On the Point Spectrum of Self-Adjoint Operators That Appears under Singular Perturbations of Finite Rank - Dudkin, M. Ye.; Koshmanenko, V. D. (Ukrains’kyi Matematychnyi Zhurnal, 2003)
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 \(\{ ψ _i \} _{i = 1}^{n < ∞ } \) satisfying the condition span |ψ i } ∩ dom A = |0} and an arbitrary collection of real numbers \(λ_i \in {\mathbb{R}}^1\) , we construct an operator \(\tilde A\) that solves the eigenvalue problem \(\tilde Aψ _i = λ_i ψ_i , i = 1, … ,n\) . We prove the uniqueness of \(\tilde A\) under the condition that rank B z = n.