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

Abstract

This paper is a direct offspring of Ref. [1] 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.2

Fractional Laplacians in bounded domains: Killed, reflected, censored, and taboo Lévy flights

“…Remark 3: Let us mention that solutions of the fractional infinite well problem, together with that of an approximating sequence of deepening fractional finite wells, based on the restricted fractional Laplacian, have been addressed in Refs. [46,[60][61][62]64] and [71][72][73]. Moreover, the fractional harmonic oscillator (including the so-called massless version.…”

“…Moreover, the fractional harmonic oscillator (including the so-called massless version. with the Cauchy generator involved)) has been addressed in a number of papers, [13,61,74] see also [75] for the quartic case.…”

“…The ground state function in the restricted case has been proposed in quite complicated analytic form (actrually it is an approximate expression for a "true" eigenfunction), [72,73]. Porspective ground state properties were also analyzed in minute detail, with a numemrical assistance, [60][61][62]. The available data justify another approximation in terms of a function ψ(x) = C α,γ [(1 − x 2 ) cos(γx)] α/2 , where C α,γ stands for the L 2 (D) normalization factor, while γ is considered to be the "best-fit" parameter, allowing to approach quite closely the computer-assisted eigenfunction outcomes, [61].…”

“…Porspective ground state properties were also analyzed in minute detail, with a numemrical assistance, [60][61][62]. The available data justify another approximation in terms of a function ψ(x) = C α,γ [(1 − x 2 ) cos(γx)] α/2 , where C α,γ stands for the L 2 (D) normalization factor, while γ is considered to be the "best-fit" parameter, allowing to approach quite closely the computer-assisted eigenfunction outcomes, [61]. This may be directly compared with the outcome for the spectral case, whose the ground state cos(πx/2) is shared with the ordinary Laplacian in the interval.…”

“…It is our purpose to follow the pragmatic routine, tested before in our investigation of fractional Laplacians with exterior Dirichlet boundary conditions, [61,62,64]. We point out the existence of alternative computation routines (based on a direct discretization of fractional Laplacians), [60,86,87].…”