Solving fractional Schroedinger-type spectral problems: Cauchy oscillator and Cauchy well

Abstract

This paper is a direct offspring of Ref. [J. Math. Phys. 54, 072103, (2013)] where basic tenets of the nonlocally induced random and quantum dynamics were analyzed. A number of mentions was maid with respect to various inconsistencies and faulty statements omnipresent in the literature devoted to so-called fractional quantum mechanics spectral problems. Presently, we give a decisive computer-assisted proof, for an exemplary finite and ultimately infinite Cauchy well problem, that spectral solutions proposed so far were plainly wrong. As a constructive input, we provide an explicit spectral solution of the finite Cauchy well. The infinite well emerges as a limiting case in a sequence of deepening finite wells. The employed numerical methodology (algorithm based on the Strang splitting method) has been tested for an exemplary Cauchy oscillator problem, whose analytic solution is available. An impact of the inherent spatial nonlocality of motion generators upon computer-assisted outcomes (potentially defective, in view of various cutoffs), i.e. detailed eigenvalues and shapes of eigenfunctions, has been analyzed.

arXiv comment20 pages, 12 figures, 7 tables

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• Eigenvalues of the fractional Laplace operator in the unit ball
• Fractional Laplacians in bounded domains: Killed, reflected, censored and taboo Lévy flights
• Symmetry-breaking transitions in quiescent and moving solitons in fractional couplers
• Killing (absorption) versus survival in random motion
• Factorization of the Quantum Fractional Oscillator
• Lévy flights in the infinite potential well as the hypersingular Fredholm problem
• Nonlocally-induced (fractional) bound states: Shape analysis in the infinite Cauchy well
• On the coherent states for a relativistic scalar particle
• Brownian motion in trapping enclosures: Steep potential wells, bistable wells and false bistability of induced Feynman-Kac (well) potentials
• Levy processes in bounded domains: Path-wise reflection scenarios and signatures of confinement
• Tamed Feynman-Kac diffusion processes: Killing-branching intertwine