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
formula_1
for a pseudodifferential operator formula_2 on a smooth manifold. It says that the propagation of singularities follows the bicharacteristic flow of the principal symbol of formula_2.
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.
Propagation of singularities theorem.
We use the following notation:
Statement.
Let formula_2 be a properly supported pseudodifferential operator of class formula_22 with a real principal symbol formula_23, which is homogeneous of degree formula_24 in formula_25. Let formula_26 be a distribution that satisfies the equation formula_27, then it follows that
formula_28
Furthermore, formula_29 is invariant under the Hamiltonian flow induced by formula_20.