2017
Том 69
№ 9

All Issues

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

Dudkin M. Ye., Koshmanenko V. D.

Full text (.pdf)


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