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Bernard Helffer

Publications and source records attributed to Bernard Helffer.

At least 73 records · Page 4Linked to original sources
arXiv2018

On the semi-classical analysis of the groundstate energy of the Dirichlet Pauli operator III: Magnetic fields that change sign

We consider the semi-classical Dirichlet Pauli operator in bounded connected domains in the plane. Rather optimal results have been obtained in previous papers by Ekholm-Kovařík-Portmann and Helffer-Sundqvist for the asymptotics of the ground state energy in the semi-classical limit when the magnetic field has constant sign. In this paper, we focus on the case when the magnetic field changes sign. We show, in particular, that the ground state energy of this Pauli operator will be exponentially small as the semi-classical parameter tends to zero and give lower bounds and upper bounds for this decay rate. Concrete examples of magnetic fields changing sign on the unit disc are discussed. Various natural conjectures are disproved and this leaves the research of an optimal result in the general case still open.

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arXiv2018

Sturm's theorem on the zeros of sums of eigenfunctions: Gelfand's strategy implemented

In the second section "Courant-Gelfand theorem" of his last published paper (Topological properties of eigenoscillations in mathematical physics, Proc. Steklov Institute Math. 273 (2011) 25--34), Arnold recounts Gelfand's strategy to prove that the zeros of any linear combination of the $n$ first eigenfunctions of the Sturm-Liouville problem $$-\, y"(s) + q(x)\, y(x) = λ\, y(x) \mbox{ in } ]0,1[\,, \mbox{ with } y(0)=y(1)=0\,,$$divide the interval into at most $n$ connected components, and concludes that "the lack of a published formal text with a rigorous proof \dots is still distressing." Inspired by Quantum mechanics, Gelfand's strategy consists in replacing the ana\-lysis of linear combinations of the $n$ first eigenfunctions by that of their Slater determinant which is the first eigenfunction of the associated $n$-particle operator acting on Fermions. In the present paper, we implement Gelfand's strategy, and give a complete proof of the above assertion. As a matter of fact, refining Gelfand's strategy, we prove a stronger property taking the multiplicity of zeros into account, a result which actually goes back to Sturm (1836).

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arXiv2018

The density of superconductivity in the bulk regime

In the asymptotic limit of a large Ginzburg-Landau parameter, we give a new asymptotic formula for the $L^2$-norm of the Ginzburg-Landau order parameter. The formula is valid in the bulk regime where the intensity of the applied magnetic field is of the same order as the Ginzburg-Landau parameter and strictly below the second critical field. Our formula complements the celebrated one of Sandier-Serfaty for the $L^4$-norm.

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arXiv2018

On Courant's nodal domain property for linear combinations of eigenfunctions, Part I

According to Courant's theorem, an eigenfunction as\-sociated with the $n$-th eigenvalue $λ\_n$ has at most $n$ nodal domains. A footnote in the book of Courant and Hilbert, states that the same assertion is true for any linear combination of eigenfunctions associated with eigenvalues less than or equal to $λ\_n$. We call this assertion the \emph{Extended Courant Property}.\smallskipIn this paper, we propose simple and explicit examples for which the extended Courant property is false: convex domains in $\R^n$ (hypercube and equilateral triangle), domains with cracks in $\mathbb{R}^2$, on the round sphere $\mathbb{S}^2$, and on a flat torus $\mathbb{T}^2$.

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arXiv2018

Sturm's theorem on zeros of linear combinations of eigenfunctions

Motivated by recent questions about the extension of Courant's nodal domain theorem, we revisit a theorem published by C. Sturm in 1836, which deals with zeros of linear combination of eigenfunctions of Sturm-Liouville problems. Although well known in the nineteenth century, this theorem seems to have been ignored or forgotten by some of the specialists in spectral theory since the second half of the twentieth-century. Although not specialists in History of Sciences, we have tried to put these theorems into the context of nineteenth century mathematics.

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arXiv2018

Inequalities for the lowest magnetic Neumann eigenvalue

We study the ground state energy of the Neumann magnetic Laplacian on planar domains. For a constant magnetic field we consider the question whether, under an assumption of fixed area, the disc maximizes this eigenvalue. More generally, we discuss old and new bounds obtained on this problem.

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arXiv2018

The spectrum of a Schrödinger operator in a wire-like domain with a purely imaginary degenerate potential in the semiclassical limit

Consider a two-dimensional domain shaped like a wire, not necessarily of uniform cross section. Let $V$ denote an electric potential driven by a voltage drop between the conducting surfaces of the wire. We consider the operator ${\mathcal A}_h=-h^2Δ+iV$ in the semi-classical limit $h\to 0$. We obtain both the asymptotic behaviour of the left margin of the spectrum, as well as resolvent estimates on the left side of this margin. We extend here previous results obtained for potentials for which the set where the current (or $\nabla V$) is normal to the boundary is discrete, in contrast with the present case where $V$ is constant along the conducting surfaces.

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arXiv2017

Nodal domains, spectral minimal partitions, and their relation to Aharonov-Bohm operators

This survey is a short version of a chapter written by the first two authors in the book [A. Henrot, editor. Shape optimization and spectral theory. Berlin: De Gruyter, 2017] (where more details and references are given) but we have decided here to put more emphasis on the role of the Aharonov-Bohm operators which appear to be a useful tool coming from physics for understanding a problem motivated either by spectral geometry or dynamics of population. Similar questions appear also in Bose-Einstein theory. Finally some open problems which might be of interest are mentioned.

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arXiv2017

Peierls' substitution for low lying spectral energy windows

We consider a $2d$ magnetic Schrödinger operator perturbed by a weak magnetic field which slowly varies around a positive mean. In a previous paper we proved the appearance of a `Landau type' structure of spectral islands at the bottom of the spectrum, under the hypothesis that the lowest Bloch eigenvalue of the unperturbed operator remained simple on the whole Brillouin zone, even though its range may overlap with the range of the second eigenvalue. We also assumed that the first Bloch spectral projection was smooth and had a zero Chern number. In this paper we extend our previous results to the only two remaining possibilities: either the first Bloch eigenvalue remains isolated while its corresponding spectral projection has a non-zero Chern number, or the first two Bloch eigenvalues cross each other.

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arXiv2017

The density of superconductivity in domains with corners

We compute the $L^2$-norm of the minimizer of the Ginzburg-Landau functional in a planar domain with a finite number of corners. Our computations are valid for a uniform applied magnetic field, large Ginzburg-Landau parameter and in the regime where superconductivity is confined near the corners of the domain.

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arXiv2017

Spectral semi-classical analysis of a complex Schrödinger operator in exterior domains

Generalizing previous results obtained for the spectrum of the Dirichlet and Neumann realizations in a bounded domain of a Schrödinger operator with a purely imaginary potential $h^2Δ+iV$ in the semiclassical limit $h\to 0$ we address the same problem in exterior domains. In particular we obtain the left margin of the spectrum, and the emptiness of the essential part of the spectrum under some additional assumptions.

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arXiv2017

On a Schrödinger operator with a purely imaginary potential in the semiclassical limit

We consider the operator ${\mathcal A}_h=-Δ+iV$ in the semi-classical $h\rightarrow 0$, where $V$ is a smooth real potential with no critical points. We obtain both the left margin of the spectrum, as well as resolvent estimates on the left side of this margin. We extend here previous results obtained for the Dirichlet realization of ${\mathcal A}_h$ by removing significant limitations that were formerly imposed on $V$. In addition, we apply our techniques to the more general Robin boundary condition and to a transmission problem which is of significant interest in physical applications.

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arXiv2017

Some nodal properties of the quantum harmonic oscillator and other Schr{ö}dinger operators in $\mathbb{R}^2$

For the spherical Laplacian on the sphere and for the Dirichlet Laplacian in the square}, Antonie Stern claimed in her PhD thesis (1924) the existence of an infinite sequence of eigenvalues whose corresponding eigenspaces contain an eigenfunction with exactly two nodal domains. These results were given complete proofs respectively by Hans Lewy in 1977, and the authors in 2014 (see also Gauthier-Shalom--Przybytkowski, 2006). In this paper, we obtain similar results for the two dimensional isotropic quantum harmonic oscillator. In the opposite direction, we construct an infinite sequence of regular eigenfunctions with as many nodal domains as allowed by Courant's theorem, up to a factor $\frac{1}{4}$. A classical question for a $2$-dimensional bounded domain is to estimate the length of the nodal set of a Dirichlet eigenfunction in terms of the square root of the energy. In the last section, we consider some Schr{ö}dinger operators $-Δ+ V$ in $\mathbb{R}^2$ and we provide bounds for the length of the nodal set of an eigenfunction with energy $λ$ in the classically permitted region $\{V(x) < λ\}$.

math.AP↗
arXiv2017

Low lying spectral gaps induced by slowly varying magnetic fields

We consider a periodic Schrödinger operator in two dimensions perturbed by a weak magnetic field whose intensity slowly varies around a positive mean. We show in great generality that the bottom of the spectrum of the corresponding magnetic Schrödinger operator develops spectral islands separated by gaps, reminding of a Landau-level structure. First, we construct an effective Hofstadter-like magnetic matrix which accurately describes the low lying spectrum of the full operator. The construction of this effective magnetic matrix does not require a gap in the spectrum of the non-magnetic operator, only that the first and the second Bloch eigenvalues do not cross but their ranges might overlap. The crossing case is more difficult and will be considered elsewhere. Second, we perform a detailed spectral analysis of the effective matrix using a gauge-covariant magnetic pseudo-differential calculus adapted to slowly varying magnetic fields. As an application, we prove in the overlapping case the appearance of spectral islands separated by gaps.

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arXiv2017

On the semi-classical analysis of the groundstate energy of the Dirichlet Pauli operator in non-simply connected domains

We consider the Dirichlet Pauli operator in bounded connected domains in the plane, with a semi-classical parameter. We show, in particular, that the ground state energy of this Pauli operator will be exponentially small as the semi-classical parameter tends to zero and estimate this decay rate. This extends our results, discussing the results of a recent paper by Ekholm--Kovařík--Portmann, to include also non-simply connected domains.

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arXiv2016

The weak Pleijel theorem with geometric control

Let $Ω\subset \mathbb R^d\,, d\geq 2$, be a bounded open set, and denote by $λ\_j(Ω), j\geq 1$, the eigenvalues of the Dirichlet Laplacian arranged in nondecreasing order, with multiplicities. The weak form of Pleijel's theorem states that the number of eigenvalues $λ\_j(Ω)$, for which there exists an associated eigenfunction with precisely $j$ nodal domains (Courant-sharp eigenvalues), is finite. The purpose of this note is to determine an upper bound for Courant-sharp eigenvalues, expressed in terms of simple geometric invariants of $Ω$. We will see that this is connected with one of the favorite problems considered by Y. Safarov.

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