Memorias de investigación
Artículos en revistas:
High-frequency propagation for the Schrödinger equation on the torus
Año:2010
Áreas de investigación
  • Ingeniería mecánica, aeronaútica y naval
Datos
Descripción
The main objective of this paper is understanding the propagation laws obeyed by high-frequency limits of Wigner distributions associated to solutions to the Schrödinger equation on the standard d-dimensional torus Td . From the point of view of semiclassical analysis, our setting corresponds to performing the semiclassical limit at times of order 1/h, as the characteristic wave-length h of the initial data tends to zero. It turns out that, in spite that for fixed h every Wigner distribution satisfies a Liouville equation, their limits are no longer uniquely determined by those of the Wigner distributions of the initial data. We characterize them in terms of a new object, the resonant Wigner distribution, which describes high-frequency effects associated to the fraction of the energy of the sequence of initial data that concentrates around the set of resonant frequencies in phase-space T ¿Td . This construction is related to that of the so-called two-microlocal semiclassical measures.We prove that any limit ¿ of theWigner distributions corresponding to solutions to the Schrödinger equation on the torus is completely determined by the limits of both theWigner distribution and the resonant Wigner distribution of the initial data; moreover, ¿ follows a propagation law described by a family of density-matrix Schrödinger equations on the periodic geodesics of Td . Finally, we present some connections with the study of the dispersive behavior of the Schrödinger flow (in particular, with Strichartz estimates). Among these, we show that the limits of sequences of position densities of solutions to the Schrödinger equation on T2 are absolutely continuous with respect to the Lebesgue measure.
Internacional
Si
JCR del ISI
Si
Título de la revista
JOURNAL OF FUNCTIONAL ANALYSIS
ISSN
0022-1236
Factor de impacto JCR
1,247
Información de impacto
Volumen
DOI
Número de revista
258
Desde la página
933
Hasta la página
955
Mes
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Participantes
Grupos de investigación, Departamentos, Centros e Institutos de I+D+i relacionados
  • Creador: Grupo de Investigación: CEHINAV (Canal de Ensayos Hidrodinámicos de la E.T.S.I. Navales)
  • Departamento: Enseñanzas Básicas de la Ingeniería Naval