On the nonlocality of the fractional Schrödinger equation

Details On the nonlocality of the fractional Schrödinger equation On the nonlocality of the fractional Schrödinger equation

2008-10-08

arxiv.org/abs/0810.1543v1

Physics

Mathematical Physics

5 pages

Scientific paper

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 groundstate, which is based on a piecewise approach, is definitely not a solution of the fractional Schr\"{o}dinger equation for general fractional parameters $\alpha$. On a more positive note, we present a solution to the fractional Schr\"{o}dinger equation for the one-dimensional harmonic oscillator with $\alpha=1$.

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