Unexpected spectral asymptotics for wave equations on certain compact spacetimes

Abstract

We study the spectral asymptotics of wave equations on certain compact spacetimes where some variant of the Weyl asymptotic law is valid. The simplest example is the spacetime S 1 × S 2 . For the Laplacian on S 1 × S 2 the Weyl asymptotic law gives a growth rate O(s 3/2 ) for the eigenvalue counting function N (s) = #{λ j : 0 ≤ λ j ≤ s}. For the wave operator there are two corresponding eigenvalue counting functions N ± (s) = #{λ j : 0 < ±λ j ≤ s} and they both have a growth rate of O(s 2 ). More precisely there is a leading term π 2 4 s 2 and a correction term of as 3/2 where the constant a is different for N ± . These results are not robust, in that if we include a speed of propagation constant to the wave operator the result depends on number theoretic properties of the constant, and generalizations to S 1 × S q are valid for q even but not q odd. We also examine some related examples. AMS Mathematics subject Classification. Primary 35P20 35L05

Linear Independence of Generalized Poincaré Series for Anti-de Sitter 3-Manifolds

“…However, it is not the case for non-Riemannian manifolds. Kobayashi [19], and later Fox-Strichartz [4], investigated the distribution of the discrete spectrum of the Laplacian M of some pseudo-Riemannian manifolds, i.e., when M is the flat pseudo-Riemannian manifold R p,q /Z p+q and is the Lorentzian manifold S 1 × S q , respectively. Let us recall some basic notions.…”

Linear Independence of Generalized Poincaré Series for Anti-de Sitter 3-Manifolds

“…However, it is not the case for non-Riemannian manifolds. Kobayashi [18], and later Fox-Strichartz [4], investigated the distribution of discrete spectrum of the Laplacian M of some pseudo-Riemannian manifolds, i.e., when M is the flat pseudo-Riemannian manifold R p,q /Z p+q and is the Lorentzian manifold S 1 × S q , respectively. Let us recall some basic notions.…”