In microlocal analysis, the propagation of singularities theorem (also called the Duistermaat–Hörmander theorem) is theorem which characterizes the wavefront set of the distributional solution of the partial (pseudo) differential equation
for a pseudodifferential operator on a smooth manifold. It says that the propagation of singularities follows the bicharacteristic flow of the principal symbol of .
The theorem appeared 1972 in a work on Fourier integral operators by Johannes Jisse Duistermaat and Lars Hörmander and since then there have been many generalizations which are known under the name propagation of singularities.[1][2]
Propagation of singularities theorem
editWe use the following notation:
- is a -differentiable manifold, and is the space of smooth functions with a compact set , such that .
- denotes the class of pseudodifferential operators of type with symbol .
- is the Hörmander symbol class.
- .
- is the space of distributions, the Dual space of .
- is the wave front set of
- is the characteristic set of the principal symbol
Statement
editLet be a properly supported pseudodifferential operator of class with a real principal symbol , which is homogeneous of degree in . Let be a distribution that satisfies the equation , then it follows that
Furthermore, is invariant under the Hamiltonian flow induced by .[3]
Bibliography
edit- Hörmander, Lars (1972). Fourier integral operators. I. Acta Mathematica. Vol. 128. Institut Mittag-Leffler. pp. 79–183. doi:10.1007/BF02392052.
- Duistermaat, Johannes Jisse; Hörmander, Lars (1972). Fourier integral operators. II. Acta Mathematica. Vol. 128. Institut Mittag-Leffler. pp. 183–269. doi:10.1007/BF02392165.
- Shubin, Mikhail A. Pseudodifferential Operators and Spectral Theory. Springer Berlin, Heidelberg. ISBN 978-3-540-41195-6.
- Taylor, Michael E. (1978). "Propagation, reflection, and diffraction of singularities of solutions to wave equations". Bulletin of the American Mathematical Society. 84 (4). American Mathematical Society: 589–611.
References
edit- ^ Duistermaat, Johannes Jisse; Hörmander, Lars (1972). Fourier integral operators. II. Acta Mathematica. Vol. 128. Institut Mittag-Leffler. pp. 183–269. doi:10.1007/BF02392165.
- ^ Shubin, Mikhail A. Pseudodifferential Operators and Spectral Theory. Springer Berlin, Heidelberg. ISBN 978-3-540-41195-6.
- ^ Duistermaat, Johannes Jisse; Hörmander, Lars (1972). Fourier integral operators. II. Acta Mathematica. Vol. 128. Institut Mittag-Leffler. p. 196. doi:10.1007/BF02392165.