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

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.

27 pages

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• Complex magnetic fields: An improved Hardy-Laptev-Weidl inequality and quasi-self-adjointness
• Solvable quantum lattices with nonlocal non-Hermitian endpoint interactions
• Quantum star-graph analogues of PT-symmetric square wells. II: Spectra
• On S-matrix of Schrodinger Operators with Non-Symmetric Zero-Range Potentials
• Wave operators, similarity and dynamics for a class of Schroedinger operators with generic non-mixed interface conditions in 1D
• Hidden symmetries in non-self-adjoint graphs
• Coupled Susy, pseudo-bosons and a deformed $\mathfrak{su}(1,1)$ Lie algebra
• Contact interactions and Kronig-Penney Models in Hermitian and PT-symmetric Quantum Mechanics
• Mathematical and physical meaning of the crossings of energy levels in $\mathcal{P}\mathcal{T}-$symmetric systems