Complex tridiagonal quantum Hamiltonians and matrix continued fractions

Quantum resonances described by non-Hermitian tridiagonal-matrix Hamiltonians H with complex energy eigenvalues are considered. The possibility is analyzed of the evaluation of quantities ${\sigma }_n$ known as the singular values of H. What is constructed are self-adjoint block-tridiagonal operators $H$ (with eigenvalues ${\sigma }_n$) and their resolvents (defined in terms of a matrix continued fraction, MCF). In an illustrative application of the formalism to the discrete version of conventional $H=-d^2/d{x}^2+V\left(x\right)$ with complex local $V\left(x\right)\ne V^{⁎}\left(x\right)$, the numerical MCF convergence is found quick and, moreover, supported also by a fixed-point-based formal proof.