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 Schrödinger equation (i∂t−H)u=0 with initial data u(0)=u0. If u0 is compactly supported the solution u(t) is smooth for small t>0, but not for all t. We determine the wavefront se...

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Bibliographic Details
Published inMathematische Nachrichten Vol. 292; no. 9; pp. 1941 - 1955
Main Author Doll, Moritz
Format Journal Article
LanguageEnglish
Published Weinheim Wiley Subscription Services, Inc 01.09.2019
Subjects
35A21
35Q40
35S05
harmonic oscillators
propagation of singularities
pseudodifferential operators
Schrodinger equation
Singularity (mathematics)
Wave fronts
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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 Schrödinger equation (i∂t−H)u=0 with initial data u(0)=u0. If u0 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 u0 and the principal and subprincipal symbol of H.
ISSN:0025-584X
1522-2616
DOI:10.1002/mana.201800034