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

Abstract

Fractional (Lévy-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 (−Δ)1/2 in the interval D = (−1, 1) ⊂ R, with a focus on functional shapes of first few eigenfunctions and their fall-off at the boundary 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.