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 $S^1 \times S^2$. For the Laplacian on $S^1 \times S^2$ the Weyl asymptotic law gives a growth rate $O(s^{3/2})$ for the eigenvalue counting function $N(s) = #{\lambda _j: 0 \leq \lambda _j \leq s}$. For the wave operator there are two corresponding eigenvalue counting functions $N^{\pm}(s) = #{\lambda _j: 0 < \pm \lambda _j \leq s}$ and they both have a growth rate of $O(s^2)$. More precisely there is a leading term $\frac{\pi^2}{4}s^2$ and a correction term of $as^{3/2}$ where the constant $a$ is different for $N^{\pm}$. 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 $S^1 \times S^q$ are valid for $q$ even but not $q$ odd. We also examine some related examples.