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

TL;DR

This paper investigates non-Hermitian Sturm-Liouville operators with Robin boundary conditions, showing they are similar to normal operators via Hilbert-Schmidt transformations, and explores their properties in quantum mechanics contexts.

Contribution

It provides explicit integral formulas for similarity transformations and constructs associated self-adjoint and charge conjugation operators for specific boundary conditions.

Findings

Transformations are sums of identity and Hilbert-Schmidt operators.

Explicit integral formulas for similarity transformations are derived.

Constructed charge conjugation operator as a fundamental symmetry.

Abstract

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.