On the point spectrum of ℋ︁ –2 ‐singular perturbations

We prove that for any self‐adjoint operator A in a separable Hilbert space ℋ and a given countable set Λ = { λ i } i ∈ℕ of real numbers, there exist ℋ –2 ‐singular perturbations à of A such that Λ ⊂ σ p ( à ). In particular, if Λ = { λ 1 ,…, λ n } is finite, then the operator à solving the eigenvalues problem, à ψ k = λ k ψ k , k = 1,…, n , is uniquely defined by a given set of orthonormal vectors { ψ k } nk =1 satisfying the condition span { ψ k } nk =1 ∩ dom (| A | 1/2 ) = {0}. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)