Evolution equations on non flat waveguides

2010/10/05 by D'Ancona, Piero, Racke, Reinhard · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.1010.0817

Abstract

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

Cited by

• Variational Principles for the Helmholtz equation: application to Finite Element and Neural Network approximations

Related

• Evolution of time-harmonic electromagnetic and acoustic waves along waveguides
• Dispersive estimates for wave and Schrödinger equations with a potential in non-trapping exterior domains
• Damping-induced dispersion in simple waveguides
• Evolution systems for paraxial wave equations of Schroedinger-type with non-smooth coefficients
• A radiation condition for uniqueness in a wave propagation problem for 2-D open waveguides