Unexpected Spectral Asymptotics for Wave Equations on certain Compact Spacetimes

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.