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

April 10, 2018

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 self-adjoint
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 into an auxiliary
Hilbert space $\mathsf{h}$, 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 $\Theta$ on $\mathsf{h}$. On the one
hand, we show that, for a natural choice of boundary triple, each $\Theta$ 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 $\Theta$, 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.

Keywords:

fundamental solutions, wave equations, manifolds with boundary