Quasi-hermitian lattices with imaginary zero-range interactions

TL;DR

This paper investigates PT-symmetric tridiagonal Hamiltonians with imaginary interactions, exploring their spectral properties, metric operators, and the conditions for physical observability, with a focus on simplifying cases for explicit solutions.

Contribution

It introduces a broad class of quasi-hermitian lattices with imaginary zero-range interactions and provides methods for constructing metric operators and analyzing their spectral domains.

Findings

Numerical analysis of domains of observability and exceptional points.

Introduction of one-parametric models with closed-form metric solutions.

Identification of conditions for physical Hilbert space construction.

Abstract

We study a general class of PT-symmetric tridiagonal Hamiltonians with purely imaginary interaction terms in the quasi-hermitian representation of quantum mechanics. Our general Hamiltonian encompasses many previously studied lattice models as special cases. We provide numerical results regarding domains of observability and exceptional points, and discuss the possibility of explicit construction of general metric operators (which in turn determine all the physical Hilbert spaces). The condition of computational simplicity for the metrics motivates the introduction of certain one-parametric special cases, which consequently admit closed-form extrapolation patterns of the low-dimensional results.