Lévy flights in an infinite potential well as a hypersingular Fredholm problem

Кириченко, Е. В.; Garbaczewski, Piotr; Stephanovich, V. A.; Żaba, Mariusz

doi:10.1103/physreve.93.052110

Cited by 25 publications

(44 citation statements)

References 57 publications

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“…Other object to apply the solutions of fractional quantum mechanical problems is oxide interfaces [29,30], where non-Gaussian quantum fluctuations occur both in phonon and electron spectra due to specific potential at the interface [31][32][33]. Here, both the above results on 1D fractional quantum oscillator and those for quantum well [17] can be well applied.…”

Section: Discussionmentioning

confidence: 99%

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Confinement of Lévy flights in a parabolic potential and fractional quantum oscillator

Кириченко

Stephanovich

2018

Phys. Rev. E

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We study Lévy flights confined in a parabolic potential. This has to do with a fractional generalization of ordinary quantum-mechanical oscillator problem. To solve the spectral problem for the fractional quantum oscillator, we pass to the momentum space, where we apply the variational method. This permits to obtain approximate analytical expressions for eigenvalues and eigenfunctions with very good accuracy. Latter fact has been checked by numerical solution of the problem. We point to the realistic physical systems ranging from multiferroics and oxide heterostructures to quantum chaotic excitons, where obtained results can be used.

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Section: Discussionmentioning

confidence: 99%

“…It turns out, however, that the matrix method, adopted in Ref. [17] for our problem, converges extremely slowly so that large (around 10 4 × 10 4 ) matrices are to be diagonalized. This, along with quite long time, needed to calculate each matrix element, renders this method unsuitable for our present problem.…”

Section: General Formalismmentioning

confidence: 99%

Confinement of Lévy flights in a parabolic potential and fractional quantum oscillator

Кириченко

Stephanovich

2018

Phys. Rev. E

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“…[59] (obtained by an alternative fractional Laplacian discretization method). Since our main purpose has been to test the computation method of [63,64] against an alternative proposal of Refs. [59,60], we refrain from a comparative listing of other eigenvalues and other α choices, see however [59].…”

Section: Reflected Motions In a Bounded Domainmentioning

confidence: 99%

“…The pertinent censored process never crosses or reaches the boundary, which is a property shared with taboo processes in the impenetrable enclosure, c.f. the fractional infnite square well spectral problem or the related taboo process in the interval (so-called ground state process), [15,[59][60][61][62][63][64]. Nonetheless, the associated stationary probability distributions appear to be very diferrent, behaving reciprocally at the boundary: quick decay with a lowering distance from the boundary ∼ (dist) α/2 (taboo process), versus blow up to infinity ∼ 1/(dist) α/2−1 in the same regime (censored process).…”

Section: Regional Fractional Laplacian: Signatures Of Reflecting Bmentioning

confidence: 99%

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

Garbaczewski

Stephanovich

2019

Phys. Rev. E

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The fractional Laplacian (−∆) α/2 , α ∈ (0, 2) has many equivalent (albeit formally different) realizations as a nonlocal generator of a family of α-stable stochastic processes in R n . On the other hand, if the process is to be restricted to a bounded domain, there are many inequivalent proposals for what a boundary-data respecting fractional Laplacian should actually be. This ambiguity holds true not only for each specific choice of the process behavior at the boundary (like e.g. absorbtion, reflection, conditioning or boundary taboos), but extends as well to its particular technical implementation (Dirchlet, Neumann, etc. problems). The inferred jump-type processes are inequivalent as well, differing in their spectral and statistical characteristics, which may strongly influence the ability of the formalism (if uncritically adopted) to provide an unambigous description of real geometrically confined physical systems with disorder. Specifically that refers to their relaxation properties and the near-equilibrium asymptotic behavior. In the present paper we focus on Lévy flight-induced jump-type processes which are constrained to stay forever inside a finite domain. That refers to a concept of taboo processes (imported from Brownian to Lévy -stable contexts), to so-called censored processes and to reflected Lévy flights whose status still remains to be unequivocally settled. As a byproduct of our fractional spectral analysis, with reference to Neumann boundary conditions, we discuss disordered semiconducting heterojunctions as the bounded domain problem. I. MOTIVATIONBrownian motion in a bounded domain is a classic problem with an ample coverage in the literature, specifically concerning the absorbing (Dirichlet) and reflecting (Neumann) boundary data (for the present purpose we disregard other boundary data choices). A coverage concerning their physical relevance is enormous as well [1,2].Anticipating further discussion, we quite inentionally point out source papers dealing with reflected Brownian motion, [3] and exposing at some length the method of eigenfuction expansions for the reflected and other boundarydata problems, [4]- [8], c.f. also [9]. The latter method is as well an indispensable tool in the analysis of spectral properties of fractional Laplacians and related jump-type processes in bounded domains. Its direct link with well developed theory of heat semigroups for jump-type processes (mostly these with absorpion/killing) allows to address the statistics of exits from the domain, like e.g. the first and mean first exit times, large time behavior, stationarity issues, probability of survival and its asymptotic decay, c.f.[10]- [15]. Compare e.g. also [16,17] (Brownian case) and [19][20][21] (Lévy-stable case), where the role of lowest eigenstates and eigenvalues (thence eigenvalue gaps) of the motion generator has appeared to be vital for the description of decay rates of killed stochastic processes. The spectral data of motion generators are relevant for quantifying long-living processes in a spatial...

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Existence theory and semi-analytical study of non-linear Volterra fractional integro-differential equations

Alrabaiah

Jamil

Shah

et al. 2020

Alexandria Engineering Journal

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Lévy flights in an infinite potential well as a hypersingular Fredholm problem

Cited by 25 publications

References 57 publications

Confinement of Lévy flights in a parabolic potential and fractional quantum oscillator

Confinement of Lévy flights in a parabolic potential and fractional quantum oscillator

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

Existence theory and semi-analytical study of non-linear Volterra fractional integro-differential equations

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