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