Bernard Helffer
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View article: On uniqueness of radial potentials for given Dirichlet spectra with distinct angular momenta
On uniqueness of radial potentials for given Dirichlet spectra with distinct angular momenta Open
We consider an inverse spectral problem for radial Schr\" odinger operators with singular potentials. First, we show that the knowledge of the Dirichlet spectra for infinitely many angular momenta~$\ell$ satisfying a Müntz-type condition u…
View article: Flux effects on Magnetic Laplace and Steklov eigenvalues in the exterior of a disk
Flux effects on Magnetic Laplace and Steklov eigenvalues in the exterior of a disk Open
We derive a three-term asymptotic expansion for the lowest eigenvalue of the magnetic Laplace and Steklov operators in the exterior of the unit disk in the strong magnetic field limit. This improves recent results of Helffer-Nicoleau (2025…
View article: On Courant and Pleijel theorems for sub-Riemannian Laplacians
On Courant and Pleijel theorems for sub-Riemannian Laplacians Open
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 large…
View article: Cantor spectrum for multidimensional quasi-periodic Schrödinger operators
Cantor spectrum for multidimensional quasi-periodic Schrödinger operators Open
In this paper, we investigate the spectrum of a class of multidimensional quasi-periodic Schrödinger operators that exhibit a Cantor spectrum, which provides a resolution to a question posed by Damanik, Fillman, and Gorodetski \cite{DFG}. …
View article: On Courant and Pleijel theorems for sub-Riemannian Laplacians
On Courant and Pleijel theorems for sub-Riemannian Laplacians Open
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 the number of nodal domains of an eigenfunc…
View article: Asymptotics for the magnetic Dirichlet-to-Neumann eigenvalues in general domains
Asymptotics for the magnetic Dirichlet-to-Neumann eigenvalues in general domains Open
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…
View article: A rigorous Peierls-Onsager effective dynamics for semimetals in long-range magnetic fields
A rigorous Peierls-Onsager effective dynamics for semimetals in long-range magnetic fields Open
We consider periodic (pseudo)differential {elliptic operators of Schrödinger type} perturbed by weak magnetic fields not vanishing at infinity, and extend our previous analysis in \cite{CIP,CHP-2,CHP-4} to the case {of a semimetal having a…
View article: Eigenvalues of the Neumann magnetic Laplacian in the unit disk
Eigenvalues of the Neumann magnetic Laplacian in the unit disk Open
In this paper, we study the first eigenvalue of the magnetic Laplacian with Neumann boundary conditions in the unit disk $\mathbb D$ in $\mathbb R^2$. There is a rather complete asymptotic analysis when the constant magnetic field tends to…
View article: Inequalities between Dirichlet and Neumann eigenvalues on Carnot groups
Inequalities between Dirichlet and Neumann eigenvalues on Carnot groups Open
We show that the $j$-th Dirichlet eigenvalue of the sub-Laplacian on an open set of a Carnot group is greater than the $(j+1)$-st Neumann eigenvalue. This extends earlier results in the Euclidean and Heisenberg case and has a remarkably si…
View article: The index of sub-laplacians: beyond contact manifolds
The index of sub-laplacians: beyond contact manifolds Open
In this paper we study the following question: do sub-Laplacian type operators have non-trivial index theory on Carnot manifolds in higher degree of nilpotency? The problem relates to characterizing the structure of the space of hypoellipt…
View article: Trace formulas for the magnetic Laplacian and Dirichlet to Neumann operator -- Explicit expansions --
Trace formulas for the magnetic Laplacian and Dirichlet to Neumann operator -- Explicit expansions -- Open
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…
View article: Global counterexamples to uniqueness for a Calderón problem with $C^k$ conductivities
Global counterexamples to uniqueness for a Calderón problem with $C^k$ conductivities Open
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)…
View article: Matrix Representation of Magnetic Pseudo-Differential Operators via Tight Gabor Frames
Matrix Representation of Magnetic Pseudo-Differential Operators via Tight Gabor Frames Open
In this paper we use some ideas from [12, 13] and consider the description of Hörmander type pseudo-differential operators on $$\mathbb {R}^d$$ ( $$d\ge 1$$ ), including the case of the magnetic pseudo-differential operators intr…
View article: Semiclassical eigenvalue estimates under magnetic steps
Semiclassical eigenvalue estimates under magnetic steps Open
International audience
View article: On Courant and Pleijel theorems for sub-Riemannian Laplacians
On Courant and Pleijel theorems for sub-Riemannian Laplacians Open
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 large…
View article: Quantum tunneling in deep potential wells and strong magnetic fields revisited
Quantum tunneling in deep potential wells and strong magnetic fields revisited Open
International audience
View article: On critical points of eigenvalues of the Montgomery family of quartic oscillators
On critical points of eigenvalues of the Montgomery family of quartic oscillators Open
We discuss spectral properties of the family of quartic oscillators\n$\\mathfrak h_{\\mathcal M}(\\alpha) =-\\frac{d^2}{dt^2} +\\Big(\\frac{1}{2} t^{2}\n-\\alpha\\Big)^2$ on the real line, where $\\alpha\\in \\mathbb{R}$ is a parameter.\nT…
View article: Helical magnetic fields and semi-classical asymptotics of the lowest eigenvalue
Helical magnetic fields and semi-classical asymptotics of the lowest eigenvalue Open
We study the three-dimensional Neumann magnetic Laplacian in the presence of a semiclassical parameter and a non-uniform magnetic field with constant intensity. We determine a sharp two term asymptotics for the lowest eigenvalue, where the…
View article: Flux and symmetry effects on quantum tunneling
Flux and symmetry effects on quantum tunneling Open
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 application…
View article: Effective operators on an attractive magnetic edge
Effective operators on an attractive magnetic edge Open
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 attr…
View article: Discussing semigroup bounds with resolvent estimates
Discussing semigroup bounds with resolvent estimates Open
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 opera…
View article: Stability estimates for semigroups in the Banach case
Stability estimates for semigroups in the Banach case Open
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).
View article: On the stability of symmetric flows in a two-dimensional channel
On the stability of symmetric flows in a two-dimensional channel Open
We consider the stability of symmetric flows in a two-dimensional channel (including the Poiseuille flow). In 2015 Grenier, Guo, and Nguyen have established instability of these flows in a particular region of the parameter space, affirmin…
View article: Matrix representation of Magnetic pseudo-differential operators via tight Gabor frames
Matrix representation of Magnetic pseudo-differential operators via tight Gabor frames Open
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 …
View article: On critical points of eigenvalues of the Montgomery family of quartic oscillators
On critical points of eigenvalues of the Montgomery family of quartic oscillators Open
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 var…
View article: Effective operators on an attractive magnetic edge
Effective operators on an attractive magnetic edge Open
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 attr…
View article: On the spectrum of some Bloch–Torrey vector operators
On the spectrum of some Bloch–Torrey vector operators Open
We consider the Bloch-Torrey operator in $L^2(I,{\\mathbb R}^3)$ where\n$I\\subseteq{\\mathbb R}$. In contrast with the $L^2(I,{\\mathbb R}^2)$ (as well\nas the $L^2({\\mathbb R}^k,{\\mathbb R}^2)$) case considered in previous works.\nWe o…
View article: Helical magnetic fields and semi-classical asymptotics of the lowest eigenvalue
Helical magnetic fields and semi-classical asymptotics of the lowest eigenvalue Open
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 in…