457 research outputs found
Magnetic calculus and semiclassical trace formulas
The aim of these notes is to show how the magnetic calculus developed in
\cite{MP, IMP1, IMP2, MPR, LMR} permits to give a new information on the nature
of the coefficients of the expansion of the trace of a function of the magnetic
Schr\"odinger operator whose existence was established in \cite{HR2}
Applications of Magnetic PsiDO Techniques to Space-adiabatic Perturbation Theory
In this review, we show how advances in the theory of magnetic
pseudodifferential operators (magnetic DO) can be put to good use in
space-adiabatic perturbation theory (SAPT). As a particular example, we extend
results of [PST03] to a more general class of magnetic fields: we consider a
single particle moving in a periodic potential which is subjectd to a weak and
slowly-varying electromagnetic field. In addition to the semiclassical
parameter \eps \ll 1 which quantifies the separation of spatial scales, we
explore the influence of additional parameters that allow us to selectively
switch off the magnetic field.
We find that even in the case of magnetic fields with components in
, e. g. for constant magnetic fields, the results of
Panati, Spohn and Teufel hold, i.e. to each isolated family of Bloch bands,
there exists an associated almost invariant subspace of and an
effective hamiltonian which generates the dynamics within this almost invariant
subspace. In case of an isolated non-degenerate Bloch band, the full quantum
dynamics can be approximated by the hamiltonian flow associated to the
semiclassical equations of motion found in [PST03].Comment: 32 page
On the third critical field in Ginzburg-Landau theory
Using recent results by the authors on the spectral asymptotics of the
Neumann Laplacian with magnetic field, we give precise estimates on the
critical field, , describing the appearance of superconductivity in
superconductors of type II. Furthermore, we prove that the local and global
definitions of this field coincide. Near only a small part, near the
boundary points where the curvature is maximal, of the sample carries
superconductivity. We give precise estimates on the size of this zone and decay
estimates in both the normal (to the boundary) and parallel variables
A generalized virial theorem and the balance of kinetic and potential energies in the semiclassical limit
We obtain two-sided bounds on kinetic and potential energies of a bound state
of a quantum particle in the semiclassical limit, as the Planck constant
\hbar\ri 0.
Proofs of these results rely on the generalized virial theorem obtained in
the paper as well as on a decay of eigenfunctions in the classically forbidden
region
Nodal and spectral minimal partitions -- The state of the art in 2015 --
In this article, we propose a state of the art concerning the nodal and
spectral minimal partitions. First we focus on the nodal partitions and give
some examples of Courant sharp cases. Then we are interested in minimal
spectral partitions. Using the link with the Courant sharp situation, we can
determine the minimal k-partitions for some particular domains. We also recall
some results about the topology of regular partitions and Aharonov-Bohm
approach. The last section deals with the asymptotic behavior of minimal
k-partition
On the semi-classical analysis of the groundstate energy of the Dirichlet Pauli operator in non-simply connected domains
We consider the Dirichlet Pauli operator in bounded connected domains in the
plane, with a semi-classical parameter. We show, in particular, that the ground
state energy of this Pauli operator will be exponentially small as the
semi-classical parameter tends to zero and estimate this decay rate. This
extends our results, discussing the results of a recent paper by
Ekholm--Kova\v{r}\'ik--Portmann, to include also non-simply connected domains.Comment: 15 pages, 4 figure
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