On the similarity of Sturm-Liouville operators with non-Hermitian boundary conditions to self-adjoint and normal operators. (arXiv:1108.4946v1 [math.SP])

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.

quant-ph updates on arXiv.org

Incoming search terms:

• hermitian adjoint of the quantities
• non hermitian sturm liouville