96 research outputs found
On the differentiable vectors for contragredient representations
We establish a few simple results on contragredient representations of Lie
groups, with a view toward applications to the abstract characterization of
some spaces of pseudo-differential operators. In particular, this method
provides an abstract approach to J.Nourrigat's recent description of the norm
closure of the pseudo-differential operators of order zero.Comment: 6 pages; the final version is published in C. R. Math. Acad. Sci.
Paris 351 (2013), no. 13-14, 513--51
A semiclassical Egorov theorem and quantum ergodicity for matrix valued operators
We study the semiclassical time evolution of observables given by matrix
valued pseudodifferential operators and construct a decomposition of the
Hilbert space L^2(\rz^d)\otimes\kz^n into a finite number of almost invariant
subspaces. For a certain class of observables, that is preserved by the time
evolution, we prove an Egorov theorem. We then associate with each almost
invariant subspace of L^2(\rz^d)\otimes\kz^n a classical system on a product
phase space \TRd\times\cO, where \cO is a compact symplectic manifold on
which the classical counterpart of the matrix degrees of freedom is
represented. For the projections of eigenvectors of the quantum Hamiltonian to
the almost invariant subspaces we finally prove quantum ergodicity to hold, if
the associated classical systems are ergodic
Semi- and Non-relativistic Limit of the Dirac Dynamics with External Fields
We show how to approximate Dirac dynamics for electronic initial states by
semi- and non-relativistic dynamics. To leading order, these are generated by
the semi- and non-relativistic Pauli hamiltonian where the kinetic energy is
related to and , respectively. Higher-order
corrections can in principle be computed to any order in the small parameter
v/c which is the ratio of typical speeds to the speed of light. Our results
imply the dynamics for electronic and positronic states decouple to any order
in v/c << 1.
To decide whether to get semi- or non-relativistic effective dynamics, one
needs to choose a scaling for the kinetic momentum operator. Then the effective
dynamics are derived using space-adiabatic perturbation theory by Panati et. al
with the novel input of a magnetic pseudodifferential calculus adapted to
either the semi- or non-relativistic scaling.Comment: 42 page
First-order hyperbolic pseudodifferential equations with generalized symbols
We consider the Cauchy problem for a hyperbolic pseudodifferential operator
whose symbol is generalized, resembling a representative of a Colombeau
generalized function. Such equations arise, for example, after a
reduction-decoupling of second-order model systems of differential equations in
seismology. We prove existence of a unique generalized solution under log-type
growth conditions on the symbol, thereby extending known results for the case
of differential operators with generalized functions as coefficients
Wodzicki Residue for Operators on Manifolds with Cylindrical Ends
We define the Wodzicki Residue TR(A) for A in a space of operators with
double order (m_1,m_2). Such operators are globally defined initially on R^n
and then, more generally, on a class of non-compact manifolds, namely, the
manifolds with cylindrical ends. The definition is based on the analysis of the
associate zeta function. Using this approach, under suitable ellipticity
assumptions, we also compute a two terms leading part of the Weyl formula for a
positive selfadjoint operator belonging the mentioned class in the case
m_1=m_2.Comment: 24 pages, picture changed, added references, corrected typo
- …