Nonlocally-induced (fractional) bound states: Shape analysis in the infinite Cauchy well

TL;DR

This paper investigates the spectral properties and eigenfunction shapes of the fractional Cauchy operator within a finite interval, providing new high-accuracy formulas and analyzing the relationship between shape fidelity and eigenvalue precision.

Contribution

It introduces novel high-accuracy formulas for eigenfunctions of the fractional Cauchy operator and explores their shape and boundary behavior in a finite domain.

Findings

New formulas for approximate eigenfunctions with high accuracy

Correlation between shape fidelity and eigenvalue evaluation finesse

Insights into boundary fall-off behavior of eigenfunctions

Abstract

Fractional (L\'{e}vy-type) operators are known to be spatially nonlocal. This becomes an issue if confronted with a priori imposed exterior Dirichlet boundary data. We address spectral properties of the prototype example of the Cauchy operator $(-\Delta )^{1/2}$ in the interval $D=(-1,1) \subset R$, with a focus on functional shapes of lowest eigenfunctions and their fall-off at the boundaries of $D$. New high accuracy formulas are deduced for approximate eigenfunctions. We analyze how their shape reproduction fidelity is correlated with the evaluation finesse of the corresponding eigenvalues.