Construction of solutions of wave equations on global hyperbolic Lorentzian manifolds via Hadamard expansion

14.04.2016, 16:15 Uhr  –  Campus Golm, Haus 9, Raum 0.14
Forschungsseminar Differentialgeometrie

Max Lewandowski

I will start again with solutions of d'Alembert's equation on n-dimensional Minkowski space fulfilling Wightman's Axioms as a prototype with regard to general wave equations on global hyperbolic Lorentzian manifolds. With an appropriate classification of distributions, which are invariant with respect to the special orthochronous Lorentz group, one can decompose these solutions into Riesz' distributions, which are based on a well understood theory, and a linear combination of symmetric and all over supported fundamental solutions of the d'Alembert operator. In order to establish a similar construction as for the Riesz distributions we will embed them into another family of holomorphic distributions, translate this family into geodesically convex subsets of global hyperbolic Lorentzian manifolds and eventually consider Hadamard's expansion in order to obtain symmetric and all over supported fundamental solutions to general normal hyperbolic operators. Thereby the local theory for the construction of solutions of wave equations is established and if there is enough time we will see that the singular part for example of the solution of Klein-Gordon's equation is in fact composed of these ingredients.