Asymptotics of Expansion of the Evolution Operator Kernel in Powers of Time Interval $Δt$

Abstract

The upper bound for asymptotic behavior of the coefficients of expansion of the evolution operator kernel in powers of the time interval $\Dt$ was obtained. It is found that for the nonpolynomial potentials the coefficients may increase as $n!$. But increasing may be more slow if the contributions with opposite signs cancel each other. Particularly, it is not excluded that for number of the potentials the expansion is convergent. For the polynomial potentials $\Dt$-expansion is certainly asymptotic one. The coefficients increase in this case as $Γ(n \frac{L-2}{L+2})$, where $L$ is the order of the polynom. It means that the point $\Dt=0$ is singular point of the kernel.