Quantum diffusion and generalized Rényi dimensions of spectral measures

Jean-Marie Barbaroux; François Germinet; Serguei Tcheremchantsev

Journées équations aux dérivées partielles (2000)

  • page 1-16
  • ISSN: 0752-0360

Abstract

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We estimate the spreading of the solution of the Schrödinger equation asymptotically in time, in term of the fractal properties of the associated spectral measures. For this, we exhibit a lower bound for the moments of order p at time T for the state ψ defined by [ 1 T ∫ 0 T ∥ | X | p / 2 e - i t H ψ ∥ 2 d t ] . We show that this lower bound can be expressed in term of the generalized Rényi dimension of the spectral measure μ ψ associated to the hamiltonian H and the state ψ . We especially concentrate on continuous models.

How to cite

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Barbaroux, Jean-Marie, Germinet, François, and Tcheremchantsev, Serguei. "Quantum diffusion and generalized Rényi dimensions of spectral measures." Journées équations aux dérivées partielles (2000): 1-16. <http://eudml.org/doc/93397>.

@article{Barbaroux2000,
abstract = {We estimate the spreading of the solution of the Schrödinger equation asymptotically in time, in term of the fractal properties of the associated spectral measures. For this, we exhibit a lower bound for the moments of order $p$ at time $T$ for the state $\psi $ defined by $[ \frac\{1\}\{T\}\int _0^T \Vert |X|^\{p/2\} \mathrm \{e\}^\{-itH\}\psi \Vert ^2\, \mathrm \{d\}t]$. We show that this lower bound can be expressed in term of the generalized Rényi dimension of the spectral measure $\mu _\psi $ associated to the hamiltonian $H$ and the state $\psi $. We especially concentrate on continuous models.},
author = {Barbaroux, Jean-Marie, Germinet, François, Tcheremchantsev, Serguei},
journal = {Journées équations aux dérivées partielles},
language = {eng},
pages = {1-16},
publisher = {Université de Nantes},
title = {Quantum diffusion and generalized Rényi dimensions of spectral measures},
url = {http://eudml.org/doc/93397},
year = {2000},
}

TY - JOUR
AU - Barbaroux, Jean-Marie
AU - Germinet, François
AU - Tcheremchantsev, Serguei
TI - Quantum diffusion and generalized Rényi dimensions of spectral measures
JO - Journées équations aux dérivées partielles
PY - 2000
PB - Université de Nantes
SP - 1
EP - 16
AB - We estimate the spreading of the solution of the Schrödinger equation asymptotically in time, in term of the fractal properties of the associated spectral measures. For this, we exhibit a lower bound for the moments of order $p$ at time $T$ for the state $\psi $ defined by $[ \frac{1}{T}\int _0^T \Vert |X|^{p/2} \mathrm {e}^{-itH}\psi \Vert ^2\, \mathrm {d}t]$. We show that this lower bound can be expressed in term of the generalized Rényi dimension of the spectral measure $\mu _\psi $ associated to the hamiltonian $H$ and the state $\psi $. We especially concentrate on continuous models.
LA - eng
UR - http://eudml.org/doc/93397
ER -

References

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