Complete Set of Inner Products for a Discrete PT-symmetric Square-well Hamiltonian

TL;DR

This paper constructs a complete set of inner products (metrics) for a discrete non-Hermitian square-well Hamiltonian, enabling a consistent quantum description across various parameters and system sizes.

Contribution

It provides a closed-form construction of all possible metrics for a discrete PT-symmetric Hamiltonian for any size and coupling within a specified range.

Findings

Explicit formulas for metrics for all N and λ

Demonstrates the existence of multiple valid inner products

Enables consistent quantum interpretation of the model

Abstract

A discrete $N-$point Runge-Kutta version $H^{(N)}({\lambda})$ of one of the simplest non-Hermitian square-well Hamiltonians with real spectrum is studied. A complete set of its possible hermitizations (i.e., of the eligible metrics $\Theta^{(N)}({\lambda})$ defining its non-equivalent physical Hilbert spaces of states) is constructed, in closed form, for any coupling ${\lambda}\in (-1,1)$ and any matrix dimension $N$.