Uniform Asymptotic Solutions of a System of two Schr\"odinger Equations with Potential-Curve-Crossing Point

TL;DR

This paper develops a uniform asymptotic solution for a coupled system of Schrödinger equations near a potential curve-crossing point, using parabolic cylinder functions, and analyzes the properties of the solution coefficients.

Contribution

It introduces a formal uniform asymptotic method for solving coupled Schrödinger equations with potential crossing points, expanding solutions in terms of special functions and analyzing their properties.

Findings

Derived asymptotic solutions using parabolic cylinder functions.

Analyzed the analytical properties of expansion coefficients.

Extended the method to cases involving potential barriers.

Abstract

A formal uniform asymptotic solution of the system of differential equations $ h^{2}\frac{d^{2}U_{1}}{dz^{2}}+\Phi_{1} U_{1}=\alpha U_{2} $ , $ h^{2}\frac{d^{2}U_{2}}{dz^{2}}+\Phi_{2} U_{2}=\alpha U_{1}$ , for $ z\in D$ and for h real, large is obtained, when D contains curve-crossing point. Asymptotic approximations for the solutions are constructed in terms of parabolic cylinder functions. Analytical properties of the expansion's coefficients are investigated.The case of potantial barier is also considered.