Poincar{é} series for analytic convex bodies

Abstract

We study Poincar{é} series associated to strictly convex bodies in the Euclidean space. These series are Laplace transforms of the distribution of lengths (measured with the Finsler metric associated to one of the bodies) from one convex body to a lattice. Assuming that the convex bodies have analytic boundaries, we prove that the Poincar{é} series, originally defined in the right complex half-plane, continues holomorphically to a conical neighborhood of this set, removing a countable set of cuts and points. The latter correspond to the spectrum of a dual elliptic operator. We describe singularities of the Poincar{é} series at each of these branching points. One of the steps of the proof consists in showing analytic continuation of the resolvent of multiplication operators by a real-valued analytic Morse function on the sphere as a branched holomorphic function, a result of independent interest.