Uniform Asymptotic Solutions of a System of two Schrödinger Equations with Potential-Curve-Crossing Point

Abstract

A formal uniform asymptotic solution of the system of differential equations $ h^{2}\frac{d^{2}U_{1}}{dz^{2}}+Φ_{1} U_{1}=αU_{2} $ , $ h^{2}\frac{d^{2}U_{2}}{dz^{2}}+Φ_{2} U_{2}=αU_{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.