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Spectral theory of translation surfaces : A short introduction

Luc Hillairet

Séminaire de théorie spectrale et géométrie

We define translation surfaces and, on these, the Laplace operator that is associated with the Euclidean (singular) metric. This Laplace operator is not essentially self-adjoint and we recall how self-adjoint extensions are chosen. There are essentially two geometrical self-adjoint extensions and we show that they actually share the same spectrum

Quantum ergodicity and quantum limits for sub-Riemannian Laplacians

Yves Colin de VerdièreLuc HillairetEmmanuel Trélat

Séminaire Laurent Schwartz — EDP et applications

This paper is a proceedings version of [6], in which we state a Quantum Ergodicity (QE) theorem on a 3D contact manifold, and in which we establish some properties of the Quantum Limits (QL). We consider a sub-Riemannian (sR) metric on a compact 3D manifold with an oriented contact distribution. There exists a privileged choice of the contact form, with an associated Reeb vector field and a canonical volume form that coincides with the Popp measure. We state a QE theorem for the eigenfunctions...

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