scholarly journals Fractional schrödinger equation with zero and linear potentials

2016 ◽  
Vol 19 (4) ◽  
Author(s):  
Saleh Baqer ◽  
Lyubomir Boyadjiev
Keyword(s):  
Schrödinger Equation ◽  
Potential Field ◽  
Schrodinger Equation ◽  
Fractional Derivatives ◽  
Free Particle ◽  
Linear Potential ◽  
Fractional Schrödinger Equation ◽  
Zero Potential ◽  
Fractional Schrodinger Equation

AbstractThis paper is about the fractional Schrödinger equation expressed in terms of the Caputo time-fractional and quantum Riesz-Feller space fractional derivatives for particle moving in a potential field. The cases of free particle (zero potential) and a linear potential are considered. For free particle, the solution is obtained in terms of the Fox

Comb-like geometric constraints leading to emergence of the time-fractional Schrödinger equation

2021 ◽  
pp. 2130005
Author(s):  
Irina Petreska ◽  
Trifce Sandev ◽  
Ervin Kaminski Lenzi
Keyword(s):  
Fractal Dimension ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Free Particle ◽  
Geometric Constraints ◽  
Nonlocal Term ◽  
Fractional Schrödinger Equation ◽  
Dirac Delta ◽  
Comb Structure ◽  
Fractional Schrodinger Equation

This paper presents an overview over several examples, where the comb-like geometric constraints lead to emergence of the time-fractional Schrödinger equation. Motion of a quantum object on a comb structure is modeled by a suitable modification of the kinetic energy operator, obtained by insertion of the Dirac delta function in the Laplacian. First, we consider motion of a free particle on two- and three-dimensional comb structures, and then we extend the study to the interacting cases. A general form of a nonlocal term, which describes the interactions of the particle with the medium, is included in the Hamiltonian, and later on, the cases of constant and Dirac delta potentials are analyzed. At the end, we discuss the case of non-integer dimensions, considering separately the case of fractal dimension between one and two, and the case of fractal dimension between two and three. All these examples show that even though we are starting with the standard time-dependent Schrödinger equation on a comb, the time-fractional equation for the Green’s functions appears, due to these specific geometric constraints.

The Numerical Computation of the Time Fractional Schrödinger Equation on an Unbounded Domain

2018 ◽  
Vol 18 (1) ◽  
pp. 77-94
Author(s):  
Dan Li ◽  
Jiwei Zhang ◽  
Zhimin Zhang
Keyword(s):  
Schrödinger Equation ◽  
Fractional Derivative ◽  
Unbounded Domain ◽  
Caputo Fractional Derivative ◽  
Schrodinger Equation ◽  
Computational Cost ◽  
Main Idea ◽  
Fractional Schrödinger Equation ◽  
Direct Scheme ◽  
Fractional Schrodinger Equation

AbstractA fast and accurate numerical scheme is presented for the computation of the time fractional Schrödinger equation on an unbounded domain. The main idea consists of two parts. First, we use artificial boundary methods to equivalently reformulate the unbounded problem into an initial-boundary value (IBV) problem. Second, we present two numerical schemes for the IBV problem: a direct scheme and a fast scheme. The direct scheme stands for the direct discretization of the Caputo fractional derivative by using the L1-formula. The fast scheme means that the sum-of-exponentials approximation is used to speed up the evaluation of the Caputo fractional derivative. The resulting fast algorithm significantly reduces the storage requirement and the overall computational cost compared to the direct scheme. Furthermore, the corresponding stability analysis and error estimates of two schemes are established, and numerical examples are given to verify the performance of our approach.

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