= Wave trace
{title2=$W(t)=\operatorname{Tr}(e^{it\sqrt P})$}
For the <positive Laplace-Beltrami operator> $P$ on a <closed manifold>, the <wave trace> is the <distribution> $\sum_j e^{it\sqrt{\lambda_j}}$, counting <multiplicities>. The series is interpreted after testing against smooth compactly supported functions, rather than as an ordinarily convergent sum. Its singularities relate the spectrum to the <length spectrum> of <closed geodesics>, through the <Duistermaat-Guillemin trace formula>.
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