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This is the first paper of a series in which we plan to study spectralasymptotics for sub-Riemannian Laplacians and to extend results that areclassical in the Riemannian case concerning Weyl measures, quantum limits,quantum ergodicity, quasi-modes, trace formulae.Even if hypoelliptic operatorshave been well studied from the point of view of PDEs, global geometrical anddynamical aspects have not been the subject of much attention. As we will see,already in the simplest case, the statements of the results in thesub-Riemannian setting are quite different from those in the Riemannian one.Let us consider a sub-Riemannian (sR) metric on a closed three-dimensionalmanifold with an oriented contact distribution. There exists a privilegedchoice of the contact form, with an associated Reeb vector field and acanonical volume form that coincides with the Popp measure. We establish aQuantum Ergodicity (QE) theorem for the eigenfunctions of any associated sRLaplacian under the assumption that the Reeb flow is ergodic. The limit measureis given by the normalized Popp measure.This is the first time that such aresult is established for a hypoelliptic operator, whereas the usual Shnirelmantheorem yields QE for the Laplace-Beltrami operator on a closed Riemannianmanifold with ergodic geodesic flow.To prove our theorem, we first establish amicrolocal Weyl law, which allows us to identify the limit measure and to provethe microlocal concentration of the eigenfunctions on the characteristicmanifold of the sR Laplacian. Then, we derive a Birkhoff normal form along thischaracteristic manifold, thus showing that, in some sense, all 3D contactstructures are microlocally equivalent. The quantum version of this normal formprovides a useful microlocal factorization of the sR Laplacian. Using thenormal form, the factorization and the ergodicity assumption, we finallyestablish a variance estimate, from which QE follows.We also obtain a secondresult, which is valid without any ergodicity assumption: every Quantum Limit(QL) can be decomposed in a sum of two mutually singular measures: the firstmeasure is supported on the unit cotangent bundle and is invariant under the sRgeodesic flow, and the second measure is supported on the characteristicmanifold of the sR Laplacian and is invariant under the lift of the Reeb flow.Moreover, we prove that the first measure is zero for most QLs.

Spectral asymptotics for sub-Riemannian Laplacians. I: quantum ergodicity and quantum limits in the 3D contact case