A unifying asymptotic approach for nonadiabatic transitions near pairs of real or complex turning points

TL;DR

This paper develops a unified asymptotic method for analyzing nonadiabatic transitions in Schrödinger equations near pairs of real or complex degeneracy points, providing explicit formulas and practical recipes.

Contribution

It introduces a unifying asymptotic framework that handles both real and complex degeneracy points using matched asymptotic expansions and parabolic cylinder functions.

Findings

Derived transition matrix connecting adiabatic modes

Constructed asymptotic expansion with parabolic cylinder functions

Provided a practical recipe for applying results to physical problems

Abstract

An asymptotic approach for a Schroedinger type equation with non selfadjoint Hamiltonian of a special type in the case of two close degeneracy (turning) points is developed. Both real and complex degeneracy points are treated by a method of matched asymptotic expansions in the context of a unifying approach. An asymptotic expansion near degeneracy point containing the parabolic cylinder functions is constructed and the transition matrix connecting the coefficients of adiabatic modes in front of and behind the degeneracy point is derived. A simple non-technical recipe is also provided, which enables one to apply results to different physical problems without performing intermediate calculations.