Reanalysis of an open problem associated with the fractional Schrödinger equation

Sayevand, Khosro; Pichaghchi, K.

doi:10.1134/s0040577917070078

Cited by 8 publications

(2 citation statements)

References 15 publications

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“…Hawkins and Schwarz [36] pointed out an error in Bayın's methodology, thereby reaffirming the original results of Jeng et al [33]. Several authors [37][38][39] adopted various types of fractional derivatives such as Weyl [38] and Caputo-Fabrizio [39] to analyze the solutions of FSE for the one-dimensional infinite potential well, again, in a piecewise continuous fashion. As such, the aforementioned controversies cast doubt on the existing results in literature making it imperative for the quantum chemistry community to be aware of such pitfalls and open problems while advancing the development of fractional quantum chemical models from a firm mathematical standpoint.…”

Section: Controversies and Open Problemsmentioning

confidence: 80%

Fractional paradigms in quantum chemistry

Mostafanejad

2021

Int J of Quantum Chemistry

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The realization of fractional quantum chemistry is presented. Adopting the integrodifferential operators of the calculus of arbitrary-order, we develop a general framework for the description of quantum nonlocal effects in the complex electronic environments. After a brief overview of the historical and fundamental aspects of the calculus of arbitrary-order, various classes of fractional Schrödinger equations are discussed and pertinent controversies and open problems around their applications to model systems are detailed. We provide a unified approach toward fractional generalization of the quantum chemical models such as Hartree-Fock and Kohn-Sham density functional theory and develop fractional variants of the fundamental molecular integrals and correlation energy. Furthermore, we offer various strategies for modeling static-and dynamic-order quantum nonlocal effects through constant-and variable-order fractional operators, respectively. Possible directions for future developments of fractional quantum chemistry are also outlined.

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Section: Controversies and Open Problemsmentioning

confidence: 80%

Fractional paradigms in quantum chemistry

Mostafanejad

2021

Int J of Quantum Chemistry

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“…A type of fractional quantum harmonic oscillator has been first discussed by Laskin in one of his breakthrough papers [1] on fractional quantum mechanics, but he tackled only a semiclassical approximation. Since then, several authors have dealt with the spatial fractional Schrödinger equation with different types of fractional derivatives and various potentials presenting contradictory results and arguments [2,3,4,5,6,7].…”

Section: Introductionmentioning

confidence: 99%

Factorization of the Riesz-Feller fractional quantum harmonic oscillators

Rosu,

Mancas

2020

Preprint

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Using the Riesz-Feller fractional derivative, we apply the factorization algorithm to the fractional quantum harmonic oscillator along the lines previously proposed by Olivar-Romero and Rosas-Ortiz, extending their results. We solve the non-Hermitian fractional eigenvalue problem in the k space by introducing in that space a new class of Hermite 'polynomials' that we call Riesz-Feller Hermite 'polynomials'. Using the inverse Fourier transform in Mathematica, interesting analytic results for the same eigenvalue problem in the x space are also obtained. Additionally, a more general factorization with two different Lévy indices is briefly introduced.

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