Effective potential from the generalized time-dependent Schrödinger equation
type: Item
instance of: scholarly article
title: Effective potential from the generalized time-dependent Schrödinger equation
zbMATH Open document ID: 1369.35077
author: Trifce Sandev
author: Irina Petreska
author: Ervin K. Lenzi
published in: Mathematics
publication date: 2017-06-22T00:00:00Z
review text: Summary: We analyze the generalized time-dependent Schrödinger equation for the force free case, as a generalization, for example, of the standard time-dependent Schrödinger equation, time fractional Schrödinger equation, distributed order time fractional Schrödinger equation, and tempered in time Schrödinger equation. We relate it to the corresponding standard Schrödinger equation with effective potential. The general form of the effective potential that leads to a standard time-dependent Schrodinger equation with the same solution as the generalized one is derived explicitly. Further, effective potentials for several special cases, such as Dirac delta, power-law, Mittag-Leffler and truncated power-law memory kernels, are expressed in terms of the Mittag-Leffler functions. Such complex potentials have been used in the transport simulations in quantum dots, and in simulation of resonant tunneling diode.
Mathematics Subject Classification ID: 35Q41
Mathematics Subject Classification ID: 81Q05
Mathematics Subject Classification ID: 34F05
Mathematics Subject Classification ID: 35R11
zbMATH DE Number: 6734251
zbMATH Keywords: Schrödinger equation
zbMATH Keywords: memory kernel
zbMATH Keywords: effective potential
zbMATH Keywords: Mittag-Leffler function
MaRDI profile type: Publication
full work available at URL: math4040059
OpenAlex ID: W2524591573
cites work: Time fractional Schrödinger equation
cites work: Fractional quantum mechanics and Lévy path integrals
cites work: Time fractional Schrödinger equation: Fox's H-functions and the effective potential
cites work: Time fractional development of quantum systems
cites work: Exact solutions of fractional Schrödinger-like equation with a nonlocal term
cites work: Solutions for a Schrödinger equation with a nonlocal term
cites work: Fractional Schrödinger equation in the presence of the linear potential
cites work: Fractional Schrödinger equation for a particle moving in a potential well
cites work: Solutions of a particle with fractional \(\delta \)-potential in a fractional dimensional space
cites work: Continuous-time random walk as a guide to fractional Schrödinger equation
cites work: Lévy Transport in Slab Geometry of Inhomogeneous Media
cites work: Time-dependent Schrödinger-like equation with nonlocal term
cites work: Space-time fractional Schrödinger equation with composite time fractional derivative
cites work: Diffusion and Fokker-Planck-Smoluchowski equations with generalized memory kernel
cites work: Comb Model with Slow and Ultraslow Diffusion
cites work: Convergence of series in three parametric Mittag-Leffler functions
cites work: Fractional Schrödinger equation with noninteger dimensions
cites work: Q5625726
cites work: On a generalization of Mittag-Leffler function and its properties
cites work: Langevin equation for a free particle driven by power law type of noises
cites work: Correlation functions for the fractional generalized Langevin equation in the presence of internal and external noise
cites work: Q5190149
DOI: 10.3390/MATH4040059
description: scientific article