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\title{CONNECTION FORMULAE FOR A SYSTEM OF TWO SCHR\"ODINGER
EQUATIONS WITH POTENTIAL-CURVE-CROSSING POINT}
\author{Irina N. Yakushina}
{St.Petersburg State Academy of Aerospace Instrumentation}

We consider the system of
differential equations
\begin{equation}
h^{2}\frac{d^{2}U_{1}}{dz^{2}}+\Phi_{1}(z)
U_{1}=\alpha(z,\delta) U_{2},
\qquad
h^{2}\frac{d^{2}U_{2}} {dz^{2}}+\Phi_{2}(z)
U_{2}=\alpha(z,\delta) U_{1},
\end{equation}
Here $\Phi_{1}, \Phi_{2} $ and $\alpha$ are functions of the
complex variable $z$ in the domain $D$ which depend on the parameter
$\delta$ in the complex domain $G$, $(0\in G)$. We assume that
$\alpha(z,0)\equiv 0$.The uniform asymptotic expansion for the
solution $U_{1}$ is constructed for small and positive values
of the parameter $h$ in the domain $\Omega=D\times G$,where
$D$ contains a crossing point $z=x_{0}$ (i.e. a point where
$\Phi_{1}(x_{0})=\Phi_{2}(x_{0})$).
The problem of energy level crossing appears in different brunches
of physics, for example in the problem of non-elastic collision
of two atoms with a different masses. The WKB-solutions of the
set of equations (1) are well known, but they are not correct in the vicinity
of the point $x_{0}$, where the energetic term cross.
The fourth-order differential equation for the function $U_{1}$ is
obtained,the coefficients and turning points of which depend on the
small parameter h. The asymptotic solution of this equation is
found in Olver's form in terms of parabolic cylinder functions.
The uniform approximations are re-approximated away from turning
points (away from a potantial-curve-crossing point) by replacing
the Weber functions by their asymptotic approximations for large
argument and for large order, yielding the W.K.B.-approximations
to the four linear independent solutions. The transition matrix,
which relate four W.K.B.-solutions for $z<-c$ and four W.K.B.-solutions
 for $z>c$ was
 constructed. The earliest investigation of transition matrix
for potantial-curve-crossing point appears to be the phase-integral
analysis of Stueckelberg (1937).It was developed by Crothers (1985)
for model case of linear potentials $\Phi_{1}$ and $\Phi_{2}$ and
constant coupling function $\alpha$. The comparison equation method
allows to find explicit asymptotic approximations for the coefficients
in the connection formulae for arbitrary potentials
and to estimate errors for the approximate
 coefficients.



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