Wave trace ($W(t)=\mathrm{Tr}(e^{it\sqrt{P}})$)

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.

For the positive Laplace-Beltrami operator on a closed manifold, the singular support of the wave trace lies in zero together with signed lengths of closed geodesics. Under the clean fixed-set hypotheses, its singularity at a periodic-geodesic length has an expansion determined by the geodesic flow and its linearized return map. In particular, when the relevant leading coefficient does not vanish through cancellation, that length is determined by the spectrum. The inclusion of singular supports alone does not justify an unconditional recovery of every geodesic length.