Operator interpretation of resonances arising in spectral problems for 2x2 matrix Hamiltonians

TL;DR

This paper develops a mathematical framework to interpret resonances in spectral problems of 2x2 matrix Hamiltonians by constructing non-selfadjoint operators, analyzing their spectra, and establishing completeness properties of their root vectors.

Contribution

It introduces a novel operator-theoretic approach to analyze resonances in 2x2 matrix Hamiltonians through operator roots of the transfer function.

Findings

Constructed non-selfadjoint operators representing operator roots of the transfer function.

Proved completeness and basis properties for the root vectors, including resonances.

Analyzed the spectrum in the unphysical sheets of the energy Riemann surface.

Abstract

We consider the analytic continuation of the transfer function for a 2x2 matrix Hamiltonian into the unphysical sheets of the energy Riemann surface. We construct non-selfadjoint operators representing operator roots of the transfer function which reproduce certain parts of its spectrum including resonances situated in the unphysical sheets neighboring the physical sheet. On this basis, completeness and basis properties for the root vectors of the transfer function (including those for the resonances) are proved.