Fractional Laplacian in bended strip

TL;DR

This paper investigates the spectral properties of the fractional Laplacian in a bent waveguide, proving the existence of eigenvalues below the continuous spectrum threshold using the Caffarelli--Silvestre extension.

Contribution

It generalizes classical results for the local Laplace operator to the non-local fractional Laplacian in curved geometries, establishing conditions for trapped modes.

Findings

Existence of eigenvalues below the continuous spectrum threshold.

Conditions on curvature for trapped modes.

Application of Caffarelli--Silvestre extension to non-local operators.

Abstract

The spectral properties of the restricted fractional Laplacian with Dirichlet boundary conditions in a smoothly bent waveguide is investigated. The existence of eigenvalues below the threshold of the continuous spectrum is proved, generalizing classical results known for the local Laplace operator. Our approach utilizes the Caffarelli--Silvestre extension, addressing the specific geometric difficulties arising from the operator non-locality. The sufficient conditions on the curvature magnitude and distribution to ensure the existence of these trapped modes is established.