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