Mode transformation for a Schrödinger type equation: Avoided and unavoidable level crossings

Abstract

Methods elaborated in quantum mechanics for the Landau–Zener problem are generalized to study the non-adiabatic transitions in a wide class of problems of wave propagation, in particular in the waveguide problems. If the properties of the waveguide slowly vary along its axis and the phase velocities of two modes have a degeneracy point or are almost degenerate near some point, the transformation of modes may occur. The conditions are formulated under which we can find formal asymptotic expansions of modes outside the vicinity of the degeneracy point and write out explicitly the transition matrix. The starting point is rewriting the governing equations in the form of the Schrödinger type equation. The Hamiltonian is assumed to be the result of a small perturbation of an operator with a degeneracy point of the crossing types of two eigenvalues. The perturbation of the Hamiltonian produces a close pair of simple degeneracy points. Two regimes of mode transformation for the Schrödinger type equation are identified: avoided crossing of eigenvalues (corresponding to complex degeneracy points) and an explicit unavoidable crossing (with real degeneracy points).