula (cf. (3.2)) A (∗) M f = A (∗) M (u + m) = Apu = Au ∀u ∈ D(Ap+1) ∀m ∈M. (4.10) Theorem 4.2. Let AN be defined by (2.1) and let (N,Γ0,Γ1) be a BVS of AN such that ker Γ1 = D(A). Then the triple (⊕Np/2+1, Γ̂0, Γ̂1), where ⊕Np/2+1 = = N⊕N⊕ . . .⊕N︸︷︷︸ p/2+1 times and Γ̂0f =  Γ0f Γ0A ∗ Nf ... Γ0(A∗ N ) p 2 f  , Γ̂1f =  Γ1(A∗ N )pf Γ1(A∗ N )p−1f ... Γ1(A∗ N ) p 2 f  ∀f ∈ Lm (4.11) ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 6 SINGULARLY PERTURBED SELF-ADJOINT OPERATORS IN SCALES OF HILBERT SPACES 739 is a quasi-BVS of the symmetric operator AM inHp. In particular, the Green identity (A(∗) M f, g)p − (f,A(∗) M g)p = (Γ̂1f, Γ̂0g)⊕Np/2+1 − (Γ̂0f, Γ̂1g)⊕Np/2+1 (4.12) is true for any f, g ∈ Lm = D(Ap+1)+̇M. Proof. It follows from Lemma 4.1 that D((A∗ N )p+1) = D(Ap+1)+̇M̃ and (A∗ N )p+1(u + m̃) = Ap+1u, where u ∈ D(Ap+1) and m̃ ∈ M̃. By virtue of (4.7), M = M̃ ∩Hp. Thus, the latter relations and (4.10) imply that A (∗) M f = A−p(A∗ N )p+1f (4.13) for any f ∈ Lm = D(A(∗) M ) = D((A∗ N )p+1) ∩Hp. Using the assumption that ker Γ1 = D(A) and relations (4.1), (4.13), we verify the abstract Green identity for any f, g ∈ Lm: ((A∗ N )p+1f, g)− (f, (A∗ N )p+1g) = (A−p(A∗ N )p+1f, g)p − (f,A−p(A∗ N )p+1g)p = = (A(∗) M f, g)p − (f,A(∗) M g)p = (Γ̂1f, Γ̂0g)⊕Np/2+1 − (Γ̂0f, Γ̂1g)⊕Np/2+1 . Let F0, F1 be an arbitrary elements from⊕Np/2+1. Since⊕Np/2+1 can be embedded into⊕Np+1 as a subspace (⊕Np/2+1)⊕0⊕ . . . ,⊕, 0︸︷︷︸ p/2 times , the elements Fi belong to⊕Np+1 and have the representations: F0 = ( η0 1 , η 0 2 , . . . , η 0 p/2+1, 0, . . . , 0︸︷︷︸ p+1 times ) , F1 = ( η1 1 , η 1 2 , . . . , η 1 p/2+1, 0, . . . , 0︸︷︷︸ p+1 times ) . Since (⊕Np, Γ̃0, Γ̃1) is a BVS of Ap+1 N constructed in Theorem 4.1, there exists f ∈ ∈ D((A∗ N )p+1) such that Γ̃0f = F0 and Γ̃1f = F1. Furthermore, it follows from (4.1) and the choice of F1 that Γ1f = . . . = Γ1(A∗ N ) p 2−1f = 0. These equalities and condition ker Γ1 = D(A) mean that f ∈ D(Ap/2) = Hp. But then, the description of Lm in (4.13) implies that f ∈ Lm. To complete the proof of Theorem 4.2 it suffices to observe that Γ̃if = Γ̂if, where Γ̂i have the form (4.11). Example 4.2 (cf. Example 4.1). Let H = L2(R), A = − d2 dx2 + I, D(A) = W 2 2 (R) and let AN be a symmetric operator defined in Example 2.1. In this case, Hp coincides with the Sobolev space W p 2 (R), p ∈ N. Further, by (4.9), the symmetric operator AM acting in W p 2 (R) has the form AM = − d2 dx2 + I, D(AM ) = { u(x) ∈W p+2 2 (R) ∣∣∣u(0) = u′(0) = . . . = u(p)(0) = u(p+1)(0) = 0 } . Here, the defect subspace M is determined by (4.7) and it coincides with a linear span of fundamental solutions m2j(x) of the equation ( − d2 dx2 + I )j m2j(x) = δ and their derivatives m2j−1(x) = m′ 2j(x) that belong to Hp. Precisely, M is a linear span of the functions m2j(x) = 1 (j − 1)!2j j−1∑ r=0 Cr 2j−2−r(2j − 3− 2r)!!\|x\|re−\|x\|, m2j−1(x) = m′ 2j(x), where index j runs the set {p/2 + 1, p/2 + 2, . . . , p + 1}. ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 6 740 S. ALBEVERIO, S. KUZHEL, L. NIZHNIK The operator AM is nondensely defined in W p 2 (R). Its quasi-adjoint A(∗) M (see (4.10) and (4.13)) is defined on the domain D(A(∗) M ) = Lm = W 2p+2 2 (R)+̇M = W p 2 (R) ∩W 2p+2 2 (R \ {0}) and acts as follows: A(∗) M f(x) = A−pf [2p+2](x) for all f(x) ∈W p 2 (R)∩W 2p+2 2 (R \ {0}). Let (C2,ΓR 0 ,Γ1) be a BVS of AN defined by (2.22). Obviously, ker Γ1 = D(A). According to Theorem 4.2, the triple (Cp+2, Γ̂0, Γ̂1), where Γ̂0f =  fr −f [1] r ... f [p] r −f [p+1] r  , Γ̂1f =  fs [2p+1] fs [2p] ... f [p+1] s f [p] s  (4.14) (f(x) ∈W p 2 (R)∩W 2p+2 2 (R \ {0})) is a quasi-BVS of the symmetric operator AM acting in W p 2 (R). The corresponding Green identity has the form (A(∗) M f, g)W p 2 (R)−(f,A(∗) M g)W p 2 (R) = p+1∑ τ=0 (−1)τf [τ ] r g [2p+1−τ ] s − p+1∑ τ=0 (−1)τf [2p+1−τ ] s g [τ ] r , where f and g are arbitrary functions from W p 2 (R) ∩W 2p+2 2 (R \ {0}) [26]. 4.3. Description of self-adjoint extensions of AM in Hp. A quasi-BVS (⊕Np/2+1, Γ̂0, Γ̂1) of AM presented in Theorem 4.2 enables one to get a simple description of self- adjoint extensions of AM inHp. Lemma 4.2. Let Γ̂0 be determined by (4.11). Then ker Γ̂0 ⊃ D(AM ) ∩ Lm. Proof. Obviously ker Γ0 ⊃ D(AN ) (since (N,Γ0,Γ1) is a BVS of AN ). But then relations (4.9) and (4.11) give that Γ̂0f = 0 for any f ∈ D(AM ) ∩ Lm. Lemma 4.2 is proved. By Lemma 4.2, the equation BΓ̂0f = Γ̂1f (f ∈ D(AM ) ∩ Lm) has only the trivial solution Γ̂0f = Γ̂1f = 0 for an arbitrary self-adjoint operator B acting in ⊕Np/2+1. So, any B is admissible with respect to the quasi-BVS (⊕Np/2+1, Γ̂0, Γ̂1). The next statement is a direct consequence of Proposition 3.3. Theorem 4.3. For an arbitrary self-adjoint operator B in ⊕Np/2+1 the formula A′ B = A (∗) M �D(A′ B), D(A′ B) = { f ∈ Lm ∣∣BΓ̂0f = Γ̂1f } , (4.15) determines an essentially self-adjoint operator in Hp and its closure is a self-adjoint extension of AM inHp. Example 4.3. Let us preserve the notation of Example 4.2 and let B be an arbitrary Hermitian matrix of the order p + 2. Then, according to Theorem 4.3, the closure of the operator A′ B defined by the rule: A′ Bf(x) = A−pf [2p+2](x), where f(x) belong to W p 2 (R) ∩W 2p+2 2 (R \ {0}) and satisfy the condition BΓ̂0f = Γ̂1f ( Γ̂i are defined by (4.14) ) ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 6 SINGULARLY PERTURBED SELF-ADJOINT OPERATORS IN SCALES OF HILBERT SPACES 741 is a self-adjoint extension AB of the nondensely defined operator AM = − d2 dx2 + I, D(AM ) = { u(x) ∈ W p+2 2 (R) ∣∣u(0) = . . . = u(p+1)(0) = 0 } acting in W p 2 (R). The operator AB can be interpreted as a one-dimensional Schrödinger operator with point interaction in the Sobolev space W p 2 (R) [26]. 4.4. Realization of self-adjoint extensions of AM in Hp by additive perturbations. In mathematical physics, the self-adjoint extensions AB,R of AN described in Theo- rem 2.2 appear naturally as self-adjoint realizations of the additive purely singular pertur- bations (1.6) in H. Our aim is to give a similar interpretation for self-adjoint extensions AB of AM defined by (4.15) in the spaceHp. In what follows, without loss of generality, we assume that an auxiliary Hilbert space N in (⊕Np/2+1, Γ̂0, Γ̂1) coincides with C n (here n = dimN). So, ⊕Np/2+1 = C n(p/2+1). In this case the operator B in (4.15) is given by an Hermitian matrix B of the order n(p/2 + 1). It follows from the relation ker Γ1 = D(A) and equalities (2.2), (4.11) that ker Γ̂1 = = D(Ap+1). Hence, the restriction Γ̂1 �M determines a one-to-one correspondence be- tween M and C n(p/2+1). Thus ( Γ̂1 �M )−1 exists and ( Γ̂1 �M )−1 maps C n(p/2+1) onto M. Putting Ψd := −Ap+1(Γ̂1 �M )−1d, where d ∈ C n(p/2+1), we determine an injective linear mapping of C n(p/2+1) toH−p−2 such thatR(Ψ) ∩H = {0}. Let us determine its adjoint Ψ∗ : Hp+2 → C n(p/2+1) by the formula 〈u,Ψd〉 = (Ψ∗u, d)Cn(p/2+1) ∀u ∈ Hp+2 = D(Ap/2+1) ∀d ∈ C n(p/2+1). (4.16) To describe Ψ∗ we set ψj = Ψej , where {ej}n(p/2+1) 1 is the canonical basis of C n(p/2+1). Setting f = u ∈ D(Ap+1) and g = A−p−1ψj = A−p−1Ψej = = −(Γ̂1 �M )−1ej in the Green identity (4.12), using (4.10), and recalling that ker Γ̂1 = = D(Ap+1), we get 〈u, ψj〉 = (Ap+1u,A−p−1ψj) = −(Γ̂0u, Γ̂1g)Cn(p/2+1) = (Γ̂0u, ej)Cn(p/2+1) . The latter relation and (4.16) imply that Ψ∗u =  〈u, ψ1〉 ... 〈u, ψn(p/2+1)〉  = Γ̂0u (4.17) for ‘smooth’ vectors u ∈ D(Ap+1) = H2p+2. The continuation of Ψ∗ ontoD(Ap/2+1) = = Hp+2 is obtained by the closure. Let us consider the formal expression Ap + n(p/2+1)∑ i,j=1 bij〈·, ψj〉ψi = Ap + ΨBΨ∗, (4.18) where B = (bij) n(p/2+1) ij is an Hermitian matrix of the order n(p/2 + 1) and Ap = = A �D(Ap/2+1) is a self-adjoint operator inHp. In general, the singular elements ψj belong toH−p−2 and hence, they are well defined on u ∈ Hp+2 . For this reason it is natural to consider the ‘potential’ V = ΨBΨ∗ in (4.18) as a singular perturbation of the ‘free’ operator Ap in Hp and, reasoning by ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 6 742 S. ALBEVERIO, S. KUZHEL, L. NIZHNIK analogy with Subsection 2.2.1, to give a meaning of the formal expression (4.18) as a self-adjoint operator extension Ã of the symmetric operator (cf. (2.9)) Asym := Ap �D(Asym), D(Asym) = { u ∈ D(Ap/2+1) ∣∣ Ψ∗u = 0 } acting inHp. It follows from (4.8) and (4.16) that Asym = AM . So, in contrast to the operator Asym = AN defined by (2.9), the operator Asym = AM is non-densely defined. There- fore, a modification of the Albeverio – Kurasov approach (see Subsection 2.2.1) is re- quired to describe self-adjoint extensions of AM by additive mixed singular perturba- tion (4.18). First of all we restrict (4.18) to the set D(Ap+1) and define the action of (4.18) on vectors from the domain of definition D(A(∗) M ) = D(Ap+1)+̇M of the quasi-adjoint op- erator A(∗) M (in other words, we construct a regularization A+ p + ΨBΨ∗ R of (4.18) defined on D(Ap+1)+̇M ). Relation (4.17) means that the extension Ψ∗ R can naturally be defined by the boundary operator Γ̂0. Namely, Ψ∗ Rf =  〈f, ψex 1 〉 ... 〈f, ψex n(p/2+1)〉  := Γ̂0f ∀f ∈ D(Ap+1)+̇M. (4.19) The extension A+ p of Ap can be defined by analogy with (2.12). Precisely, we only need to indicate the action of A+ p on M. Assuming that A+ p �M acts as the isometric mapping Ap+1 in A-scale (see Subsection 2.2), we get A+ p f = Apu + Ap+1m = A (∗) M f + Ap+1m ∀f = u + m ∈ D(A(∗) M ). (4.20) After such a preparation work, the operator realization Ã of (4.18) in Hp is determined by the formula (cf. (2.11)) Ã = [A+ p + ΨBΨ∗ R] �D(Ã), D(Ã) = {f ∈ D(Ap+1)+̇M \| A+ p f + ΨBΨ∗ Rf ∈ Hp}. (4.21) Theorem 4.4. Let B be an Hermitian matrix of the order n(p/2 + 1). Then the operator Ã is essentially self-adjoint inHp and it can be also defined by the formula A′ B = A (∗) M �D(A′ B), D(A′ B) = { f ∈ D(Ap+1)+̇M ∣∣BΓ̂0f = Γ̂1f } . (4.22) Proof. By the definition of Ψ, Ap+1m = −ΨΓ̂1m = −ΨΓ̂1f for any m ∈ M and f = u + m (u ∈ D(Ap+1)). The obtained expression, (4.19), and (4.20) yield [A+ p + ΨBΨ∗ R]f = A (∗) M f + Ψ[BΓ̂0 − Γ̂1]f (∀f ∈ D(Ap+1)+̇M). (4.23) The latter equality means f ∈ D(Ã) ⇐⇒ BΓ̂0f = Γ̂1f (since R(Ψ) ∩ H = {0} and hence, R(Ψ) ∩Hp = {0}). Combining this fact with (4.21) – (4.23) we conclude that Ã coincides with A′ B. The property of the operator Ã to be essentially self-adjoint follows from Theorem 4.3. ISSN 1027-3190. Ukr. mat. 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Singularly perturbed self-adjoint operators in scales of Hilbert spaces

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