Long-time behavior of spreading solutions of Schrödinger and diffusion equations

C. Anteneodo, J. C. Dias, and R. S. Mendes
Phys. Rev. E 73, 051105 – Published 9 May 2006

Abstract

We investigate the asymptotic time behavior of the solutions of a large class of linear differential equations that generalize the free-particle Schrödinger and diffusion equations, containing the standard ones as particular cases. We find general scalings that depend only on characteristic features of both the arbitrary initial condition and the Green function associated with the evolution equation. Basically, the amplitude of a long-time solution can be expressed in terms of low order moments of the initial condition (if finite) and low order spatial derivatives of the Green function. These derivatives can also be of the fractional type, which naturally arise when moments are divergent. We apply our results to a large class of differential equations that includes the fractional Schrödinger and Lévy diffusion equations. In particular, we show that, except for threshold cases, the amplitude of a packet may follow the asymptotic law tα, with arbitrary positive α.

  • Received 9 November 2005

DOI:https://doi.org/10.1103/PhysRevE.73.051105

©2006 American Physical Society

Authors & Affiliations

C. Anteneodo*

  • Departamento de Física, Pontifícia Universidade Católica do Rio de Janeiro, CP 38071, 22452-970, Rio de Janeiro, Brazil

J. C. Dias and R. S. Mendes

  • Departamento de Física, Universidade Estadual de Maringá, Av. Colombo 5790, 87020-900 Maringá-Paraná, Brazil

  • *Electronic address: celia@fis.puc-rio.br, celia@cbpf.br
  • Electronic address: jocrisdias@yahoo.com.br
  • Electronic address: rsmendes@dfi.uem.br

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Issue

Vol. 73, Iss. 5 — May 2006

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