Unexpected spectral asymptotics for wave equations on certain compact spacetimes |
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Authors: | Jonathan Fox Robert S Strichartz |
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Institution: | 1.Department of Mathematics,Cornell University,Cornell,USA |
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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 S1×S2. For the Laplacian on S1×S2, theWeyl asymptotic law gives a growth rate O(s3/2) for the eigenvalue counting function N(s) = #{λj: 0 ≤ λj ≤ s}. For the wave operator, there are two corresponding eigenvalue counting functions: N±(s) = #{λj: 0 < ±λj ≤ s}, and they both have a growth rate of O(s2). More precisely, there is a leading term π2s2/4 and a correction term of as3/2, 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 S1 × Sq are valid for q even but not q odd. We also examine some related examples. |
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