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# Contribution d'orbites périodiques diffractives à la formule de trace

Abstract : The trace formula is a relation between the lengths of the periodic geodesic of a riemannian compact manifold and the spectrum of the laplacian ; it is therefore a useful tool for the study of the inverse spectral problem. We show here the va lidity of such a formula in two settings where punctual singularities occur. In both cases, we use the wave equation and the propagation of singularities to prove the trace formula. On a riemmanian $3-$dimensional manifold, we consider a Dirac potential located at a point $p$. It means that we define the laplacian on $\Cinf(M \sans \{ p\})$ and that we take another selfadjoint extension than the usual one. We then construct the propagator of the wave equation, showing that it reflects multiple diffractions at $p$. The propagation of singularities and the trace formula are deduced from this construction. We show in particular that curves consisting of successive geodesic paths joigning $p$ to $p$ contribute to the trace formula, and we compute the principal part of such a contribution. On a euclidean surface with conical singularities, we have first to redefine the geodesics in order to allow diffractions at conical points. The neighborhood of a geodesic $g$ in the set of all geodesics of same length, is described by a natural number (that we call classical complexity'') that we compute according to the angles of diffraction along $g$. We then show that the propagation of singularities involve these generalized geodesics. Periodic diffractive geodesic contribute to the trace formula, and under some hypothesis we compute the principal part of these contributions.
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https://tel.archives-ouvertes.fr/tel-00001664
Contributor : Arlette Guttin-Lombard <>
Submitted on : Thursday, September 5, 2002 - 3:20:09 PM
Last modification on : Friday, November 13, 2020 - 4:50:03 PM
Long-term archiving on: : Tuesday, September 11, 2012 - 5:55:24 PM

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• HAL Id : tel-00001664, version 1

### Citation

Luc Hillairet. Contribution d'orbites périodiques diffractives à la formule de trace. Mathématiques [math]. Université Joseph-Fourier - Grenoble I, 2002. Français. ⟨tel-00001664⟩

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