On the similarity of Sturm-Liouville operators with non-Hermitian boundary conditions to self-adjoint and normal operators

Siegl, Petr; Železný, Jakub; Krejčiřík, David (2014). On the similarity of Sturm-Liouville operators with non-Hermitian boundary conditions to self-adjoint and normal operators. Complex analysis and operator theory, 8(1), pp. 255-281. Birkhäuser 10.1007/s11785-013-0301-y

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We consider one-dimensional Schrödinger-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 Schrödinger operator and also find the associated “charge conjugation” operator, which plays the role of fundamental symmetry in a Krein-space reformulation of the problem.

Item Type:

Journal Article (Original Article)

Division/Institute:

08 Faculty of Science > Department of Mathematics and Statistics > Institute of Mathematics

UniBE Contributor:

Siegl, Petr

Subjects:

500 Science > 510 Mathematics

ISSN:

1661-8254

Publisher:

Birkhäuser

Language:

English

Submitter:

Mario Amrein

Date Deposited:

14 Apr 2015 15:51

Last Modified:

14 Apr 2015 15:51

Publisher DOI:

10.1007/s11785-013-0301-y

BORIS DOI:

10.7892/boris.66714

URI:

https://boris.unibe.ch/id/eprint/66714

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