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 $\Psi$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 $C_b^{\infty}(\R^d)$, 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 $L^2(\R^d)$ 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, $H_{C_3}$, describing the appearance of superconductivity in superconductors of type II. Furthermore, we prove that the local and global definitions of this field coincide. Near $H_{C_3}$ 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

Eigenfunctions decay for magnetic pseudodifferential operators

We prove rapid decay (even exponential decay under some stronger assumptions) of the eigenfunctions associated to discrete eigenvalues, for a class of self-adjoint operators in $L^2(\mathbb{R}^d)$ defined by ``magnetic'' pseudodifferential operators (studied in \cite{IMP1}). This class contains the relativistic Schr\"{o}dinger operator with magnetic field