Recurrence of singularities for second order isotropic pseudodifferential operators

Let \(H\) be a self-adjoint isotropic elliptic pseudodifferential operator of order \(2\). Denote by \(u(t)\) the solution of the Schr\"odinger equation \((i\partial_t - H)u = 0\) with initial data \(u(0) = u_0\). If \(u_0\) is compactly supported the solution \(u(t)\) is smooth for small \(t &...

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Bibliographic Details
Published inarXiv.org
Main Author Doll, Moritz
Format Paper Journal Article
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 28.12.2017
Subjects
Mathematics - Analysis of PDEs
Mathematics - Mathematical Physics
Physics - Mathematical Physics
Singularities
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Summary:Let \(H\) be a self-adjoint isotropic elliptic pseudodifferential operator of order \(2\). Denote by \(u(t)\) the solution of the Schr\"odinger equation \((i\partial_t - H)u = 0\) with initial data \(u(0) = u_0\). If \(u_0\) is compactly supported the solution \(u(t)\) is smooth for small \(t > 0\), but not for all \(t\). We determine the wavefront set of \(u(t)\) in terms of the wavefront set of \(u_0\) and the principal and subprincipal symbol of \(H\).
ISSN:2331-8422
DOI:10.48550/arxiv.1712.09935