Complex tridiagonal quantum Hamiltonians and matrix continued fractions
Miloslav Znojil
TL;DR
The paper addresses resonances in open quantum systems described by non-Hermitian tridiagonal Hamiltonians with complex spectra by introducing a Hermitian block-tridiagonal partner $\mathbb{H}$ whose eigenvalues are the singular values $\sigma_n$ of $H$. It develops a matrix continued-fraction (MCF) framework to express the resolvent of $\mathbb{H}$, enabling efficient computation of $\sigma_n$ and spectral-related quantities, and provides a fixed-point-based proof of convergence in a discretized model. The approach links analytic continued fractions for scalar problems to a matrix-valued analogue, offers concrete convergence criteria, and demonstrates rapid numerical convergence in a discrete complex potential example, while outlining conditions under which convergence can fail. This method advances practical analysis of resonances in open quantum systems and suggests broad applicability to block-structured non-Hermitian problems and their analytic properties.
Abstract
Quantum resonances described by non-Hermitian tridiagonal-matrix Hamiltonians $H$ with complex energy eigenvalues are considered. The method of evaluation of quantities $σ_n$ known as the singular values of $H$ is proposed. Its basic idea is that the quantities $σ_n$ can be treated as eigenvalues of an auxiliary self-adjoint operator $\mathbb{H}$. As long as such an operator can be given a block-tridiagonal matrix form, we finally expand its resolvent in terms of a matrix continued fraction (MCF). In an illustrative application, a discrete version of conventional Hamiltonian $H=-d^2/dx^2+V(x)$ with complex local $V(x) \neq V^*(x)$ is considered. The numerical MCF convergence is found quick, supported also by a fixed-point-based formal proof.