On the Point Spectrum of Self-Adjoint Operators That Appears under Singular Perturbations of Finite Rank

Abstract

We discuss purely singular finite-rank perturbations of a self-adjoint operator A in a Hilbert space ℋ. The perturbed operators \(\tilde A\) are defined by the Krein resolvent formula \((\tilde A - z)^{ - 1} = (A - z)^{ - 1} + B_z \) , Im z ≠ 0, where B _z are finite-rank operators such that dom B _z ∩ dom A = |0}. For an arbitrary system of orthonormal vectors \(\{ \psi _i \} _{i = 1}^{n < \infty } \) satisfying the condition span |ψ _i } ∩ dom A = |0} and an arbitrary collection of real numbers \({\lambda}_i \in {\mathbb{R}}^1\) , we construct an operator \(\tilde A\) that solves the eigenvalue problem \(\tilde A\psi _i = {\lambda}_i {\psi}_i , i = 1, \ldots ,n\) . We prove the uniqueness of \(\tilde A\) under the condition that rank B _z = n