A generalized family of discrete PT-symmetric square wells

TL;DR

This paper introduces a family of discrete PT-symmetric square-well Hamiltonians with real spectra, exploring their properties and explicitly constructing the metrics needed for unitarity, emphasizing their computational simplicity.

Contribution

It presents a systematic set of discrete PT-symmetric Hamiltonians with explicit metric construction, extending the understanding of non-Hermitian quantum models with boundary-localized non-Hermiticity.

Findings

Hamiltonians have real spectra and localized non-Hermiticity

Explicit polynomial forms of the metric are obtainable

The Hamiltonians are computationally straightforward to analyze

Abstract

N-site-lattice Hamiltonians H are introduced and perceived as a set of systematic discrete approximants of a certain PT-symmetric square-well-potential model with the real spectrum and with a non-Hermiticity which is localized near the boundaries of the interval. Its strength is controlled by one, two or three parameters. The problem of the explicit construction of a nontrivial metric which makes the theory unitary is then addressed. It is proposed and demonstrated that due to the not too complicated tridiagonal-matrix form of our input Hamiltonians the computation of the metric is straightforward and that its matrix elements prove obtainable, non-numerically, in elementary polynomial forms.