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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.
©2015 American Institute of Physics

##### MSC:
 34L10 Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions of ordinary differential operators 34A08 Fractional ordinary differential equations and fractional differential inclusions 34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations 34D15 Singular perturbations of ordinary differential equations 34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
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