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Bernard Helffer

Publications and source records attributed to Bernard Helffer.

At least 91 records · Page 5Linked to original sources
arXiv2016

On the semi-classical analysis of the groundstate energy of the Dirichlet Pauli operator

We discuss the results of a recent paper by Ekholm, Kovařík and Portmann in connection with a question of C. Guillarmou about the semiclassical expansion of the lowest eigenvalue of the Pauli operator with Dirichlet conditions. We exhibit connections with the properties of the torsion function in mechanics, the exit time of a Brownian motion and the analysis of the low eigenvalues of some Witten Laplacian.

math.SP↗
arXiv2016

Pleijel's theorem for Schrödinger operators with radial potentials

In 1956 $Å$. Pleijel gave his celebrated theorem showing that the inequality in Courant's theorem on the number of nodal domains is strict for large eigenvalues of the Laplacian. This was a consequence of a stronger result giving an asymptotic upper bound for the number of nodal domains of the eigenfunctions as the eigenvalues tend to $+\infty$. A similar question occurs naturally for Schr"\odinger operators. The first significant result has been obtained recently by the first author for the harmonic oscillator. The purpose of this paper is to consider more general potentials which are radial. We will analyze the case when the potential tends to $+\infty$ and the case when the potential is negative and tends to zero, where the considered eigenfucntion are associated to the eigenvalues below the essential spectrum.

math.SP↗
arXiv2016

Decay of superconductivity away from the magnetic zero set

We establish exponential bounds on the Ginzburg-Landau order parameter away from the curve where the applied magnetic field vanishes. In the units used in this paper, the estimates are valid when the parameter measuring the strength of the applied magnetic field is comparable with the Ginzburg-Landau parameter. This completes a previous work by the authors analyzing the case when this strength was much higher. Our results display the distribution of surface and bulk superconductivity and are valid under the assumption that the magnetic field is Hölder continuous.

math.AP↗
arXiv2016

On nodal domains in Euclidean balls

Å. Pleijel (1956) has proved that in the case of the Laplacian with Dirichlet condition, the equality in the Courant nodal theorem (Courant sharp situation) can only be true for a finite number of eigenvalues when the dimension is $\geq 2$. Recently Polterovich extended the result to the Neumann problem in two dimensions in the case when the boundary is piecewise analytic. A question coming from the theory of spectral minimal partitions has motivated the analysis of the cases when one has equality in Courant's theorem. In this note, we identify the Courant sharp eigenvalues for the Dirichlet and Neumann Laplacians in balls in $\mathbb R^d$, $d\geq 2$. The result for the Dirichlet Laplacian in the disc in $\mathbb R^2$ was obtained by B. Helffer, T. Hoffmann-Ostenhof and S. Terracini.

math.SP↗
arXiv2016

Tunneling for the Robin Laplacian in smooth planar domains

We study the low-lying eigenvalues of the semiclassical Robin Laplacian in a smooth planar domain symmetric with respect to an axis. In the case when the curvature of the boundary of the domain attains its maximum at exactly two points away from the axis of symmetry, we establish an explicit asymptotic formula for the splitting of the first two eigenvalues. This is a rigorous derivation of the semiclassical tunneling effect induced by the domain's geometry. Our approach is close to the Born-Oppenheimer one and yields, as a byproduct, a Weyl formula of independent interest.

math.AP↗
arXiv2015

On a magnetic characterization of spectral minimal partitions

Given a bounded open set $Ω$ in $ \mathbb R^n$ (or in a Riemannian manifold) and a partition of $Ω$ by $k$ open sets $D_j$, we consider the quantity $\max_j λ(D_j)$ where $λ(D_j)$ is the ground state energy of the Dirichlet realization of the Laplacian in $D_j$. If we denote by $ \mathfrak L_k(Ω)$ the infimum over all the $k$-partitions of $ \max_j λ(D_j)$, a minimal $k$-partition is then a partition which realizes the infimum. When $k=2$, we find the two nodal domains of a second eigenfunction, but the analysis of higher $k$'s is non trivial and quite interesting. In this paper, we give the proof of one conjecture formulated previously by V. Bonnaillie-Noel and B. Helffer about a magnetic characterization of the minimal partitions when $n=2$.

math.SP↗
arXiv2015

A review on large k minimal spectral k-partitions and Pleijel's Theorem

In this survey, we review the properties of minimal spectral $k$-partitions in the two-dimensional case and revisit their connections with Pleijel's Theorem. We focus on the large $k$ problem (and the hexagonal conjecture) in connection with two recent preprints by J. Bourgain and S. Steinerberger on the Pleijel Theorem. This leads us also to discuss some conjecture by I. Polterovich, in relation with square tilings. We also establish a Pleijel Theorem for Aharonov-Bohm Hamiltonians and deduce from it, via the magnetic characterization of the minimal partitions, some lower bound for the number of critical points of a minimal partition.

math.SP↗
arXiv2015

Minimal partitions for anisotropic tori

We analyze spectral minimal $k$-partitions for the torus. In continuation with what we have obtained for thin annuli or thin strips on a cylinder (Neumann case), we get similar results for anisotropic tori.

math.SP↗
arXiv2015

A. Stern's analysis of the nodal sets of some families of spherical harmonics revisited

In this paper, we revisit the analyses of Antonie Stern (1925) and Hans Lewy (1977) devoted to the construction of spherical harmonics with two or three nodal domains. Our method yields sharp quantitative results and a better understanding of the occurrence of bifurcations in the families of nodal sets.This paper is a natural continuation of our critical reading of A. Stern's results for Dirichlet eigenfunctions in the square, see arXiv:14026054.

math.DG↗
arXiv2015

Courant-sharp eigenvalues for the equilateral torus, and for the equilateral triangle

We address the question of determining the eigenvalues $λ\_n$ (listed in nondecreasing order, with multiplicities) for which Courant's nodal domain theorem is sharp i.e., for which there exists an associated eigenfunction with $n$ nodal domains (Courant-sharp eigenvalues). Following ideas going back to Pleijel (1956), we prove that the only Courant-sharp eigenvalues of the flat equilateral torus are the first and second, and that the only Courant-sharp Dirichlet eigenvalues of the equilateral triangle are the first, second, and fourth eigenvalues. In the last section we sketch similar results for the right-angled isosceles triangle and for the hemiequilateral triangle.

math.AP↗
arXiv2015

Dirichlet eigenfunctions on the cube, sharpening the Courant nodal inequality

This paper is devoted to the refine analysis of Courant's theorem for the Dirichlet Laplacian. Many papers (and some of them quite recent) have investigated in which cases this inequality in Courant's theorem is an equality: Pleijel, Helffer--Hoffmann-Ostenhof--Terracini, Helffer--Hoffmann-Ostenhof, Bérard-Helffer, Helffer--Persson-Sundqvist, Léna, Leydold. All these results were devoted to $(2D)$-cases in open sets in $\mathbb R^2$ or in surfaces like $\mathbb S^2$ or $\mathbb T^2$. The aim of the current paper is to look for analogous results for domains in $\mathbb{R}^3$ and, as $Å.$Pleijel was suggesting in his 1956 founding paper, for the simplest case of the cube. More precisely, we will prove that the only eigenvalues of the Dirichlet Laplacian which are Courant sharp are the two first eigenvalues.

math.SP↗
arXiv2015

Mixed normal-superconducting states in the presence of strong electric currents

We study the Ginzburg-Landau equations in the presence of large electric currents, that are smaller than the critical current where the normal state losses its stability. For steady-state solutions in the large $κ$ limit, we prove that the superconductivity order parameter is exponentially small in a significant part of the domain, and small in the rest of it. Similar results are obtained for the time-dependent problem, in continuation of the paper by the two first authors [3]. We conclude by obtaining some weaker results, albeit similar, for steady-state solutions in the large domain limit.

math-ph↗
arXiv2015

Magnetic Wells in Dimension Three

This paper deals with semiclassical asymptotics of the three-dimensional magnetic Laplacian in presence of magnetic confinement. Using generic assumptions on the geometry of the confinement, we exhibit three semiclassical scales and their corresponding effective quantum Hamiltonians, by means of three microlocal normal forms à la Birkhoff. As a consequence, when the magnetic field admits a unique and non degenerate minimum, we are able to reduce the spectral analysis of the low-lying eigenvalues to a one-dimensional $\hbar$-pseudo-differential operator whose Weyl's symbol admits an asymptotic expansion in powers of $\hbar^{\frac1 2}$.

math-ph↗
arXiv2015

Eigenvalues for the Robin Laplacian in domains with variable curvature

We determine accurate asymptotics for the low-lying eigenvalues of the Robin Laplacian when the Robin parameter goes to $-\infty$. The two first terms in the expansion have been obtained by K. Pankrashkin in the $2D$-case and by K. Pankrashkin and N. Popoff in higher dimensions. The asymptotics display the influence of the scalar curvature and the splitting between every two consecutive eigenvalues. The analysis is based on the approach developed by Fournais-Helffer for the semi-classical magnetic Laplacian. We also propose a WKB construction as a candidate for the ground state energy.

math.SP↗
arXiv2015

Lower bound for the number of critical points of minimal spectral k-partitions for k large

In a recent paper with Thomas Hoffmann-Ostenhof, we proved that the number of critical points in the boundary set of a k-minimal partition tends to infinity as k tends to infinity. In this note, we show that this number increases linearly with k as suggested by a hexagonal conjecture about the asymptotic behavior of the energy of these minimal partitions. As the original proof by Pleijel, this involves Faber-Krahn's inequality and Weyl's formula, but this time, due to the magnetic characterization of the minimal partitions, we have to establish a Weyl's formula for Aharonov-Bohm operator controlled with respect to a k-dependent number of poles.

math.AP↗
arXiv2015

From constant to non-degenerately vanishing magnetic fields in superconductivity

We explore the relationship between two reference functions arising in the analysis of the Ginzburg-Landau functional. The first function describes the distribution of superconductivity in a type II superconductor subjected to a constant magnetic field. The second function describes the distribution of superconductivity in a type II superconductor submitted to a variable magnetic field that vanishes non-degenerately along a smooth curve.

math.AP↗