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 $(-\Delta {)}^{1/2}$ 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.

21 pp. 15 figures, 3 tables

References in corpus (4)

• Ten equivalent definitions of the fractional Laplace operator
• Fractional Laplacian in Bounded Domains
• Levy flights and nonlocal quantum dynamics
• Solving fractional Schroedinger-type spectral problems: Cauchy oscillator and Cauchy well

Cited by in corpus (12)

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• Brownian motion in trapping enclosures: Steep potential wells, bistable wells and false bistability of induced Feynman-Kac (well) potentials
• Ultrarelativistic (Cauchy) spectral problem in the infinite well
• Levy flights in steep potential wells: Langevin modeling versus direct response to energy landscapes
• Levy processes in bounded domains: Path-wise reflection scenarios and signatures of confinement
• Ultrarelativistic bound states in the spherical well
• Fractional Laplacians and Levy flights in bounded domains
• Ultrarelativistic bound states in the shallow spherical well
• Tamed Feynman-Kac diffusion processes: Killing-branching intertwine