Bernard Helffer

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We study the energy of relativistic electrons and positrons interacting through the second quantized Coulomb interaction and a self-generated magnetic eld. As states we allow generalized Hartree-Fock states in the Fock space. Our main result is the assertion of positivity of the energy, if the atomic numbers and the ne structure constant are not too big. We(More)
Motivated by the problem of analytic hypoellipticity, we show that a special family of compact non selfadjoint operators has a non zero eigenvalue. We recover old results obtained by ordinary differential equations techniques and show how it can be applied to the higher dimensional case. This gives in particular a new class of hypoelliptic, but not analytic(More)
  • Bernard Helffer, Maria Hoffmann–Ostenhof, Thomas Hoffmann–Ostenhof, Mark M. Owen, Maria Hoffmann-Ostenhof, Thomas Hoffmann-Ostenhof
  • 2001
This is a survey on [HHOO] and further developments of the theory [He4]. We explain in detail the origin of the problem in superconductivity as first presented in [BeRu], recall the results of [HHOO] and explain the extension to the Dirichlet case. As illustration of the theory, we detail some semi-classical aspects and give examples where our estimates are(More)
We consider a periodic magnetic Schrödinger operator H, depending on the semiclassical parameter h > 0, on a noncompact Riemannian manifold M such that H(M,R) = 0 endowed with a properly discontinuous cocompact isometric action of a discrete group. We assume that there is no electric field and that the magnetic field has a periodic set of compact magnetic(More)
Let H = ? + V be a two-dimensional Schrr odinger operator de-ned on a bounded domain R 2 with Dirichlet boundary conditions on @. Suppose that H commutes with the actions of the dihedral group D 2n , the group of the regular n-gone. We analyze completely the multiplicity of the groundstate eigenvalues associated to the different symmetry subspaces related(More)
We consider the linearization of the time-dependent Ginzburg-Landau near the normal state. We assume that an electric current is applied through the sample, which captures the whole plane, inducing thereby, a magnetic field. We show that independently of the current, the normal state is always stable. Using Fourier analysis the detailed behaviour of(More)
Given a bounded open set in R2 (or in a Riemannian manifold), and a partition of Ω by k open sets ωj, we consider the quantity maxj λ(ωj), where λ(ωj) is the ground state energy of the Dirichlet realization of the Laplacian in ωj. We denote by Lk(Ω) the infimum of maxj λ(ω) over all k-partitions. A minimal k-partition is a partition that realizes the(More)