This is the first paper of a series in which we plan to study spectral asymptotics for sub-Riemannian Laplacians and to extend results that are classical in the Riemannian case concerning Weyl measures, quantum limits, quantum ergodicity, quasi-modes
, trace formulae.Even if hypoelliptic operators have been well studied from the point of view of PDEs, global geometrical and dynamical aspects have not been the subject of much attention. As we will see, already in the simplest case, the statements of the results in the sub-Riemannian setting are quite different from those in the Riemannian one. Let us consider a sub-Riemannian (sR) metric on a closed three-dimensional 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 establish a Quantum Ergodicity (QE) theorem for the eigenfunctions of any associated sR Laplacian under the assumption that the Reeb flow is ergodic. The limit measure is given by the normalized Popp measure.This is the first time that such a result is established for a hypoelliptic operator, whereas the usual Shnirelman theorem yields QE for the Laplace-Beltrami operator on a closed Riemannian manifold with ergodic geodesic flow.To prove our theorem, we first establish a microlocal Weyl law, which allows us to identify the limit measure and to prove the microlocal concentration of the eigenfunctions on the characteristic manifold of the sR Laplacian. Then, we derive a Birkhoff normal form along this characteristic manifold, thus showing that, in some sense, all 3D contact structures are microlocally equivalent. The quantum version of this normal form provides a useful microlocal factorization of the sR Laplacian. Using the normal form, the factorization and the ergodicity assumption, we finally establish a variance estimate, from which QE follows.We also obtain a second result, which is valid without any ergodicity assumption: every Quantum Limit (QL) can be decomposed in a sum of two mutually singular measures: the first measure is supported on the unit cotangent bundle and is invariant under the sR geodesic flow, and the second measure is supported on the characteristic manifold 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.
In this paper, I describe the weak limits of the measures associated to the eigenfunctions of the Laplacian on a Quantum graph for a generic metric in terms of the Gauss map of the determinant manifold. I describe also all the limits with minimal support (the scars).
We consider a smooth hyper-surface Z of a closed Riemannian manifold X. Let P be the Poisson operator associating to a smooth function on Z its harmonic extension on XZ. If A is a pseudo-differential operator on X of degree <3, we prove that B=P^* A
P is a pseudo-differential operator on Z and calculate the principal symbol of B.
We describe the spectral theory of the adjacency operator of a graph which is isomorphic to homogeneous trees at infinity. Using some combinatorics, we reduce the problem to a scattering problem for a finite rank perturbation of the adjacency operato
r on an homogeneous tree. We developp this scattering theory using the classical recipes for Schrodinger operators in Euclidian spaces.
In this note, we present a natural proof of a recent and surprising result of Gregory Berkolaiko (arXiv 1110.5373) interpreting the Courant nodal defect of a Schrodinger operator on a finite graph as a Morse index associated to the deformations of th
e operator by switching on a magnetic field. This proof is inspired by a nice paper of Miroslav Fiedler published in 1975.
We prove a 2-terms Weyl formula for the counting function N(mu) of the spectrum of the Laplace operator in the Euclidean disk with a sharp remainder estimate O(mu^2/3).
in the recent paper [Journal of Physics A, 43474-0288 (2011)], B. Helffer and R. Purice compute the second term of a semi-classical trace formula for a Schrodinger operator with magnetic field. We show how to recover their formula by using the method
s developped by the geometers in the seventies for the heat expansions.
In this paper we give an exact relation between the Greens function in a scattering problem for a wave equation and the correlation of scattered plane waves. This general relation was proved in a special case by Sanchez-Sesma and al.
