Fox's H-Functions and the Time Fractional Schrödinger Equation

Details Fox's H-Functions and the Time Fractional Schrödinger Equation Fox's H-Functions and the Time Fractional Schrödinger Equation

2011-03-16

arxiv.org/abs/1103.3295v3

Physics

Mathematical Physics

Scientific paper

After introducing the general space and time fractional Schr\"odinger equation, we concentrate on the time fractional Schr\"odinger equation and present some new results via the Fox's H-function. We are first going to show that writing the time dependent part of the wave function, which in separable coordinates is a Mittag-Leffler function with an imaginary argument, as a purely oscillatory plus a purely uniformly decaying piece is not possible. In the light of this, we are going to reconsider the time fractional Schr\"odinger equation. We discuss the particle in a box problem. We also give the free particle solution for the time and space fractional Schr\"odinger equation in terms of H-functions. We introduce the effective potential approach for the solutions of the fractional Schr\"odinger equations in terms of the ordinary Schr\"odinger equation and discuss implications of these results. PACS numbers: 03.65.Ca, 02.50.Ey, 02.30.Gp, 03.65.Db

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