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Analytic smoothing effect for the Schrödinger equation with long‐range perturbation
Author(s) -
Martinez Andre,
Nakamura Shu,
Sordoni Vania
Publication year - 2006
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.20112
Subject(s) - mathematics , smoothing , mathematical analysis , perturbation (astronomy) , laplace operator , phase space , commutator , microlocal analysis , operator (biology) , pure mathematics , operator theory , fourier integral operator , physics , quantum mechanics , algebra over a field , statistics , biochemistry , lie conformal algebra , chemistry , repressor , transcription factor , gene
We study the microlocal analytic singularity of solutions to the Schrödinger equation with analytic coefficients. Using microlocal weight estimates developed for estimating phase space tunneling, we prove microlocal smoothing estimates that generalize results by Robbiano and Zuily. We show that the exponential decay of the initial state in a cone in the phase space implies microlocal analytic regularity of the solution at a positive time. We suppose the Schrödinger operator is a long‐range‐type perturbation of the Laplacian, and we employ positive commutator‐type estimates to prove the smoothing property. © 2005 Wiley Periodicals, Inc.