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

Seminário do GFM

IIIUL, B2-01

2012-02-24 14:00
2012-02-24 15:00
2012-02-24
14:00
..
15:00

by Petr Siegl (GFMUL, Portugal)

We consider one-dimensional Schroedinger-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 in detail. We show that they can be expressed as the sum of the identity and an integral Hilbert-Schmidt operator. In the case of parity and time reversal boundary conditions, we establish closed integral-type formulae for the similarity transformations, derive the similar self-adjoint operator and also find the associated “charge conjugation” operator, which plays the role of fundamental symmetry in a Krein space reformulation of the problem.

Further generalisations to operators in two dimensional manifolds will be mentioned.

The talk is based on joint works with D. Krejcirik (NPI ASCR, Rez) and J. Zelezny} (FZU ASCR, Prague):

[1] D. Krejcirik, P. Siegl and J. Zelezny: On the similarity of Sturm-Liouville operators with non-Hermitian boundary conditions to self-adjoint and normal operators, preprint available at arXiv:1108.4946;

[2] D. Krejcirik and P. Siegl: PT-symmetric models in curved manifolds}, Journal of Physics A: Mathematical and Theoretical, 43 (2010), 485204 (30pp).

Further generalisations to operators in two dimensional manifolds will be mentioned.

The talk is based on joint works with D. Krejcirik (NPI ASCR, Rez) and J. Zelezny} (FZU ASCR, Prague):

[1] D. Krejcirik, P. Siegl and J. Zelezny: On the similarity of Sturm-Liouville operators with non-Hermitian boundary conditions to self-adjoint and normal operators, preprint available at arXiv:1108.4946;

[2] D. Krejcirik and P. Siegl: PT-symmetric models in curved manifolds}, Journal of Physics A: Mathematical and Theoretical, 43 (2010), 485204 (30pp).