Convergence in $L^p$ for Feynman path integrals

TL;DR

This paper proves sharp convergence results in $L^p$ for Feynman path integrals associated with Schrödinger equations with specific time-dependent potentials, even over long times without smoothing effects.

Contribution

It establishes the first sharp $L^p$ convergence results for Feynman path integrals with long-time validity and no smoothing, using phase space wave packet techniques.

Findings

Convergence in $L^p$ with loss of derivatives is achieved for Schrödinger equations with linear and quadratic growth potentials.

Results are sharp and valid for long time intervals.

Techniques involve phase space decomposition and reconstruction of functions and operators.

Abstract

We consider a class of Schrodinger equations with time-dependent smooth magnetic and electric potentials having a growth at infinity at most linear and quadratic, respectively. We study the convergence in $L^p$ with loss of derivatives, $1<p<\infty$, of the time slicing approximations of the corresponding Feynman path integral. The results are completely sharp and hold for long time, where no smoothing effect is available. The techniques are based on the decomposition and reconstruction of functions and operators with respect to certain wave packets in phase space.