Zig-zag-matrix algebras and solvable quasi-Hermitian quantum models

TL;DR

This paper explores a novel zig-zag-matrix representation for non-Hermitian quantum Hamiltonians with real spectra, aiming to simplify the analysis of bound states in quasi-Hermitian quantum models.

Contribution

It introduces the concept of zig-zag-matrix algebras as a new sparse matrix framework for non-Hermitian Hamiltonians with real spectra.

Findings

Proposes a zig-zag-matrix algebra structure for Hamiltonians.

Suggests this approach simplifies spectral analysis.

Provides a conjecture on the transferability of diagonalization to zig-zag matrices.

Abstract

It is well known that the unitary evolution of a closed $M-$level quantum system can be generated by a non-Hermitian Hamiltonian $H$ with real spectrum. Its Hermiticity can be restored via an amended inner-product metric $\Theta$. In Hermitian cases the evaluation of the spectrum (i.e., of the bound-state energies) is usually achieved by the diagonalization of the Hamiltonian. In the non-Hermitian (or, more precisely, in the $\Theta-$quasi-Hermitian) quantum mechanics we conjecture that the role of the diagonalized-matrix solution of the quantum bound-state problem could be transferred to a maximally sparse ``zig-zag-matrix'' representation of the Hamiltonians.