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

Кириченко, Е. В.; Stephanovich, V. A.

doi:10.1103/physreve.98.052127

Cited by 14 publications

(24 citation statements)

References 30 publications

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“…In the linear regime, the 1D problems of fractional quantum well 21 and quantum oscillator 22 had been solved. Possible physical realizations of 2D 23 and 3D hydrogen atom 24 as well as 2D hydrogen atom with Rytova–Keldysh (screened) interaction 25 had been pointed out along with solutions of corresponding problems.…”

Section: Introductionmentioning

confidence: 99%

Stabilization of 1D solitons by fractional derivatives in systems with quintic nonlinearity

Stephanovich

Olchawa

2022

Sci Rep

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We study theoretically the properties of a soliton solution of the fractional Schrödinger equation with quintic nonlinearity. Under “fractional” we understand the Schrödinger equation, where ordinary Laplacian (second spatial derivative in 1D) is substituted by its fractional counterpart with Lévy index $$\alpha$$ α . We speculate that the latter substitution corresponds to phenomenological account for disorder in a system. Using analytical (variational and perturbative) and numerical arguments, we have shown that while in the case of Schrödinger equation with the ordinary Laplacian (corresponding to Lévy index $$\alpha =2$$ α = 2 ), the soliton is unstable, even infinitesimal difference $$\alpha$$ α from 2 immediately stabilizes the soliton texture. Our analytical and numerical investigations of $$\omega (N)$$ ω ( N ) dependence ($$\omega$$ ω is soliton frequency and N its mass) show (within the famous Vakhitov–Kolokolov criterion) the stability of our soliton texture in the fractional $$\alpha <2$$ α < 2 case. Direct numerical analysis of the linear stability problem of soliton texture also confirms this point. We show analytically and numerically that fractional Schrödinger equation with quintic nonlinearity admits the existence of (stable) soliton textures at $$2/3<\alpha <2$$ 2 / 3 < α < 2 , which is in accord with existing literature data. These results may be relevant to both Bose–Einstein condensates in cold atomic gases and optical solitons in the disordered media.

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Section: Introductionmentioning

confidence: 99%

Stabilization of 1D solitons by fractional derivatives in systems with quintic nonlinearity

Stephanovich

Olchawa

2022

Sci Rep

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“…However, its application to quantum mechanics was started by the pioneering work of the Laskin and others . Based on this outstanding idea, intense investigations have been carried out in different fields of studies, for example, energy band structure for the periodic potential, position‐dependent mass fractional Schrödinger equation, fractional quantum oscillator, nuclear dynamics of the H 2 + molecular ion, propagation dynamics of a light beam, spatial soliton propagation, solitons in the fractional Schrödinger equation with parity‐time‐symmetric lattice potential, gap solitons, Rabi oscillations in a fractional Schrödinger equation, self‐focusing, and wave collapse, elliptic solitons, light propagation in honeycomb lattice, and so on. These studies are based on the different methods such as domain decomposition method, energy conservative difference scheme, conservative finite element method, fractional Fan subequation method, split‐step Fourier spectral method, transfer‐matrix method, and so on.…”

Section: Introductionmentioning

confidence: 99%

Quantum information entropies of multiple quantum well systems in fractional Schrödinger equations

Solaimani

Dong

2019

Int J of Quantum Chemistry

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In this work, we study the position and momentum information entropies of multiple quantum well systems in fractional Schrödinger equations, which, to the best of our knowledge, have not so far been studied. Through a confining potential, their shape and number of wells (NOW) can be controlled by using a few tuning parameters; we present some interesting quantum effects that only appear in the fractional Schrödinger equation systems. One of the parameters denoted by the Ld can affect the position and momentum probability densities if the system is fractional (1 < α < 2). We find that the position (momentum) probability density tends to be more severely localized (delocalized) in more fractional systems (ie, in smaller values of α). Affecting the Ld on the position and momentum probability densities is a quantum effect that only appears in the fractional Schrödinger equations. Finally, we show that the Beckner Bialynicki‐Birula‐Mycieslki (BBM) inequality in the fractional Schrödinger equation is still satisfied by changing the confining potential amplitude Vconf, the NOW, the fractional parameter α, and the confining potential parameter Ld.

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“…It tells us that the solution to the problem exists only at 1 < µ ≤ 2. This is at odds with 1D fractional quantum mechanical problems of infinite potential well [20] and oscillator [21], where the solution exist for all admissible 0 < µ < 2. Second is the lifting of orbital degeneracy in fractional (µ < 2) case.…”

Section: Discussionmentioning

confidence: 97%

On the discreet spectrum of fractional quantum hydrogen atom in two dimensions

Stephanovich¹

2019

Phys. Scr.

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We consider a fractional generalization of two-dimensional (2D) quantum-mechanical Kepler problem corresponding to 2D hydrogen atom. Our main finding is that the solution for discreet spectrum exists only for µ > 1 (more specifically 1 < µ ≤ 2, where µ = 2 corresponds to "ordinary" 2D hydrogenic problem), where µ is the Lévy index. We show also that in fractional 2D hydrogen atom, the orbital momentum degeneracy is lifted so that its energy starts to depend not only on principal quantum number n but also on orbital m. To solve the spectral problem, we pass to the momentum representation, 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 also found the new integral representation (in terms of complete elliptic integrals) of Schrödinger equation for fractional hydrogen atom in momentum space. We point to the realistic physical systems like bulk semiconductors as well as their heterostructures, where obtained results can be used.

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

Cited by 14 publications

References 30 publications

Stabilization of 1D solitons by fractional derivatives in systems with quintic nonlinearity

Stabilization of 1D solitons by fractional derivatives in systems with quintic nonlinearity

Quantum information entropies of multiple quantum well systems in fractional Schrödinger equations

On the discreet spectrum of fractional quantum hydrogen atom in two dimensions

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