On the point spectrum of ℋ︁_–2‐singular perturbations

Abstract

We prove that for any self-adjoint operator A in a separable Hilbert space H and a given countable set Λ = {λi} i∈N of real numbers, there exist H−2-singular perturbationsà of A such that Λ ⊂ σp`ô. In particular, if Λ = {λ1, . . . , λn} is finite, then the operatorà solving the eigenvalues problem,Ãψ k = λ k ψ k , k = 1, . . . , n, is uniquely defined by a given set of orthonormal vectors {ψ k } n k=1 satisfying the condition span{ψ k } n k=1 ∩ dom`|A| 1/2´= {0}.

Jacobi matrices associated with the inverse eigenvalue problem in the theory of singular perturbations of self-adjoint operators

“…Note that the singular perturbation α 1 〈 ⋅, ω 1 〉 ω 1 of rank one and the matrix element b 1 of the Jacobi matrix J 0 constructed here are uniquely determined by the operator A and a given pair E 1 , ψ 1 (for a detailed proof of this fact, see [5,6,9]). Note that we associate the number b 1 with the operator A 1 , despite the fact that relation (6) contains the operator A 0 .…”

“…where cl means closure in H. As follows from the results of [5,6] (see also [8,9]), for every finite n there exists a unique singularly perturbed self-adjoint operator A n ∈ P ws n A ( ) that solves the eigenvalue problem…”

“…For a given unbounded self-adjoint operator A ≥ 1 in a Hilbert space H, an arbitrary sequence of real numbers E j ∈ R, j = 1, 2, … , and a sequence of vectors ψ j ∈ H 1 ( A ) \ D ( A ) orthonormal in H and such that condition (1) is satisfied, the recursive procedure of the solution of the inverse eigenvalue problem (2) by using relations (5), (7), (9), and (11) [or (15) if equality (14) holds at a certain step] generates a sequence of singularly perturbed operators A n ∈ P ws n A ( ), which are, in turn, associated with a sequence of Jacobi matrices J n consistent with one another and converging to the matrix…”

Dual pair of eigenvalues in rank one singular nonsymmetric perturbations

“…But many standard facts of the singular perturbation theory of self-adjoint operators ( [5,14]) and their spectral properties have also corresponding analogies for expressions (2) and (3).…”

“…For realization of our ideas we use the mathematical tools from [9]. The tools in the case of self-adjoint operator (Ã =Ã * ) were used in [2].…”