### 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 language | English (US) |
---|---|

Article number | 002006JMP |

Journal | Journal of Mathematical Physics |

Volume | 51 |

Issue number | 6 |

DOIs | |

State | Published - Jun 2010 |

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### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Journal of Mathematical Physics*,

*51*(6), [002006JMP]. https://doi.org/10.1063/1.3430552

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

Research output: Contribution to journal › Article

*Journal of Mathematical Physics*, vol. 51, no. 6, 002006JMP. https://doi.org/10.1063/1.3430552

}

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 -