Euler solutions of pseudodifferential equations |
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Authors: | Bert-Wolfgang Schulze Nikolai Tarkhanov |
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Institution: | (1) Institut für Mathematik, Universität Potsdam, Postfach 60 15 53, 14415 Potsdam, Germany |
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Abstract: | We consider a homogeneous pseudodifferential equation on a cylinderC= ×X over a smooth compact closed manifoldX whose symbol extends to a meromorphic function on the complex plane with values in the algebra of pseudodifferential operators overX. When assuming the symbol to be independent on the variablet , 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 sinceC can be thought of as a stretched manifold with conical points att=– andt= . When compared with the general theory, our approach is constructive while highlighting all the features of this latter. |
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Keywords: | primary: 45N05 secondary: 34G10 |
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