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Physica Scripta

From degenerate solution spaces to unique states: exploring fractional operators’ impact on fractional Schrödinger equation in linear potential

Xueling Duan, Ke Deng

Published Mar 26, 2026• 0 citations
Publisher

Abstract

Abstract This study systematically investigates the physical significance and advantages of infinite-memory fractional operators in the fractional Schrödinger equation for a linear potential. We derive and compare analytical solutions for both infinite-memory and finite-memory fractional operators, highlighting their distinct mathematical structures and physical implications. For infinite-memory operators, we identify a rich degeneracy in the solution space due to non-local historical dependence, whereas finite-memory operators yield unique solutions. Furthermore, we demonstrate that the unique solutions generated by finite-memory operators represent a certain degenerate form of the solutions produced by infinite-memory operators. Additionally, we analyze their convergence and physical admissibility. Our results indicate that the choice of fractional operator significantly influences quantum behavior, which holds important implications for the development and application of fractional quantum mechanics.

DOI
10.1088/1402-4896/ae57e0

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Abstract This study systematically investigates the physical significance and advantages of infinite-memory fractional operators in the fractional Schrödinger equation for a linear potential. We derive and compare analytical solutions for both infinite-memory and finite-memory fractional operators, highlighting their distinct mathematical structures and physical implications. For infinite-memory operators, we identify a rich degeneracy in the solution space due to non-local historical dependence, whereas finite-memory operators yield unique solutions. Furthermore, we demonstrate that the unique solutions generated by finite-memory operators represent a certain degenerate form of the solutions produced by infinite-memory operators. Additionally, we analyze their convergence and physical admissibility. Our results indicate that the choice of fractional operator significantly influences quantum behavior, which holds important implications for the development and application of fractional quantum mechanics.
In this paper, the fractional Schrödinger equation that contains the quantum Riesz fractional derivative instead of the Laplace operator is revisited for the case of a particle moving in the infinite potential well. In the recent papers [M. Jeng, S.-L.-Y. Xu, E. Hawkins, and J. M. Schwarz, “On the nonlocality of the fractional Schrödinger equation,” J. Math. Phys. 51, 062102 (2010)10.1063/1.3430552] and [S. S. Bayin, “On the consistency of the solutions of the space fractional Schrödinger equation,” J. Math. Phys. 53, 042105 (2012)10.1063/1.4705268] published in this journal, controversial opinions regarding solutions to the fractional Schrödinger equation for a particle moving in the infinite potential well that were derived by Laskin [“Fractals and quantum mechanics,” Chaos 10, 780–790 (2000)10.1063/1.1050284] have been given. In this paper, a thorough mathematical treatment of these matters is provided. The problem under consideration is reformulated in terms of three integral equations with the power kernels. Even if the equations look not very complicated, no solution to these equations in explicit form is known. Still, the obtained equations are used to show that the eigenvalues and eigenfunctions of the fractional Schrödinger equation for a particle moving in the infinite potential well given by Laskin [“Fractals and quantum mechanics,” Chaos 10, 780–790 (2000)10.1063/1.1050284] and many other papers by different authors cannot be valid as has been first stated by Jeng et al. [“On the nonlocality of the fractional Schrödinger equation,” J. Math. Phys. 51, 062102 (2010)10.1063/1.3430552].
A significant obstacle impeding the advancement of the time fractional Schrödinger equation lies in the challenge of determining its precise mathematical formulation. In order to address this, we undertake an exploration of the time fractional Schrödinger equation within the context of a non-Markovian environment. By leveraging a two-level atom as an illustrative case, we find that the choice to raise i to the order of the time derivative is inappropriate. In contrast to the conventional approach used to depict the dynamic evolution of quantum states in a non-Markovian environment, the time fractional Schrödinger equation, when devoid of fractional-order operations on the imaginary unit i, emerges as a more intuitively comprehensible framework in physics and offers greater simplicity in computational aspects. Meanwhile, we also prove that it is meaningless to study the memory of time fractional Schrödinger equation with time derivative 1 < α ≤ 2. It should be noted that we have not yet constructed an open system that can be fully described by the time fractional Schrödinger equation. This will be the focus of future research. Our study might provide a new perspective on the role of time fractional Schrödinger equation.
We suggest a new perturbed form of the quantum potential and investigate the possible solutions of Schrodinger equation. The new form can be written as a finite or infinite continued fraction. a comparison has been given between the continued fractional potential and the non-perturbed potential. We suggest the validity of this continued fractional quantum form in some quantum systems. As the order of the continued fraction increases the difference between the perturbed and the ordinary potentials decreases. The physically acceptable solutions critically depend on the values of the continued fraction coefficients αi.
Fractional Quantum Mechanics (FQM) has emerged as a fascinating theoretical framework extending traditional quantum mechanics to describe physical systems with non-local or long-range interactions. In this paper, we delve into the realm of FQM, focusing on stability analysis and wave propagation in coupled Schrödinger equations. We begin with a comprehensive overview of FQM, elucidating its fundamental principles and mathematical formalism. Subsequently, we conduct stability analysis of coupled fractional Schrödinger equations, exploring the conditions under which these systems exhibit stable behavior. Furthermore, we investigate wave propagation phenomena within such systems, shedding light on the unique characteristics of fractional quantum waves. Our findings not only contribute to advancing the theoretical understanding of FQM but also offer insights into potential applications in diverse fields ranging from condensed matter physics to quantum information processing.
The space-fractional Schrödinger equation for a local potential is difficult to solve because the fractional Riesz operator in it is nonlocal. In fractional quantum mechanics, the infinite wall is a basic problem and has not been solved yet. In this paper, we consider a free particle in an infinite wall region. Making use of the Lévy path integral method, we derive the Lévy path integral amplitude of the particle and take advantage of it to get the solution of the fractional Schrödinger equation for an infinite wall. Then, the infinite wall potential with a delta-function perturbation is also studied. According to Fox’s H-function, we get an equation of bound state energies of a free particle moving in an infinite wall region perturbed by the δ-function for two situations, E < 0 and E > 0. We give a transcendental equation that determines energy levels when E < 0 and prove that there is no bound states when E > 0. The asymptotic behaviors of the center of the delta function approaching to the origin and the infinite are also shown. The results of this paper include special cases in standard quantum mechanics.
A time fractional quantum framework has been introduced into quantum mechanics.A new version of the space-time fractional Schrödinger equation has been launched.The introduced space-time fractional Schrödinger equation has a new scale parameter, which is a fractional generalization of Planck's constant in quantum physics.It has been shown that the presence of a time fractional time derivative in the space-time fractional Schrödinger equation significantly impacts quantum mechanical fundamentals.Time fractional quantum mechanical operators of coordinate, momentum and angular momentum were defined and their commutation relationships were established.The pseudo-Hamilton operator was introduced and its Hermicity has been proven.Two new functions related to the Mittag-Leffler function have been introduced to solve the space-time fractional Schrödinger equation.Energy of a time fractional quantum system has been defined and calculated in terms of the newly introduced functions.It has been found that in the framework of time fractional quantum mechanics there are no stationary states, and the eigenvalues of the pseudo-Hamilton operator are not the energy levels of the time fractional quantum system.A free particle solution to the space-time fractional Schrödinger equation was found.A free particle space-time fractional quantum mechanical kernel has been found and expressed in terms of the H-function.Renormalization properties of a free particle solution and the space-time fractional quantum kernel were established.Some particular cases of time fractional quantum mechanics have been analyzed and discussed.