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 $H=-d^2/d{x}^2+V(x)$ with complex local $V(x)\ne V^{\ast }(x)$ is considered. The numerical MCF convergence is found quick, supported also by a fixed-point-based formal proof.

15 pp

References in corpus (3)

• Making Sense of Non-Hermitian Hamiltonians
• Mathematical and physical aspects of complex symmetric operators
• Dynamical regimes of a quantum swap gate beyond the Fermi Golden Rule

Cited by in corpus (3)

• Few-grid-point simulations of Big Bang singularity in quantum cosmology
• Resonances and continued-fraction Green's functions in non-Hermitian Bose-Hubbard-like quantum models
• Non-Hermitian Bose-Hubbard-like quantum models