Lévy processes and Schrödinger equation |
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Authors: | Nicola Cufaro Petroni Modesto Pusterla |
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Institution: | a Dipartimento di Matematica and TIRES, Università di Bari, INFN Sezione di Bari, via E. Orabona 4, 70125 Bari, Italy b Dipartimento di Fisica, Università di Padova, INFN Sezione di Padova, via F. Marzolo 8, 35100 Padova, Italy |
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Abstract: | We analyze the extension of the well known relation between Brownian motion and the Schrödinger equation to the family of the Lévy processes. We consider a Lévy-Schrödinger equation where the usual kinetic energy operator-the Laplacian-is generalized by means of a selfadjoint, pseudodifferential operator whose symbol is the logarithmic characteristic of an infinitely divisible law. The Lévy-Khintchin formula shows then how to write down this operator in an integro-differential form. When the underlying Lévy process is stable we recover as a particular case the fractional Schrödinger equation. A few examples are finally given and we find that there are physically relevant models-such as a form of the relativistic Schrödinger equation-that are in the domain of the non stable Lévy-Schrödinger equations. |
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Keywords: | 02 50 Ey 02 50 Ga 05 40 Fb |
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