Resonances and continued-fraction Green's functions in non-Hermitian Bose-Hubbard-like quantum models

TL;DR

This paper introduces a method to analyze resonances in non-Hermitian Bose-Hubbard-like models by computing singular values as poles of a Green's function expressed through matrix continued fractions, simplifying the localization of complex eigenvalues.

Contribution

It presents a novel approach to find singular values in non-Hermitian quantum models using continued fractions, under specific constraints like Hamiltonian tridiagonality.

Findings

Singular values can be obtained as poles of a Green's function.

The method applies to Bose-Hubbard-like models with complex Hamiltonians.

The approach simplifies the analysis of resonances in non-Hermitian systems.

Abstract

With resonances treated as eigenstates of a non-Hermitian quantum Hamiltonian, the task of localization of the complex energy eigenvalues is considered. The paper is devoted to the reduced version of this task in which one only computes the real quantities called singular values. It is shown that in such an approach (and under suitable constraints including the tridiagonality of the Hamiltdonian) the singular values can be sought as poles of an auxiliary Green's function expressible in terms of a doublet of matrix continued fractions. A family of multi-bosonic Bose-Hubbard-like complex Hamiltonians is recalled for illustration purposes.