On the Similarity of Sturm-Liouville Operators with Non-Hermitian Boundary Conditions to Self-Adjoint and Normal Operators

David Krejčiřík*, Petr Siegl, Jakub Železný

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

24 Citations (Scopus)
188 Downloads (Pure)

Abstract

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.

Original languageEnglish
Pages (from-to)255-281
Number of pages27
JournalComplex Analysis and Operator Theory
Volume8
Issue number1
Early online date03 May 2013
DOIs
Publication statusPublished - Jan 2014

Keywords

  • C operator
  • Complex symmetric operator
  • Discrete spectral operator
  • Hilbert-Schmidt operators
  • Metric operator
  • Non-symmetric Robin boundary conditions
  • PT-symmetry
  • Similarity to normal or self-adjoint operators
  • Sturm-Liouville operators

ASJC Scopus subject areas

  • Computational Theory and Mathematics
  • Computational Mathematics
  • Applied Mathematics

Fingerprint Dive into the research topics of 'On the Similarity of Sturm-Liouville Operators with Non-Hermitian Boundary Conditions to Self-Adjoint and Normal Operators'. Together they form a unique fingerprint.

Cite this