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Similarity problem for non-self-adjoint extensions of symmetric operators. (English) Zbl 1176.47013
Janas, Jan (ed.) et al., Methods of spectral analysis in mathematical physics. Conference on operator theory, analysis and mathematical physics (OTAMP), Lund, Sweden, June 15–22, 2006. Basel: Birkhäuser (ISBN 978-3-7643-8754-9/hbk). Operator Theory: Advances and Applications 186, 267-283 (2009).
Let $$A$$ be a closed densely defined symmetric operator with equal deficiency indices. We denote by $$\tilde{A}_B$$ the almost solvable (non-selfadjoint) extension of $$A$$ which corresponds to a (non-selfadjoint) bounded linear operator $$B$$. V. Ryzhov [Oper. Theory, Adv. Appl. 174, 117–158 (2007; Zbl 1133.47008)] followed the construction of B. S. Pavlov [Izv. Akad. Nauk SSSR, Ser. Mat. 39, 123–148 (1975; Zbl 0317.47006)] (cf. also B. Sz.-Nagy and C. Foiaş [“Analyse harmonique des opérateurs de l’espace de Hilbert” (Budapest: Akadémiaí Kiadó; Paris: Masson et Cie) (1967; Zbl 0157.43201)] and S. N. Naboko [Proc. Steklov Inst. Math. 147, 85–116 (1981; Zbl 0463.47009)]) to develop a functional model for the dissipative operator $$\tilde{A}_{B_+}$$ which corresponds to the “parameter” $$B_+=\operatorname{Re} B+i|\text{Im}\,B|$$.
This model is used in the paper under review to find necessary and sufficient conditions for $$\tilde{A}_B$$ (provided that it has absolute continuous spectrum) to be similar to a selfadjoint operator. The author obtains analogous results regarding the similarity problem also for the restrictions of $$\tilde{A}_B$$ to its spectral invariant subspaces which correspond to arbitrary Borel sets of the real line. The theory is illustrated by the study of the similarity problem for operators that arise when considering Schrödinger equations with potentials of zero radius in quantum mechanics. The results extend the ones established by A. V. Kiselev and M. M. Faddeev in [Funct. Anal. Appl. 34, No. 2, 143–145 (2000); translation from Funkts. Anal. Prilozh. 34, No. 2, 78–81 (2000; Zbl 0985.47017)].
For the entire collection see [Zbl 1148.46002].

##### MSC:
 47A45 Canonical models for contractions and nonselfadjoint linear operators 47A10 Spectrum, resolvent 47A20 Dilations, extensions, compressions of linear operators 47A48 Operator colligations (= nodes), vessels, linear systems, characteristic functions, realizations, etc. 47A55 Perturbation theory of linear operators 47B25 Linear symmetric and selfadjoint operators (unbounded)