Complex tridiagonal quantum Hamiltonians and matrix continued fractions

Abstract

Quantum resonances described by non-Hermitian tridiagonal-matrix Hamiltonians $H$ with complex energy eigenvalues are considered. The method of evaluation of quantities $\sigma_n$ known as the singular values of $H$ is proposed. Its basic idea is that the quantities $\sigma_n$ 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.