Pseudo-Hermitian Quantum Mechanics with Unbounded Metric Operators

TL;DR

This paper extends pseudo-Hermitian quantum mechanics to include unbounded metric operators, detailing the construction of the physical Hilbert space, observables, and equivalent Hermitian Hamiltonian for specific spectral conditions.

Contribution

It introduces a framework for handling unbounded metric operators in pseudo-Hermitian quantum mechanics, expanding the applicability of the theory.

Findings

Constructed the physical Hilbert space for unbounded eta

Defined observables and equivalent Hermitian Hamiltonian in this setting

Ensured the spectrum is real and discrete with eigenvectors in the domain of eta

Abstract

We extend the formulation of pseudo-Hermitian quantum mechanics to eta-pseudo-Hermitian Hamiltonian operators H with an unbounded metric operator eta. In particular, we give the details of the construction of the physical Hilbert space, observables, and equivalent Hermitian Hamiltonian for the case that H has a real and discrete spectrum and its eigenvectors belong to the domain of eta and consequently its positive square root.