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Bernard Helffer

Publications and source records attributed to Bernard Helffer.

At least 37 records · Page 2Linked to original sources
arXiv2025

Asymptotics for the magnetic Dirichlet-to-Neumann eigenvalues in general domains

Inspired by a paper by T. Chakradhar, K. Gittins, G. Habib and N. Peyerimhoff, we analyze their conjecture that the ground state energy of the magnetic Dirichlet-to-Neumann operator tends to infinity as the magnetic field tends to infinity. More precisely, we prove refined conjectures for general two dimensional domains, based on the analysis in the case of the half-plane and the disk by two of us (B.H. and F.N.). We also extend our analysis to the three dimensional case, and explore a connection with the eigenvalue asymptotics of the magnetic Robin Laplacian.

math-ph↗
arXiv2024

Quantum Tunneling and the Aharonov-Bohm effect

We investigate a Hamiltonian with radial potential wells and an Aharonov-Bohm vector potential with two poles. Assuming that the potential wells are symmetric, we derive the semi-classical asymptotics of the splitting between the ground and second state energies. The flux effects due to the Aharonov-Bohm vector potential are of lower order compared to the contributions coming from the potential wells.

math.SP↗
arXiv2024

Global counterexamples to uniqueness for a Calderón problem with $C^k$ conductivities

Let $Ω\subset R^n$, $n \geq 3$, be a fixed smooth bounded domain, and let $γ$ be a smooth conductivity in $\overlineΩ$. Consider a non-zero frequency $λ_0$ which does not belong to the Dirichlet spectrum of $L_γ= -{\rm div} (γ\nabla \cdot)$. Then, for all $k \geq 1$, there exists an infinite number of pairs of non-isometric $C^k$ conductivities $(γ_1, γ_2)$ on $\overlineΩ$, which are close to $γ$ such that the associated DN maps at frequency $λ_0$ satisfy \begin{equation*} Λ_{γ_1,λ_0} = Λ_{γ_2,λ_0}. \end{equation*}

math.AP↗
arXiv2024

Trace formulas for the magnetic Laplacian and Dirichlet to Neumann operator -- Explicit expansions --

Inspired by a recent paper of G. Liu and X. Tan (2023), we would like to measure how the magnetic effect appears in the heat trace formula associated with the magnetic Laplacian and the magnetic Dirichlet-to-Neumann operator. We propose to the reader an overview of magnetic heat trace formulas through explicit examples. On the way we obtain new formulas and in particular we calculate explicitely some non local terms and logarithmic terms appearing in the Steklov heat trace asymptotics.

math.AP↗
arXiv2024

On Courant and Pleijel theorems for sub-Riemannian Laplacians

We are interested in the number of nodal domains of eigenfunctions of sub-Laplacians on sub-Riemannian manifolds. Specifically, we investigate the validity of Pleijel's theorem, which states that, as soon as the dimension is strictly larger than 1, the number of nodal domains of an eigenfunction corresponding to the k-th eigenvalue is strictly (and uniformly, in a certain sense) smaller than k for large k. In the first part of this paper we reduce this question from the case of general sub-Riemannian manifolds to that of nilpotent groups. In the second part, we analyze in detail the case where the nilpotent group is a Heisenberg group times a Euclidean space. Along the way we improve known bounds on the optimal constants in the Faber-Krahn and isoperimetric inequalities on these groups.

math.SP↗
arXiv2024

Matrix representation of Magnetic pseudo-differential operators via tight Gabor frames

In this paper we use some ideas from \cite{FG-97, G-06} and consider the description of Hörmander type pseudo-differential operators on $\mathbb{R}^d$ ($d\geq1$), including the case of the magnetic pseudo-differential operators introduced in \cite{IMP-1, IMP-19}, with respect to a tight Gabor frame. We show that all these operators can be identified with some infinitely dimensional matrices whose elements are strongly localized near the diagonal. Using this matrix representation, one can give short and elegant proofs to classical results like the Calder{ó}n-Vaillancourt theorem and Beals' commutator criterion, and also establish local trace-class criteria.

math.AP↗
arXiv2023

Flux and symmetry effects on quantum tunneling

Motivated by the analysis of the tunneling effect for the magnetic Laplacian, we introduce an abstract framework for the spectral reduction of a self-adjoint operator to a hermitian matrix. We illustrate this framework by three applications, firstly the electro-magnetic Laplacian with constant magnetic field and three equidistant potential wells, secondly a pure constant magnetic field and Neumann boundary condition in a smoothed triangle, and thirdly a magnetic step where the discontinuity line is a smoothed triangle. Flux effects are visible in the three aforementioned settings through the occurrence of eigenvalue crossings. Moreover, in the electro-magnetic Laplacian setting with double well radial potential, we rule out an artificial condition on the distance of the wells and extend the range of validity for a recently established tunneling approximation, thereby settling the problem of electro-magnetic tunneling under constant magnetic field and a sum of translated radial electric potentials.

math.SP↗
arXiv2023

Stability estimates for semigroups in the Banach case

The purpose of this paper is to revisit previous works of the author with J. Sjöstrand (2010--2021) proved in the Hilbert case by considering the Banach case at the light of a paper by Y.~Latushkin and V.~Yurov (2013).

math.FA↗
arXiv2023

Discussing semigroup bounds with resolvent estimates

The purpose of this paper is to revisit the proof of the Gearhart-Prüss-Huang-Greiner theorem for a semigroup $S(t)$, following the general idea of the proofs that we have seen in the literature and to get an explicit estimate on the operator norm of $S(t)$ in terms of bounds on the resolvent of the generator. In \cite{HelSj} by the first two authors, this was done and some applications in semiclassical analysis were given. Some of these results have been subsequently published in two books written by the two first authors \cite{He1,Sj2}. A second work \cite{HeSj21} by the first two authors presents new improvements partially motivated by a paper of D. Wei \cite{W}. In this third paper, we continue the discussion on whether the aforementioned results are optimal, and whether one can improve these results through iteration. Numerical computations will illustrate some of the abstract results.

math.FA↗
arXiv2022

On critical points of eigenvalues of the Montgomery family of quartic oscillators

We discuss spectral properties of the family of quartic oscillators $\mathfrak h_{\mathcal M}(α) =-\frac{d^2}{dt^2} +\Big(\frac{1}{2} t^{2} -α\Big)^2$ on the real line, where $α\in \mathbb{R}$ is a parameter. This operator appears in a variety of applications coming from quantum mechanics to harmonic analysis on Lie groups, Riemannian geometry and superconductivity. We study the variations of the eigenvalues $λ_j(α)$ of $\mathfrak h_{\mathcal M}(α)$ as functions of the parameter $α$.We prove that for $j$ sufficiently large, $α\mapsto λ_j(α)$ has a unique critical point, which is a nondegenerate minimum.We also prove that the first eigenvalue $λ_1(α)$ enjoys the same property and give a numerically assisted proof that the same holds for the second eigenvalue $λ_2(α)$. The proof for excited states relies on a semiclassical reformulation of the problem. In particular, we develop a method permitting to differentiate with respect to the semiclassical parameter, which may be of independent interest.

math.AP↗
arXiv2022

Effective operators on an attractive magnetic edge

The semiclassical Laplacian with discontinuous magnetic field is considered in two dimensions. The magnetic field is sign changing with exactly two distinct values and is discontinuous along a smooth closed curve, thereby producing an attractive magnetic edge. Various accurate spectral asymptotics are established by means of a dimensional reduction involving a microlocal phase space localization allowing to deal with the discontinuity of the field.

math.SP↗
arXiv2022

Helical magnetic fields and semi-classical asymptotics of the lowest eigenvalue

We study the 3D Neuman magnetic Laplacian in the presence of a semi-classical parameter and a non-uniform magnetic field with constant intensity. We determine a sharp two term asymptotics for the lowest eigenvalue, where the second term involves a quantity related to the magnetic field and the geometry of the domain. In the special case of the unit ball and a helical magnetic field, the concentration takes place on two symmetric points of the unit sphere.

math.SP↗
arXiv2022

Semi-classical eigenvalue estimates under magnetic steps (Former title: Hearing the shape of a magnetic edge in the semiclassical limit)

We establish accurate eigenvalue asymptotics and, as a by-product, sharp estimates of the splitting between two consecutive eigenvalues, for the Dirichlet magnetic Laplacian with a non-uniform magnetic field having a jump discontinuity along a smooth curve. The asymptotics hold in the semiclassical limit which also corresponds to a large magnetic field limit, and is valid under a geometric assumption on the curvature of the discontinuity curve.

math-ph↗
arXiv2022

Tunneling effect induced by a curved magnetic edge

Experimentally observed magnetic fields with nanoscale variations are theoretically modeled by a piece-wise constant function with jump discontinuity along a smooth curve, the magnetic edge. Assuming the edge is a closed curve with an axis of symmetry and the field is sign changing and with exactly two distinct values, we prove that semi-classical tunneling occurs and calculate the magnitude of this tunneling effect.

math.SP↗
arXiv2022

Computing nodal deficiency with a refined Dirichlet-to-Neumann map

Recent work of the authors and their collaborators has uncovered fundamental connections between the Dirichlet-to-Neumann map, the spectral flow of a certain family of self-adjoint operators, and the nodal deficiency of a Laplacian eigenfunction (or an analogous deficiency associated to a non-bipartite equipartition). Using a refined construction of the Dirichlet-to-Neumann map, we strengthen all of these results, in particular getting improved bounds on the nodal deficiency of degenerate eigenfunctions. Our framework is very general, allowing for non-bipartite partitions, non-simple eigenvalues, and non-smooth nodal sets. Consequently, the results can be used in the general study of spectral minimal partitions, not just nodal partitions of generic Laplacian eigenfunctions.

math.SP↗
arXiv2021

Spectral analysis near a Dirac type crossing in a weak non-constant magnetic field

This is the last paper in a series of three in which we have studied the Peierls substitution in the case of a weak magnetic field. Here we deal with two $2d$ Bloch eigenvalues which have a conical crossing. It turns out that in the presence of an almost constant weak magnetic field, the spectrum near the crossing develops gaps which remind of the Landau levels of an effective mass-less magnetic Dirac operator.

math-ph↗
arXiv2021

Courant-sharp eigenvalues of compact flat surfaces: Klein bottles and cylinders

The question of determining for which eigenvalues there exists an eigenfunction which has the same number of nodal domains as the label of the associated eigenvalue (Courant-sharp property) was motivated by the analysis of minimal spectral partitions. In previous works, many examples have been analyzed corresponding to squares, rectangles, disks, triangles, tori, Möbius strips,\ldots . A natural toy model for further investigations is the flat Klein bottle, a non-orientable surface with Euler characteristic $0$, and particularly the Klein bottle associated with the square torus, whose eigenvalues have higher multiplicities. In this note, we prove that the only Courant-sharp eigenvalues of the flat Klein bottle associated with the square torus (resp. with square fundamental domain) are the first and second eigenvalues. We also consider the flat cylinders $(0,π) \times \mathbb{S}^1_r$ where $r \in \{0.5,1\}$ is the radius of the circle $\mathbb{S}^1_r$, and we show that the only Courant-sharp Dirichlet eigenvalues of these cylinders are the first and second eigenvalues.

math.SP↗