Fractional Laplacian in V-shaped waveguide

TL;DR

This paper investigates the spectral properties of the fractional Laplacian in V-shaped waveguides, revealing the existence of discrete spectra at all junction angles and their dependence on the angle.

Contribution

It establishes the presence of discrete spectra at any junction angle and analyzes how these spectra vary with the angle, advancing understanding of fractional Laplacians in complex domains.

Findings

Discrete spectrum exists at all junction angles.

Discrete spectrum depends monotonically on the junction angle.

Continuous spectrum starts at the smallest eigenvalue of cross-sectional problems.

Abstract

The spectral properties of the restricted fractional Dirichlet Laplacian in ${\sf V}$-shaped waveguides are studied. The continuous spectrum for such domains with cylindrical outlets is known to occupy the ray $[\Lambda_\dagger, +\infty)$ with the threshold corresponding to the smallest eigenvalue of the cross-sectional problems. In this work the presence of a discrete spectrum at any junction angle is established along with the monotonic dependence of the discrete spectrum on the angle.