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.