Fractional Schrödinger equation.

By Nick Laskin. Physical review. E, Statistical, nonlinear, and soft matter physics, 2002

1,535 citations recorded by the source corpus.

Some properties of the fractional Schrödinger equation are studied. We prove the Hermiticity of the fractional Hamilton operator and establish the parity conservation law for fractional quantum mechanics. As physical applications of the fractional Schrödinger equation we find the energy spectra of a hydrogenlike atom (fractional "Bohr atom") and of a fractional oscillator in the semiclassical approximation. An equation for the fractional probability current density is developed and discussed. We also discuss the relationships between the fractional and standard Schrödinger equations.

Subfields: Atomic and Molecular Physics, and Optics, Modeling and Simulation, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/47d4b3bfd7ab4e6f317de1cff50b3bd8e5b8a433

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Atomic and Molecular Physics, and Optics

In Modeling and Simulation

In Statistical and Nonlinear Physics

Citations in the corpus

On Fractional Schrodinger Equations in sobolev spaces (2015, methodology): “The previous equation is a fundamental equation of fractional quantum mechanics, a generalization of the standard quantum mechanics extending the Feynman path integral to Levy processes [23].”
Time-space fabric underlying anomalous diffusion (2005, methodology): “In the literature [20–24], the above fractional Schrödinger equation (11) were derived either based on the quantum integral over the Lévy paths in contrast to the conventional Feynman Gaussian path integral [20,21] or the fractional time derivative representation of the fractional Brownian motion [2”
Fractional powers of second order partial differential operators: extension problem and regularity theory (2010, methodology): “S olo por mencionar algunos m as, adem as de los modelos en Matem atica Financiera [29] y en mec anica de uidos [56, 27], hay aplicaciones en la F sica Moderna, por ejemplo, cuando se consideran cin eticas fraccionarias y transporte an omalo [90, 70], cin eticas raras [64], mec anica cu antica fracc”
Fractional Schrödinger–Poisson Systems with a General Subcritical or Critical Nonlinearity (2015, methodology): “4 [28] N. Laskin, Fractional Schrödinger equation, Physical Review E, 66 (2002), 056108.”
Positive solutions of fractional differential equations with derivative terms (2012, neutral): “Apparently, the nonlinear term f(t, u(t), u′(t)) containing the derivative is a more general case, and the study of such fractional differential equations is of significance theoretically and practically [7].”
Multiplicity results for the non-homogeneous fractional p-Kirchhoff equations with concave–convex nonlinearities (2015, background): “Note that Laskin in [9,27] first proposed the nonlinear fractional Schrödinger model such as (1.”
Concentration phenomenon for fractional nonlinear Schr\"{o}dinger equations (2013, background): “Recently, Laskin extended the fractality concept and formulated fractional quantum mechanics as a path integral over the Lévy flights paths [50, 49, 52, 51].”
Global Well-Posedness for the Fractional Nonlinear Schrödinger Equation (2010, background): “[6] Laskin, N. (2002).”
Multiple solutions for nonhomogeneous Schrödinger–Kirchhoff type equations involving the fractional p-Laplacian in RN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setl (2015, background): “In the context of fractional quantummechanics, nonlinear fractional Schrödinger equation has been proposed by Laskin [30,31] as a result of expanding the Feynman path integral, from the Brownian like to the Lévy like quantum mechanical paths.”