Balanced distribution-energy inequalities and related entropy bounds

Let $A$ be a self-adjoint operator acting over a space $X$ endowed with a partition. We give lower bounds on the energy of a mixed state $\rho$ from its distribution in the partition and the spectral density of $A$. These bounds improve with the refinement of the partition, and generalize inequalities by Li-Yau and Lieb--Thirring for the Laplacian in $\R^n$. They imply an uncertainty principle, giving a lower bound on the sum of the spatial entropy of $\rho$, as seen from $X$, and some spectral entropy, with respect to its energy distribution. On $\R^n$, this yields lower bounds on the sum of the entropy of the densities of $\rho$ and its Fourier transform. A general log-Sobolev inequality is also shown. It holds on mixed states, without Markovian or positivity assumption on $A$.Comment: 21 page

Linear differential operators on contact manifolds

We consider differential operators between sections of arbitrary powers of the determinant line bundle over a contact manifold. We extend the standard notions of the Heisenberg calculus: noncommutative symbolic calculus, the principal symbol, and the contact order to such differential operators. Our first main result is an intrinsically defined "subsymbol" of a differential operator, which is a differential invariant of degree one lower than that of the principal symbol. In particular, this subsymbol associates a contact vector field to an arbitrary second order linear differential operator. Our second main result is the construction of a filtration that strengthens the well-known contact order filtration of the Heisenberg calculus

Analytic torsions on contact manifolds

We propose a definition for analytic torsion of the contact complex on contact manifolds. We show it coincides with Ray-Singer torsion on any 3-dimensional CR Seifert manifold equipped with a unitary representation. In this particular case we compute it and relate it to dynamical properties of the Reeb flow. In fact the whole spectral torsion function we consider may be interpreted on CR Seifert manifolds as a purely dynamical function through Selberg-type trace formulae.Comment: 40 page

An entropic uncertainty principle for positive operator valued measures

Extending a recent result by Frank and Lieb, we show an entropic uncertainty principle for mixed states in a Hilbert space relatively to pairs of positive operator valued measures that are independent in some sense. This yields spatial-spectral uncertainty principles and log-Sobolev inequalities for invariant operators on homogeneous spaces, which are sharp in the compact case.Comment: 14 pages. v2: a technical assumption removed in main resul

Spectral density and Sobolev inequalities for pure and mixed states

We prove some general Sobolev-type and related inequalities for positive operators A of given ultracontractive spectral decay, without assuming e^{-tA} is submarkovian. These inequalities hold on functions, or pure states, as usual, but also on mixed states, or density operators in the quantum mechanical sense. This provides universal bounds of Faber-Krahn type on domains, that apply to their whole Dirichlet spectrum distribution, not only the first eigenvalue. Another application is given to relate the Novikov-Shubin numbers of coverings of finite simplicial complexes to the vanishing of the torsion of some l^{p,2}-cohomology