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

January 25, 2021

We consider the Klein-Gordon operator on an $n$-dimensional asymptotically
anti-de Sitter spacetime $(M,g)$ together with arbitrary boundary conditions
encoded by a self-adjoint pseudodifferential operator on $\partial M$ of order
up to $2$. Using techniques from $b$-calculus and a propagation of
singularities theorem, we prove that there exist advanced and retarded
fundamental solutions, characterizing in addition their structural and
microlocal properties. We apply this result to the problem of constructing
Hadamard two-point distributions. These are bi-distributions which are weak
bi-solutions of the underlying equations of motion with a prescribed form of
their wavefront set and whose anti-symmetric part is proportional to the
difference between the advanced and the retarded fundamental solutions. 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.

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