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
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