On the nonlocality of the fractional Schrödinger equation

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 groundstate, which is based on a piecewise approach, is definitely not a solution of the fractional Schrödinger equation for general fractional parameters α. On a more positive note, we present a solution to the fractional Schrödinger equation for the one-dimensional harmonic oscillator with α = 1.

Solving fractional Schrödinger-type spectral problems: Cauchy oscillator and Cauchy well

“…[9,10]. Its neglect in solution procedures for potentially simplest infinite well problem has led to erroneous formulas for both eigenvalues and eigenvectors of the generator.…”

“…See e.g. [1,[9][10][11] for a discussion of some of those issues, specifically in connection with a correct spatial shape of the pertinent eigenfunctions.…”

Consistency problem of the solutions of the space fractional Schrödinger equation

“…The first definition is given in the frequency (momentum) domain while the others are in the configuration space. For the infinite square well, the controversy proposed in [9,12] is based on the use of the momentum space definition in Equation (95). In Section II and III, we have shown that if the relevant integrals are evaluated as Cauchy principal value integrals, there is no inconsistency.…”

Comment on “On the consistency of the solutions of the space fractional Schrödinger equation” [J. Math. Phys. 53, 042105 (2012)]

“…5 One of the first solutions of this intriguing theory, which followed naturally from the Feynman path integral formulation of quantum mechanics over Lévy paths, was again given by Laskin for the infinite square well problem. [5][6][7][8] In 2010, based on a contradiction they observed in the ground state wave function of the infinite square well problem, Jeng et al 9 argued that solutions obtained for the space fractional Schrödinger equation in a piecewise fashion are not valid. [5][6][7][8] In Refs.…”