Fundamental solutions for the wave operator on static Lorentzian manifolds with timelike boundary

Abstract

We consider the wave operator on static, Lorentzian manifolds with timelike boundary and we discuss the existence of advanced and retarded fundamental solutions in terms of boundary conditions. By means of spectral calculus we prove that answering this question is equivalent to studying the selfadjoint extensions of an associated elliptic operator on a Riemannian manifold with boundary (M, g). The latter is diffeomorphic to any, constant time hypersurface of the underlying background. In turn, assuming that (M, g) is of bounded geometry, this problem can be tackled within the framework of boundary triples. These consist of the assignment of two surjective, trace operators from the domain of the adjoint of the elliptic operator onto an auxiliary Hilbert space h, which is the third datum of the triple. Self-adjoint extensions of the underlying elliptic operator are in one-to-one correspondence with self-adjoint operators Θ on h. On the one hand, we show that, for a natural choice of boundary triple, each Θ can be interpreted as the assignment of a boundary condition for the original wave operator. On the other hand, we prove that, for each such Θ, there exists a unique advanced and retarded fundamental solution. In addition, we prove that these share the same structural property of the counterparts associated to the wave operator on a globally hyperbolic spacetime.

PROPAGATION OF SINGULARITIES ON AdS SPACETIMES FOR GENERAL BOUNDARY CONDITIONS AND THE HOLOGRAPHIC HADAMARD CONDITION

“…We remark that a similar statement was obtained in [DF2,Section 4] in the case of the Poincaré patch of AdS; cf. [DDF,Proposition 34] for static spacetimes with smooth timelike boundary.…”

Evolution of confined quantum scalar fields in curved spacetime. Part I

“…Global well-posedness has been proven for the Dirac equation under MIT-boundary conditions, which are the analogous to Neumann boundary conditions for a scalar field [49]. In the case of Robin boundary conditions, global well-posedness is proven in [50] for the Klein-Gordon equation in static spacetimes of bounded geometry (for a wider class of boundary conditions including the Robin case). It is probable that global well-posedness of the Cauchy problem for the Klein-Gordon equation can be proven for the boundary conditions considered in this article as an extension of the results cited, but approaching such task is not in the aim of the present work, which goals are mainly computational.…”

“…In other words, the method is of general practical applicability to compute the scattering matrix between two physically valid Fock representations of the field, and it is indeed an extremely powerful tool for such purpose; while, on the other hand, we have no intention of providing a fully consistent field quantisation in any region of a general spacetime (see the approaches to such problem in e.g. [32][33][34][35][36]).…”

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

“…Yet, as strongly advocated in [10], the class of boundary conditions which are of interest in concrete models is greater than the one considered in [26], a notable example in this direction being the so-called Wentzell boundary conditions, see e.g. [9,13,19,38,42].…”