On the Similarity of Sturm–Liouville Operators with Non-Hermitian Boundary Conditions to Self-Adjoint and Normal Operators
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- 作者：David Krej?i?ík (1) (2)
Petr Siegl (1) (3) (4)
Jakub ?elezny (1) (3) - 关键词：Sturm–Liouville operators ; Non ; symmetric Robin boundary conditions ; Similarity to normal or self ; adjoint operators ; Discrete spectral operator ; Complex symmetric operator ; $$\mathcal{PT }$$ ; symmetry ; Metric operator ; $$\mathcal{C }$$ operator ; Hilbert–Schmidt operators ; Primary 34B24 ; 47B40 ; 34L10 ; Secondary 34L40 ; 34L05 ; 81Q12
- 刊名：Complex Analysis and Operator Theory
- 出版年：2014
- 出版时间：January 2014
- 年：2014
- 卷：8
- 期：1
- 页码：255-281
- 全文大小：361 KB
- 作者单位：David Krej?i?ík (1) (2)
Petr Siegl (1) (3) (4)
Jakub ?elezny (1) (3)
1. Department of Theoretical Physics, Nuclear Physics Institute, Czech Academy of Sciences, ?e?, Czech Republic
2. IKERBASQUE, Basque Foundation for Science, Alameda Urquijo, 36, 5, 48011, Bilbao, Spain
3. Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Prague, Czech Republic
4. Laboratoire Astroparticule et Cosmologie, Université Paris 7, Paris, France - ISSN：1661-8262
文摘
We consider one-dimensional Schr?dinger-type operators in a bounded interval with non-self-adjoint Robin-type boundary conditions. It is well known that such operators are generically conjugate to normal operators via a similarity transformation. Motivated by recent interests in quasi-Hermitian Hamiltonians in quantum mechanics, we study properties of the transformations and similar operators in detail. In the case of parity and time reversal boundary conditions, we establish closed integral-type formulae for the similarity transformations, derive a non-local self-adjoint operator similar to the Schr?dinger operator and also find the associated “charge conjugation-operator, which plays the role of fundamental symmetry in a Krein-space reformulation of the problem.