On the nonlocality of the fractional Schrödinger equation

M. Jeng, S. L Y Xu, E. Hawkins, Jennifer M Schwarz

Research output: Contribution to journalArticle

44 Citations (Scopus)

Abstract

A number of papers over the past eight years have claimed to solve the fractional Schrödinger equation for systems ranging from the one-dimensional infinite square well to the Coulomb potential to one-dimensional scattering with a rectangular barrier. However, some of the claimed solutions ignore the fact that the fractional diffusion operator is inherently nonlocal, preventing the fractional Schrödinger equation from being solved in the usual piecewise fashion. We focus on the one-dimensional infinite square well and show that the purported ground state, which is based on a piecewise approach, is definitely not a solution of the fractional Schrödinger equation for the general fractional parameter α. On a more positive note, we present a solution to the fractional Schrödinger equation for the one-dimensional harmonic oscillator with α=1.

Original languageEnglish (US)
Article number002006JMP
JournalJournal of Mathematical Physics
Volume51
Issue number6
DOIs
StatePublished - Jun 2010

Fingerprint

Nonlocality
Fractional
square wells
Coulomb potential
Fractional Diffusion
Coulomb Potential
harmonic oscillators
Harmonic Oscillator
Ground State
operators
ground state
Scattering
scattering
Operator

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

Cite this

On the nonlocality of the fractional Schrödinger equation. / Jeng, M.; Xu, S. L Y; Hawkins, E.; Schwarz, Jennifer M.

In: Journal of Mathematical Physics, Vol. 51, No. 6, 002006JMP, 06.2010.

Research output: Contribution to journalArticle

@article{7941540dc6ba4d43a1793edad06a4710,
title = "On the nonlocality of the fractional Schr{\"o}dinger equation",
abstract = "A number of papers over the past eight years have claimed to solve the fractional Schr{\"o}dinger equation for systems ranging from the one-dimensional infinite square well to the Coulomb potential to one-dimensional scattering with a rectangular barrier. However, some of the claimed solutions ignore the fact that the fractional diffusion operator is inherently nonlocal, preventing the fractional Schr{\"o}dinger equation from being solved in the usual piecewise fashion. We focus on the one-dimensional infinite square well and show that the purported ground state, which is based on a piecewise approach, is definitely not a solution of the fractional Schr{\"o}dinger equation for the general fractional parameter α. On a more positive note, we present a solution to the fractional Schr{\"o}dinger equation for the one-dimensional harmonic oscillator with α=1.",
author = "M. Jeng and Xu, {S. L Y} and E. Hawkins and Schwarz, {Jennifer M}",
year = "2010",
month = "6",
doi = "10.1063/1.3430552",
language = "English (US)",
volume = "51",
journal = "Journal of Mathematical Physics",
issn = "0022-2488",
publisher = "American Institute of Physics Publising LLC",
number = "6",

}

TY - JOUR

T1 - On the nonlocality of the fractional Schrödinger equation

AU - Jeng, M.

AU - Xu, S. L Y

AU - Hawkins, E.

AU - Schwarz, Jennifer M

PY - 2010/6

Y1 - 2010/6

N2 - A number of papers over the past eight years have claimed to solve the fractional Schrödinger equation for systems ranging from the one-dimensional infinite square well to the Coulomb potential to one-dimensional scattering with a rectangular barrier. However, some of the claimed solutions ignore the fact that the fractional diffusion operator is inherently nonlocal, preventing the fractional Schrödinger equation from being solved in the usual piecewise fashion. We focus on the one-dimensional infinite square well and show that the purported ground state, which is based on a piecewise approach, is definitely not a solution of the fractional Schrödinger equation for the general fractional parameter α. On a more positive note, we present a solution to the fractional Schrödinger equation for the one-dimensional harmonic oscillator with α=1.

AB - A number of papers over the past eight years have claimed to solve the fractional Schrödinger equation for systems ranging from the one-dimensional infinite square well to the Coulomb potential to one-dimensional scattering with a rectangular barrier. However, some of the claimed solutions ignore the fact that the fractional diffusion operator is inherently nonlocal, preventing the fractional Schrödinger equation from being solved in the usual piecewise fashion. We focus on the one-dimensional infinite square well and show that the purported ground state, which is based on a piecewise approach, is definitely not a solution of the fractional Schrödinger equation for the general fractional parameter α. On a more positive note, we present a solution to the fractional Schrödinger equation for the one-dimensional harmonic oscillator with α=1.

UR - http://www.scopus.com/inward/record.url?scp=77954562023&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77954562023&partnerID=8YFLogxK

U2 - 10.1063/1.3430552

DO - 10.1063/1.3430552

M3 - Article

AN - SCOPUS:77954562023

VL - 51

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

SN - 0022-2488

IS - 6

M1 - 002006JMP

ER -