Bernard Helffer

Hypoelliptic estimates and spectral theory for Fokker-Planck operators and Witten Laplacians

Kohns Proof of the Hypoellipticity of the Hormander Operators.- Compactness Criteria for the Resolvent of Schrodinger Operators.- Global Pseudo-differential Calculus.- Analysis of some Fokker-Planck Operator.- Return to Equillibrium for the Fokker-Planck Operator.- Hypoellipticity and Nilpotent Groups.- Maximal Hypoellipticity for Polynomial of Vector Fields and Spectral Byproducts.- On Fokker-Planck Operators and Nilpotent Techniques.- Maximal Microhypoellipticity for Systems and Applications to Witten Laplacians.- Spectral Properties of the Witten-Laplacians in Connection with Poincare Inequalities for Laplace Integrals.- Semi-classical Analysis for the Schrodinger Operator: Harmonic Approximation.- Decay of Eigenfunctions and Application to the Splitting.- Semi-classical Analysis and Witten Laplacians: Morse Inequalities.- Semi-classical Analysis and Witten Laplacians: Tunneling Effects.- Accurate Asymptotics for the Exponentially Small Eigenvalues of the Witten Laplacian.- Application to the Fokker-Planck Equation.- Epilogue.- Index.

Spectral methods in surface superconductivity

Preface.- Notation.- Part I Linear Analysis.- 1 Spectral Analysis of Schr..odinger Operators.- 2 Diamagnetism.- 3 Models in One Dimension.- 4 Constant Field Models in Dimension 2: Noncompact Case.- 5 Constant Field Models in Dimension 2: Discs and Their Complements.- 6 Models in Dimension 3: R3 or R3,+.- 7 Introduction to Semiclassical Methods for the Schr..odinger Operator with a Large Electric Potential.- 8 Large Field Asymptotics of the Magnetic Schr..odinger Operator: The Case of Dimension 2.- 9 Main Results for Large Magnetic Fields in Dimension 3.- Part II Nonlinear Analysis.-10 The Ginzburg-Landau Functional.- 11 Optimal Elliptic Estimates.- 12 Decay Estimates.- 13 On the Third Critical Field HC3.- 14 Between HC2 and HC3 in Two Dimensions.- 15 On the Problems with Corners.- 16 On Other Models in Superconductivity and Open Problems.- A Min-Max Principle.- B Essential Spectrum and Perssons Theorem.- C Analytic Perturbation Theory.- D About the Curl-Div System.- E Regularity Theorems and Precise Estimates in Elliptic PDE.- F Boundary Coordinates.- References.- Index.

Equation de Schrödinger avec champ magnétique et équation de Harper

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Calcul fonctionnel par la transformation de Mellin et opérateurs admissibles

Mellins transform is used to establish a functional calculus of a class of pseudodifferential-operators depending on a small parameter h > 0. We apply for exeample this result to prove the semi-classical behaviour of the discrete spectrum of Schrodinger operators −h2 · Δ + V, and of Dirac operators h ∑j = 13 αjDj + α4 − V.

Ergodicité et limite semi-classique

Consider a self adjoint quantic hamiltonian:P(h)=p(x, hDx) whereh>0 is the Plancks constant andp some smooth classical observable on the phase space R2n. When the classical flow on a compact energy shell {p=λ} is ergodic we prove that in the limith ↓ 0 almost all the eigenfunctions ofP(h) whose energy is near of λ are distributed according to the Liouville measure on {p=λ}.In the high energy case (λ →+∞) this sort of problem was considered by A. Schnirelman, S. Zelditch, and Y. Colin de Verdière.

Puits multiples en mécanique semi-classique. IV: Etude du complexe de Witten

On essaie de justifier et de preciser la methode developpee par E. Witten pour obtenir une demonstration analytique des inegalites de Morse

On the correlation for Kac-like models in the Convex case

The aim of this paper is to stu the behavior asm tends to ∞ of a family of measures exp[-Φ(m)(x)]dx(m) on ℝm, whereΦ(m) is a potential on ℝm which is a perturbation “in a suitable sense” of the harmonic potential Σjxj2.

Nodal Sets for Groundstates of Schrödinger Operators with Zero Magnetic Field in Non Simply Connected Domains

Abstract:We investigate nodal sets of magnetic Schrödinger operators with zero magnetic field, acting on a non simply connected domain in ℝ2. For the case of circulation 1/2 of the magnetic vector potential around each hole in the region, we obtain a characterisation of the nodal set, and use this to obtain bounds on the multiplicity of the groundstate. For the case of one hole and a fixed electric potential, we show that the first eigenvalue takes its highest value for circulation 1/2.

Upper critical field and location of surface nucleation of superconductivity

Abstract In this paper we improve the estimate obtained by Lu–Pan on the value of the upper critical field HC3(κ) for a cylindrical superconductor with cross section Ω being an arbitrary 2-dimensional smooth bounded domain. We also show that, when a homogeneous magnetic field is applied along the axis of the cylinder with magnitude of the field close to HC3, superconductivity nucleates first at the surface of the sample where the curvature of ∂ Ω is maximal.

On spectral minimal partitions: the case of the sphere

In continuation of previous work, we analyze the properties of spectral minimal partitions and focus in this paper our analysis on the case of the sphere. We prove that a minimal 3-partition for the sphere \(\mathbb{S}^2\) is up to a rotation the so-called Y-partition. This question is connected to a celebrated conjecture of Bishop in harmonic analysis.