On the Pseudo-Hermiticity of a Class of PT-Symmetric Hamiltonians in One Dimension

TL;DR

This paper demonstrates that a broad class of PT-symmetric Hamiltonians in one dimension are pseudo-Hermitian by explicitly constructing an invertible antilinear operator, without assuming diagonalizability or discrete spectrum.

Contribution

It provides a general construction of an antilinear operator showing PT-symmetric Hamiltonians are pseudo-Hermitian, extending previous results beyond diagonalizable cases.

Findings

Constructed an invertible antilinear operator u for u-anti-pseudo-Hermiticity.

Explicit form of a linear Hermitian operator for PT-symmetric Hamiltonians.

Results apply without assuming Hamiltonian diagonalizability or discrete spectrum.

Abstract

For a given standard Hamiltonian H=[p-A(x)]^2/(2m)+V(x) with arbitrary complex scalar potential V and vector potential A, with x real, we construct an invertible antilinear operator \tau such that H is \tau-anti-pseudo-Hermitian, i.e., H^\dagger=\tau H\tau^{-1}. We use this result to give the explicit form of a linear Hermitian invertible operator with respect to which any standard PT-symmetric Hamiltonian with a real degree of freedom is pseudo-Hermitian. Our results do not make use of the assumption that H is diagonalizable or that its spectrum is discrete.