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Hanne Van Den Bosch

Publications and source records attributed to Hanne Van Den Bosch.

At least 19 recordsLinked to original sources
arXiv2026

Asymptotics for eigenvalues of one-dimensional Dirac operators in the weak coupling limit

In this paper, we derive new results on the asymptotic behavior of eigenvalues of perturbed one-dimensional massive Dirac operators in the weak coupling limit. Two classes of potentials are considered. For bounded Hermitian potentials $V$ satisfying $|V(x)| \lesssim |x|^{-1}$ for large $|x|$, we recover the leading term, which may include a logarithmic correction if $V(x) \sim |x|^{-1}$ at infinity. For possibly non-Hermitian $L^1$ potentials satisfying a suitable moment condition, we obtain the second term in the asymptotic expansion. The first result is based on a min-max principle adapted to the non-relativistic limit, while the second result is obtained via the Birman-Schwinger principle and resolvent expansions.

math-ph↗
arXiv2025

Edge states for tight-binding operators with soft walls

We study one- and two-dimensional periodic tight-binding models under the presence of a potential that grows to infinity in one direction, hence preventing the particles to escape in this direction (the soft wall). We prove that a spectral flow appears in these corresponding edge models, as the wall is shifted. We identity this flow as a number of Bloch bands, and provide a lower bound for the number of edge states appearing in such models.

math-ph↗
arXiv2025

Quantitative Stability for Yamabe minimizers on manifolds with boundary

This paper addresses the quantitative stability for a Yamabe-type functional on compact manifolds with boundary introduced by Escobar. Minimizers of the functional correspond to scalar-flat metrics with constant mean curvature on the boundary. We prove that the deficit controls the distance to the minimizing set to a suitable power by reducing the problem to the analogous question for an effective functional on the boundary.

math.DG↗
arXiv2023

Keller and Lieb-Thirring estimates of the eigenvalues in the gap of Dirac operators

We estimate the lowest eigenvalue in the gap of the essential spectrum of a Dirac operator with mass in terms of a Lebesgue norm of the potential. Such a bound is the counterpart for Dirac operators of the Keller estimates for the Schrödinger operator, which are equivalent to Gagliardo-Nirenberg-Sobolev interpolation inequalities. Domain, self-adjointness, optimality and critical values of the norms are addressed, while the optimal potential is given by a Dirac equation with a Kerr nonlinearity. A new critical bound appears, which is the smallest value of the norm of the potential for which eigenvalues may reach the bottom of the gap in the essential spectrum. The Keller estimate is then extended to a Lieb-Thirring inequality for the eigenvalues in the gap. Most of our result are established in the Birman-Schwinger reformulation.

math.AP↗
arXiv2022

A Block-diagonal form for four-component operators describing Graphene Quantum Dots

We consider four-component Dirac operators on domains in the plane. With suitable boundary conditions, these operators describe graphene quantum dots. The most general boundary conditions are defined by a matrix depending on four real parameters. For operators with constant boundary parameters we show that the Hamiltonian is unitary equivalent to two copies of the two-component operator. This allows to extend the known results for this type of operators to the four-component case. As an application, we identify the boundary conditions from the tight-binding model for graphene that give rise to a block-diagonal operator in the continuum limit.

math-ph↗
arXiv2022

Results on the spectral stability of standing wave solutions of the Soler model in 1-D

We study the spectral stability of the nonlinear Dirac operator in dimension $1+1$, restricting our attention to nonlinearities of the form $f(\langleψ,βψ\rangle_{\mathbb{C}^2}) β$. We obtain bounds on eigenvalues for the linearized operator around standing wave solutions of the form $e^{-iωt} ϕ_0$. For the case of power nonlinearities $f(s)= s |s|^{p-1}$, $p>0$, we obtain a range of frequencies $ω$ such that the linearized operator has no unstable eigenvalues on the axes of the complex plane. As a crucial part of the proofs, we obtain a detailed description of the spectra of the self-adjoint blocks in the linearized operator. In particular, we show that the condition $\langleϕ_0,βϕ_0\rangle_{\mathbb{C}^2} > 0$ characterizes groundstates analogously to the Schrödinger case.

math-ph↗
arXiv2022

Mixing in an anharmonic potential well

We prove phase-space mixing for solutions to Liouville's equation for integrable systems. Under a natural non-harmonicity condition, we obtain weak convergence of the distribution function with rate $\langle \mathrm{time} \rangle^{-1}$. In one dimension, we also study the case where this condition fails at a certain energy, showing that mixing still holds but with a slower rate. When the condition holds and functions have higher regularity, the rate can be faster.

math-ph↗
arXiv2020

A sufficient condition for asymptotic stability of kinks in general (1+1)-scalar field models

We study stability properties of kinks for the (1+1)-dimensional nonlinear scalar field theory models \begin{equation*} \partial_t^2ϕ-\partial_x^2ϕ+ W'(ϕ) = 0, \quad (t,x)\in\mathbb{R}\times\mathbb{R}. \end{equation*} The orbital stability of kinks under general assumptions on the potential $W$ is a consequence of energy arguments. Our main result is the derivation of a simple and explicit sufficient condition on the potential $W$ for the asymptotic stability of a given kink. This condition applies to any static or moving kink, in particular no symmetry assumption is required. Last, motivated by the Physics literature, we present applications of the criterion to the $P(ϕ)_2$ theories and the double sine-Gordon theory.

math.AP↗
arXiv2019

Self-Adjointness of two dimensional Dirac operators on corner domains

We investigate the self-adjointness of the two-dimensional Dirac operator $D$, with quantum-dot and Lorentz-scalar $δ$-shell boundary conditions, on piecewise $C^2$ domains with finitely many corners. For both models, we prove the existence of a unique self-adjoint realization whose domain is included in the Sobolev space $H^{1/2}$, the formal form domain of the free Dirac operator. The main part of our paper consists of a description of the domain of $D^*$ in terms of the domain of $D$ and the set of harmonic functions that verify some mixed boundary conditions. Then, we give a detailed study of the problem on an infinite sector, where explicit computations can be made: we find the self-adjoint extensions for this case. The result is then translated to general domains by a coordinate transformation.

math.AP↗
arXiv2018

Existence and non-existence of minimizers for Poincaré-Sobolev inequalities

In this paper we study the existence and non-existence of minimizers for a type of (critical) Poincaré-Sobolev inequalities. We show that minimizers do exist for smooth domains in $\mathbb{R}^d$, an also for some polyhedral domains. On the other hand, we prove the non-existence of minimizers in the rectangular isosceles triangle in $\mathbb{R}^2$.

math-ph↗
arXiv2018

Gagliardo-Nirenberg-Sobolev inequalities for convex domains in $\mathbb{R}^d$

A special type of Gagliardo-Nirenberg-Sobolev (GNS) inequalities in $\mathbb{R}^d$ has played a key role in several proofs of Lieb-Thirring inequalities. Recently, a need for GNS inequalities in convex domains of $\mathbb{R}^d$, in particular for cubes, has arised. The purpose of this manuscript is two-fold. First we prove a GNS inequality for convex domains, with explicit constants which depend on the geometry of the domain. Later, using the discrete version of Rumin's method, we prove GNS inequalities on cubes with improved constants.

math-ph↗
arXiv2017

Spectral gaps of Dirac operators describing graphene quantum dots

The two-dimensional Dirac operator describes low-energy excitations in graphene. Different choices for the boundary conditions give rise to qualitative differences in the spectrum of the resulting operator. For a family of boundary conditions, we find a lower bound to the spectral gap around zero, proportional to $|Ω|^{-1/2}$, where $Ω\subset \mathbb{R}^2$ is the bounded region where the Dirac operator acts. This family contains the so-called infinite mass and armchair cases used in the physics literature for the description of graphene quantum dots.

math-ph↗
arXiv2017

The ionization conjecture in Thomas-Fermi-Dirac-von Weizsäcker theory

We prove that in Thomas-Fermi-Dirac-von Weizsäcker theory, a nucleus of charge $Z>0$ can bind at most $Z+C$ electrons, where $C$ is a universal constant. This result is obtained through a comparison with Thomas-Fermi theory which, as a by-product, gives bounds on the screened nuclear potential and the radius of the minimizer. A key ingredient of the proof is a novel technique to control the particles in the exterior region, which also applies to the liquid drop model with a nuclear background potential.

math-ph↗
arXiv2016

Nonexistence in Thomas-Fermi-Dirac-von Weizsäcker theory with small nuclear charges

We study the ionization problem in the Thomas-Fermi-Dirac-von Weizsäcker theory for atoms and molecules. We prove the nonexistence of minimizers for the energy functional when the number of electrons is large and the total nuclear charge is small. This nonexistence result also applies to external potentials decaying faster than the Coulomb potential. In the case of arbitrary nuclear charges, we obtain the nonexistence of stable minimizers and radial minimizers.

math-ph↗
arXiv2016

The maximal excess charge in Müller density-matrix-functional theory

We consider an atom described by Müller theory, which is similar to Hartree-Fock theory, but with a modified exchange term. We prove that a nucleus of charge Z can bind at most Z+C electrons, where C is a universal constant. Our proof proceeds by comparison with Thomas-Fermi theory and a key ingredient is a novel bound on the number of electrons far from the nucleus.

math-ph↗
arXiv2015

A criterion for the existence of zero modes for the Pauli operator with fastly decaying fields

We consider the Pauli operator in $\mathbb R^3$ for magnetic fields in $L^{3/2}$ that decay at infinity as $|x|^{-2-β}$ with $β> 0$. In this case we are able to prove that the existence of a zero mode for this operator is equivalent to a quantity $δ(\mathbf B)$, defined below, being equal to zero. Complementing a result from [Balinsky, Evans, Lewis (2001)], this implies that for the class of magnetic fields considered, Sobolev, Hardy and CLR inequalities hold whenever the magnetic field has no zero mode.

math-ph↗