TY - JOUR
T1 - On the Similarity of Sturm-Liouville Operators with Non-Hermitian Boundary Conditions to Self-Adjoint and Normal Operators
AU - Krejčiřík, David
AU - Siegl, Petr
AU - Železný, Jakub
N1 - Funding Information:
Acknowledgments D.K. acknowledges the hospitality of the Deusto Public Library in Bilbao. This work has been partially supported by the Czech Ministry of Education, Youth, and Sports within the project LC06002 and by the GACR grant No. P203/11/0701. P.S. appreciates the support by GACR grant No. 202/08/H072 and by the Grant Agency of the Czech Technical University in Prague, grant No. SGS OHK4-010/10. J.Ž. appreciates the support by the Czech Ministry of Education, Youth, and Sports within the project LC527.
PY - 2014/1
Y1 - 2014/1
N2 - 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.
AB - 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.
KW - C operator
KW - Complex symmetric operator
KW - Discrete spectral operator
KW - Hilbert-Schmidt operators
KW - Metric operator
KW - Non-symmetric Robin boundary conditions
KW - PT-symmetry
KW - Similarity to normal or self-adjoint operators
KW - Sturm-Liouville operators
UR - http://www.scopus.com/inward/record.url?scp=84891624474&partnerID=8YFLogxK
U2 - 10.1007/s11785-013-0301-y
DO - 10.1007/s11785-013-0301-y
M3 - Article
AN - SCOPUS:84891624474
SN - 1661-8254
VL - 8
SP - 255
EP - 281
JO - Complex Analysis and Operator Theory
JF - Complex Analysis and Operator Theory
IS - 1
ER -