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

0425780 - ÚJF 2015 RIV CH eng J - Článek v odborném periodiku
Krejčiřík, David - Siegl, Petr - Železný, Jakub
On the Similarity of Sturm-Liouville Operators with Non-Hermitian Boundary Conditions to Self-Adjoint and Normal Operators.
Complex Analysis and Operator Theory. Roč. 8, č. 1 (2014), s. 255-281. ISSN 1661-8254. E-ISSN 1661-8262
Grant CEP: GA MŠMT LC06002; GA MŠMT LC527; GA ČR GAP203/11/0701
Grant ostatní: GA ČR(CZ) GD202/08/H072
Institucionální podpora: RVO:61389005
Klíčová slova: Sturm-Liouville operators * non-symmetric Robin boundary conditions * similarity to normal or self-adjoint operators * discrete spectral operator * complex symmetric operator * PT-symmetry * metric operator * C operator * Hilbert-Schmidt operators
Kód oboru RIV: BE - Teoretická fyzika
Impakt faktor: 0.545, rok: 2014

We consider one-dimensional Schrodinger-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 Schrodinger operator and also find the associated "charge conjugation" operator, which plays the role of fundamental symmetry in a Krein-space reformulation of the problem.
Trvalý link: http://hdl.handle.net/11104/0231575