Euler solutions of pseudodifferential equations

  • We consider a homogeneous pseudodifferential equation on a cylinder C = IR x X over a smooth compact closed manifold X whose symbol extends to a meromorphic function on the complex plane with values in the algebra of pseudodifferential operators over X. When assuming the symbol to be independent on the variable t element IR, we show an explicit formula for solutions of the equation. Namely, to each non-bijectivity point of the symbol in the complex plane there corresponds a finite-dimensional space of solutions, every solution being the residue of a meromorphic form manufactured from the inverse symbol. In particular, for differential equations we recover Euler's theorem on the exponential solutions. Our setting is model for the analysis on manifolds with conical points since C can be thought of as a 'stretched' manifold with conical points at t = -infinite and t = infinite.

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Metadaten
Author:Bert-Wolfgang Schulze, Nikolai N. Tarkhanov
URN:urn:nbn:de:kobv:517-opus-25211
Series (Serial Number):Preprint ((1998) 09 )
Document Type:Preprint
Language:English
Date of Publication (online):2008/10/30
Year of Completion:1998
Publishing Institution:Universität Potsdam
Release Date:2008/10/30
Tag:meromorphic family; pseudodifferential operator; residue
RVK - Regensburg Classification:SI 990
Organizational units:Mathematisch-Naturwissenschaftliche Fakultät / Institut für Mathematik
Dewey Decimal Classification:5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik
Collections:Universität Potsdam / Schriftenreihen / Preprint / Universität Potsdam, Institut für Mathematik, Arbeitsgruppe Partielle Differentialgleichungen und Komplexe Analysis
Universität Potsdam / Schriftenreihen / Preprint / Universität Potsdam, Institut für Mathematik, Arbeitsgruppe Partielle Differentialgleichungen und Komplexe Analysis / 1998
Licence (German):License LogoKeine Nutzungslizenz vergeben - es gilt das deutsche Urheberrecht
Notes extern:
Die Printversion kann in der Universitätsbibliothek Potsdam eingesehen werden:
Preprint / Universität Potsdam, Institut für Mathematik, Arbeitsgruppe Partielle Differentialgleichungen und Komplexe Analysis, 1997-

Die Online-Fassung wird auf der Homepage des Instituts für Mathematik veröffentlicht.