On the trace of the wave group and regularity of potentials

TL;DR

This paper investigates how the smoothness of a potential function influences the behavior of the wave group trace near zero time, establishing a precise link between potential regularity and trace expansion order.

Contribution

It provides a sharp relation connecting Sobolev regularity of potentials to the existence of finite order wave group trace expansions as time approaches zero.

Findings

Established a precise relation between potential regularity and trace expansion order.

Proved that higher regularity of potential implies more accurate trace expansions.

Identified the minimal regularity needed for specific trace expansion orders.

Abstract

For the wave equation $\partial_t^2-\Delta+V$ on $\mathbb{R}^d$ with compactly supported, real valued potential $V$, we establish a sharp relation between Sobolev regularity of $V$ and the existence of finite order expansions as $t\rightarrow 0$ for the relative trace of the wave group.