A simple diffractive boundary value problem on an asymptotically anti-de Sitter space

Abstract

In this paper, we study the propagation of singularities (in the sense of $\mathcal{C}^{\infty}$ wave front set) of the solution of a model case initial-boundary value problem with glancing rays for a concave domain on an asymptotically anti-de Sitter manifold. The main result addresses the diffractive problem and establishes that there is no propagation of singularities into the shadow for the solution, i.e. the diffractive result for codimension-1 smooth boundary holds in this setting. The approach adopted is motivated by the work done for a conformally related diffractive model problem by Friedlander, in which an explicit solution was constructed using the Airy function. This work was later generalized by Melrose and by Taylor, via the method of parametrix construction. Our setting is a simple case of asympotically anti-de Sitter spaces, which are Lorentzian manifolds modeled on anti-de Sitter space at infinity but whose boundary are not totally geodesic (unlike the exact anti-de Sitter space). Most technical difficulties of the problem reduce to studying and constructing a global resolvent for a semiclassical ODE on $\RR^+$, which at one end is a b-operator (in the sense of Melrose) while having a scattering behavior at infinity. We use different techniques near 0 and infinity to analyze the local problems: near infinity we use local resolvent bounds and near zero we build a local semiclassical parametrix. After this step, the `gluing' method by Datchev-Vasy serves to combine these local estimates to get the norm bound for the global resolvent.