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On the nonlocality of the fractional Schrödinger equation. (English) Zbl 1311.81114
Summary: 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 \(\alpha\). On a more positive note, we present a solution to the fractional Schrödinger equation for the one-dimensional harmonic oscillator with \(\alpha\)=1.
©2010 American Institute of Physics

MSC:
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
35R11 Fractional partial differential equations
81U05 \(2\)-body potential quantum scattering theory
60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
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