Fundamental solutions and Hadamard states for a scalar field with arbitrary boundary conditions on an asymptotically AdS spacetime.

Claudio Dappiaggi (U Pavia)

Jul 13. 2023, 14:00 — 14:50

We consider the Klein-Gordon operator on an n-dimensional asymptotically anti-deSitter spacetime (M,g) together with arbitrary boundary conditions encoded by a self-adjoint pseudodifferential operator on the boundary of M of order up to 2. Using techniques from b-calculus we prove a propagation of singularities theorem which generalizes a recent result by O. Gannot and M. Wrochna, who considered Robin boundary conditions. In addition, we prove that there exist advanced and retarded fundamental solutions,characterizing in addition their structural and microlocal properties. As an application we apply this result to the problem of constructing Hadamard two-point distributions. In particular, under a suitable restriction of the class of admissible boundary conditions and setting to zero the mass, we prove their existence extending to the case under scrutiny a deformation argument which is typically used on globally hyperbolic spacetimes with empty boundary. This is joint work with Alessio Marta (U. of Milan).

Further Information
ESI Boltzmann Lecture Hall
Associated Event:
Spectral Theory and Mathematical Relativity (Thematic Programme)
Piotr T. Chruściel (U of Vienna)
Peter Hintz (ETH Zurich)
Alexander Strohmaier (U Leeds)
Steven Morris Zelditch † (Northwestern U Evanston)