On inverse spectral theory for singularly perturbed operators: point spectrum

Abstract

Let A be a self-adjoint operator in a separable Hilbert space H. Let {ψ k : k ≥ 1} be a given (finite or infinite) orthonormal system such that span{ψ k : k ≥ 1} ∩ dom(A) = {0} and let Λ := {λ k : k ≥ 1} be an arbitrary sequence of real numbers. Conditions for the existence of a pure singular per-turbationà of A solving the inverse eigenvalue problemÃψ k = λ k ψ k , k ≥ 1 are given.

Point Spectrum of Singularly Perturbed Self-Adjoint Operators

“…In the present paper (see also [10]), we generalize and develop the results of [7 -9] to the case of infinite Λ and Ψ. In Sec.…”

“…It is easy to see (see [10]) that the condition of the denseness of the set D 0 in H is equivalent to the condition that, for a certain (and, hence, for any) λ ∈ ρ( A ), the following relation is true:…”

“…Conversely, for infinite Λ and Ψ the set D 0 may be nondense in H. Below, we give a sufficient condition for the denseness of D 0 (see also [10]). …”

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

“…The matrix elements of J can be expressed via the given sequence of numbers E j , j = 1, 2, … , vectors ψ j , and operators A n according to relations (6), (8), (10), and (13).…”

“…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…”