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Nonlocally induced (fractional) bound states: shape analysis in the infinite Cauchy well. (English) Zbl 1332.34133
Summary: 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 $$(-\Delta)^{1/2}$$ in the interval $$D = (-1, 1) \subset R$$, 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.