Evolution equations on non flat waveguides

TL;DR

This paper studies dispersive evolution equations on non-flat, unbounded waveguides with a focus on operators with a potential, establishing sharp estimates and spectral properties for such geometries.

Contribution

It introduces the concept of a repulsive waveguide and proves sharp estimates for the Helmholtz equation, along with smoothing and Strichartz estimates for related evolution equations.

Findings

Proves sharp Helmholtz estimates on repulsive waveguides

Establishes smoothing estimates for Schrödinger and wave equations

Shows the operator has no eigenvalues in this setting

Abstract

We investigate the dispersive properties of evolution equations on waveguides with a non flat shape. More precisely we consider an operator $H=-\Delta_{x}-\Delta_{y}+V(x,y)$ with Dirichled boundary condition on an unbounded domain $\Omega$, and we introduce the notion of a \emph{repulsive waveguide} along the direction of the first group of variables $x$. If $\Omega$ is a repulsive waveguide, we prove a sharp estimate for the Helmholtz equation $Hu-\lambda u=f$. As consequences we prove smoothing estimates for the Schr\"odinger and wave equations associated to $H$, and Strichartz estimates for the Schr\"odinger equation. Additionally, we deduce that the operator $H$ does not admit eigenvalues.