2019
Том 71
№ 8

# 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.

English version (Springer): Ukrainian Mathematical Journal 55 (2003), no. 9, pp 1532-1541.

Citation Example: Dudkin M. Ye., Koshmanenko V. D. On the Point Spectrum of Self-Adjoint Operators That Appears under Singular Perturbations of Finite Rank // Ukr. Mat. Zh. - 2003. - 55, № 9. - pp. 1269-1276.

Full text