Semiclassical Asymptotics Beyond All Orders for Simple Scattering Systems

摘要: The semiclassical limit $\varepsilon \to 0$ of the scattering matrix S associated with equation $i\varepsilon \frac{{d\varphi (t)}} {{dt}} = A(t)\varphi (t)$ is considered. If $A(x)$ an analytic $n \times n$ whose eigenvalues are real andnondegenerate for all $x \in {\bf R}$, computed asymptotically up to errors $O(e^{\kappa \varepsilon ^{ - 1} } )$, $\kappa > 0$. Moreover, case 2$ and under further assumptions on behavior continuations $A(x)$, exponentially small off diagonal elements given by asymptotic expression accurate relative )$. adiabatic transition probability time-dependent Schrodinger equation, above barrier reflection coefficient stationary total variation invariant a classical oscillator illustrate results.