Poisson relation for manifolds with boundary
Vesselin M. Petkov · 2016
This chapter is devoted to the analysis of the singularities of the distribution σ(t) = Σj cos λj t, where λ2 j=1 are the eigenvalues of the Laplacian in a bounded domain Ω with Dirichlet boundary condition on ∂Ω. The proof of the Poisson relation is reduced to the analysis of the trace of a distribution B(t, x, y), on the manifold without boundary ∂Ω. The chapter examines the singularities of B(t, x, y)and those of the trace B(t, x, x),x∈∂Ω. The analysis of the singularities of B(t, x, y) for non-convex Ω leads to some difficulties. For this reason for general domains, the singularities of E(t, x, y)for x, y∈Ω ° are studied. The chapter also presents Poisson relation for arbitrary domains by following the proof in the convex case and establishes the inclusion and considers the generalized bicharacteristics passing through all points (s, z)∈R×∂Ω. The integrals are interpreted in the sense of distributions.