Wave equation with slowly decaying potential: asymptotics and wave operators

TL;DR

This paper analyzes the long-time behavior of solutions to the one-dimensional wave equation with slowly decaying potentials, establishing asymptotics and the existence of wave operators, with results showing ballistic wave propagation under certain conditions.

Contribution

It proves the existence of modified wave operators and characterizes wave propagation for potentials that are square summable, providing sharp results in the field.

Findings

Wave solutions exhibit ballistic behavior when potential is square summable.

Existence of modified wave operators is established for the problem.

Long-time asymptotics of solutions are characterized precisely.

Abstract

For the one-dimensional case, we establish the long-time asymptotics of solution to Cauchy problem and prove existence of modified wave operators. In particular, we show that the part of the wave travels ballistically if the potential is square summable and this result is sharp.