On the point spectrum of ℋ–2-singular perturbations

✍ Scribed by Sergio Albeverio; Mykola Dudkin; Alexei Konstantinov; Volodymyr Koshmanenko

• Publisher: John Wiley and Sons
• Year: 2007
• Tongue: English
• Weight: 132 KB
• Volume: 280
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.200410461

✦ Synopsis

Abstract

We prove that for any self‐adjoint operator A in a separable Hilbert space ℋ︁ and a given countable set Λ = {λ ~i~ }~i ∈ℕ~ of real numbers, there exist ℋ︁~–2~‐singular perturbations à of A such that Λ ⊂ σ ~p~ (Ã). 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~ }^n^ ~k =1~ satisfying the condition span {ψ ~k~ }^n^ ~k =1~ ∩ dom (|A |^1/2^) = {0}. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)