Ultrarelativistic (Cauchy) Spectral Problem in the Infinite Well

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

doi:10.5506/aphyspolb.47.1273

Cited by 6 publications

(32 citation statements)

References 29 publications

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“…This spectral issue has an ample coverage in the literature, and with regard to a detailed analysis of various eigenvalue problems for restricted fractional Laplacians, especially of the fully computable 1D case in the interval (−1, 1) = D ⊂ R, we mention Refs. [58], [59] - [64], see also [56].…”

Section: Restricted (Hypersingular) Fractional Laplacianmentioning

confidence: 99%

“…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.…”

Section: Spectral Fractional Laplacianmentioning

confidence: 99%

“…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].…”

Section: Reflected Motions In a Bounded Domainmentioning

confidence: 99%

“…The latter topic, for the Dirichlet fractional Laplacians, has been addressed before in a number of papers in the whole stability parameter range 0 < α < 2 and in more detail, for the distinguished α = 1 value (Cauchy process and motion generator) in Refs. [33][34][35]37], [46,[60][61][62]64] and [55,74].…”

Section: Reflected Motions In a Bounded Domainmentioning

confidence: 99%

“…The emergent spectrum is discrete E ≡ E n , n = 1, 2..., non-degenerate and positive E n > 0, see e.g. [58,63,64] and references therein. The regional fractional Laplacian is defined similarly to Eq.…”

Section: Reflected Motions In a Bounded Domainmentioning

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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Section: Restricted (Hypersingular) Fractional Laplacianmentioning

confidence: 99%

Section: Spectral Fractional Laplacianmentioning

confidence: 99%

Section: Reflected Motions In a Bounded Domainmentioning

confidence: 99%

Section: Reflected Motions In a Bounded Domainmentioning

confidence: 99%

Section: Reflected Motions In a Bounded Domainmentioning

confidence: 99%

See 3 more Smart Citations

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

Garbaczewski

Stephanovich

2019

Phys. Rev. E

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Lévy processes in bounded domains: path-wise reflection scenarios and signatures of confinement

Garbaczewski

Żaba

2022

J. Phys. A: Math. Theor.

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We discuss an impact of various (path-wise) reflection-from-the barrier scenarios upon confining properties of a paradigmatic family of symmetric $\alpha $-stable L\'{e}vy processes, whose permanent residence in a finite interval on a line is secured by a two-sided reflection. Depending on the specific reflection "mechanism", the inferred jump-type processes differ in their spectral and statistical characteristics, like e.g. relaxation properties, and functional shapes of invariant (equilibrium, or asymptotic near-equilibrium) probability density functions in the interval. The analysis is carried out in conjunction with attempts to give meaning to the notion of a reflecting L\'{e}vy process, in terms of the domain of its motion generator, to which an invariant pdf (actually an eigenfunction) does belong.

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Fractional Laplacians and Levy flights in bounded domains

Garbaczewski

2018

Preprint

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We address Lévy-stable stochastic processes in bounded domains, with a focus on a discrimination between inequivalent proposals for what a boundary data-respecting fractional Laplacian (and thence the induced random process) should actually be. Versions considered are: restricted Dirichlet, spectral Dirichlet and regional (censored) fractional Laplacians. The affiliated random processes comprise: killed, reflected and conditioned Lévy flights, in particular those with an infinite life-time. The related concept of quasi-stationary distributions is briefly mentioned. I. MOTIVATIONJump-type Lévy processes in a bounded domain are a subject of an active study both in physics and mathematics communities. The physics-oriented research is conducted with some disregard to an ample coverage of the topic in the past and modern mathematical literature. The reason is rooted not only in a methodological gap between the practitioners' pragmatism and the mathematically rigorous reasoning. An important factor is a scarce (or even lack of) communication between various research groups and research streamlines. This refers not only to rather residual physics-mathematics interplay, but also to the mathematics community per se: relevant publications are scattered in a large number of highly specialized journals and easily escape the attention of potentially interested parties.Recently an attempt has been made to establish a common conceptual basis for varied frameworks in which fractional Laplacians appear. Formally looking different, but actually equivalent, definitions of fractional Laplacians, appropriate for the description of Lévy stable processes in R n , n ≥ 1, have been collected and their mutual relationships analyzed in minute detail in Ref. [1].There is a general consensus that the standard Fourier multiplier definition appears to be defective, if one passes to Lévy flights in a bounded domain. This is a consequence of an inherent nonlocality of Lévy-stable generators. Different proposals for boundary-data-respecting fractional Laplacians were given in the literature. Often, with a view towards more efficient computer-assisted calculations (that mostly in connection with nonlinear fractional differential equations, [11,12]).However, in contrast to the situation in R n , these proposals are known to be inequivalent, c.f. Refs.[2]- [16], see also [18,19]. Likewise, the induced jump-type processes are inequivalent and have different statistical characteristics. That in particular refers to a standard physical inventory, adapted directly from the Brownian motion studies [20]: the statistics of exits from the domain, e.g. first and mean first exit times, probability of survival, its long time behavior an ultimate asymptotic decay [21]- [27].Interestingly, the existence problem for jump-type processes with an infinite life-time in a bounded domain, seems to have been left aside in the physics literature (compare e.g. Ref. [28] in connection with diffusion processes and Ref. [30] for a preliminary discussion of the Cauchy process...

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Ultrarelativistic (Cauchy) Spectral Problem in the Infinite Well

Cited by 6 publications

References 29 publications

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

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

Lévy processes in bounded domains: path-wise reflection scenarios and signatures of confinement

Fractional Laplacians and Levy flights in bounded domains

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