Dispersion for 1-d Scrodinger and wave equation with BV coefficients

TL;DR

This paper investigates how the variation of BV coefficients affects dispersion in 1D wave and Schrödinger equations, establishing conditions for dispersion occurrence and highlighting differences between the two equations.

Contribution

It provides a complete characterization of dispersion for the wave equation based on BV coefficient variation and extends the analysis to Schrödinger equations under similar conditions.

Findings

Dispersion occurs if the variation of the logarithm of the coefficient is small.

Dispersion may fail for the wave equation when variation exceeds a threshold.

Dispersion for the Schrödinger equation holds under small variation, but larger variation effects remain unknown.

Abstract

In this paper we analyze the dispersion for one dimensional wave and Schrodinger equations with BV coefficients. In the case of the wave equation we give a complete answer in terms of the variation of the logarithm of the coefficient showing that dispersion occurs if this variation is small enough but it may fail when the variation goes beyond a sharp threshold. For the Schrodigner equation we prove that the dispersion holds under the same smallness assumption on the variation of the coefficient. But, whether dispersion may fail for larger coefficients is unknown for the Schrodinger equation.