TY - JOUR

T1 - Mode transformation for a Schrödinger type equation

T2 - Avoided and unavoidable level crossings

AU - Fialkovsky, Ignat

AU - Perel, Maria

N1 - Publisher Copyright: © 2021 Cambridge University Press. All rights reserved. Copyright: Copyright 2021 Elsevier B.V., All rights reserved.

PY - 2020

Y1 - 2020

N2 - 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).

AB - 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).

UR - http://www.scopus.com/inward/record.url?scp=85104880522&partnerID=8YFLogxK

U2 - 10.1063/1.5129795

DO - 10.1063/1.5129795

M3 - Article

AN - SCOPUS:85104880522

VL - 61

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

SN - 0022-2488

IS - 4

M1 - 043506

ER -