Positivity of scalar Hadamard-bisolutions for the wave equation on globally hyperbolic Lorentzian manifolds

08.11.2018, 16:15  –  Haus 9, Raum 0.14
Forschungsseminar Differentialgeometrie

Max Lewandowski

Let W be a scalar Hadamard-bisolution for the wave equation on a globally hyperbolic Lorentzian manifold M, then positivity W[φ,φ]≥0 only affects the symmetric part W^s of W, which is essentially given by the difference of Feynman and Anti-Feynman-propagator. Therefore Theorem 6.6.2 of [1] can be applied, which ensures the existence of some f∈C^∞(M×M), s.t. W^s+f is a positive, symmetric bidistribution, but in general not a bisolution anymore. We will establish a well-posed Cauchy problem for symmetric bisolutions and use W^s+f as initial value in order to construct a global bisolution, from which we then show positivity.

[1] Duistermaat, J. J., Hörmander, L. - Fourier Integral Operators II, Acta Mathematica 128, 183-269 (1972)