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EnglishWe 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 self-adjoint, pseudo-differential 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.
Lévy processes and Schrödinger equation / CUFARO PETRONI N; PUSTERLA M. - In: PHYSICA. A. - ISSN 0378-4371. - 388(2009), pp. 824-836.
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| Titolo: | Lévy processes and Schrödinger equation |
| Autori: | |
| Data di pubblicazione: | 2009 |
| Rivista: | |
| Citazione: | Lévy processes and Schrödinger equation / CUFARO PETRONI N; PUSTERLA M. - In: PHYSICA. A. - ISSN 0378-4371. - 388(2009), pp. 824-836. |
| 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 self-adjoint, pseudo-differential 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. |
| Handle: | http://hdl.handle.net/11586/11855 |
| Appare nelle tipologie: | 1.1 Articolo in rivista |
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