Asymptotics of Expansion of the Evolution Operator Kernel in Powers of Time Interval $\Delta t$

TL;DR

This paper investigates the asymptotic behavior of the expansion coefficients of the evolution operator kernel in powers of the time interval, revealing factorial growth for nonpolynomial potentials and asymptotic divergence for polynomial potentials.

Contribution

It provides bounds on the coefficients' growth and discusses conditions under which the expansion converges or diverges, highlighting the singularity at zero time interval.

Findings

Coefficients may grow as n! for nonpolynomial potentials.

For polynomial potentials, the coefficients grow as a gamma function, indicating an asymptotic expansion.

The point at zero time interval is a singular point of the kernel.

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 $\Gamma(n \frac{L-2}{L+2})$, where $L$ is the order of the polynom. It means that the point $\Dt=0$ is singular point of the kernel.