Carleman estimates and necessary conditions for the existence of waveguides

TL;DR

This paper uses Carleman estimates to determine the optimal exponential decay rates for waveguide solutions to a perturbed Laplace equation, establishing a necessary condition based on the potential and vector field at infinity.

Contribution

It introduces a sharp decay estimate for waveguide solutions and derives a quantitative necessary condition involving the potential and vector field.

Findings

Identifies the sharpest exponential decay rate for waveguide solutions.

Establishes a necessary condition depending on the behavior of V and W at infinity.

Provides a quantitative criterion linking decay rate to potential and vector field.

Abstract

We study via Carleman estimates the sharpest possible exponential decay for {\it waveguide} solutions to the Laplace equation $$(\partial^2_t+\triangle)u=Vu+W\cdot(\partial_t,\nabla)u,$$ and find a necessary quantitative condition on the exponential decay in the spatial-variable of nonzero waveguides solutions which depends on the size of $V$ and $W$ at infinity.