Discrete quantum square well of the first kind

TL;DR

This paper introduces a toy-model quantum system with a non-Hermitian Hamiltonian characterized by Chebyshev polynomial spectra, and constructs associated metrics to ensure physical self-adjointness in a modified inner product space.

Contribution

It presents a novel discrete quantum square well model with explicit metric construction ensuring self-adjointness for non-Hermitian Hamiltonians.

Findings

Explicit Chebyshev polynomial spectrum for the Hamiltonian.

Closed-form band-matrix metrics ensuring self-adjointness.

Parameter-dependent metrics within the positivity domain.

Abstract

A toy-model quantum system is proposed. At a given integer $N$ it is defined by the pair of $N$ by $N$ real matrices $(H,\Theta)$ of which the first item $H$ specifies an elementary, diagonalizable non-Hermitian Hamiltonian $H \neq H^\dagger$ with the real and explicit spectrum given by the zeros of the $N-$th Chebyshev polynomial of the first kind. The second item $\Theta\neq I$ must be (and is being) constructed as the related Hilbert-space metric which specifies the (in general, non-unique) physical inner product and which renders our toy-model Hamiltonian selfadjoint, i.e., compatible with the Dieudonne equation $H^\dagger \Theta= \Theta\,H$. The elements of the (in principle, complete) set of the eligible metrics are then constructed in closed band-matrix form. They vary with our choice of the $N-$plet of optional parameters, $\Theta=\Theta(\vec{\kappa})>0$ which must be (and are being) selected as lying in the positivity domain of the metric, $\vec{\kappa} \in {\cal D}^{(physical)}$.