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We consider the Klein-Gordon operator on an $n$-dimensional asymptoticallyanti-de Sitter spacetime $(M,g)$ together with arbitrary boundary conditionsencoded by a self-adjoint pseudodifferential operator on $\partial M$ of orderup to $2$. Using techniques from $b$-calculus and a propagation ofsingularities theorem, we prove that there exist advanced and retardedfundamental solutions, characterizing in addition their structural andmicrolocal properties. We apply this result to the problem of constructingHadamard two-point distributions. These are bi-distributions which are weakbi-solutions of the underlying equations of motion with a prescribed form oftheir wavefront set and whose anti-symmetric part is proportional to thedifference between the advanced and the retarded fundamental solutions. Inparticular, under a suitable restriction of the class of admissible boundaryconditions and setting to zero the mass, we prove their existence extending tothe case under scrutiny a deformation argument which is typically used onglobally hyperbolic spacetimes with empty boundary.

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