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arXiv · 1503.07458

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

Mariusz Żaba·Piotr Garbaczewski
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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.

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Mariusz Żaba, Piotr Garbaczewski. 2015-03-25. Nonlocally-induced (fractional) bound states: Shape analysis in the infinite Cauchy well. https://doi.org/10.1063/1.4936645

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