Hanne Van Den Bosch

• PhD
• Professor (Assistant) at University of Chile

We consider Dirac-type operators on manifolds with boundary, and set out to determine all local smooth boundary conditions that give rise to (strongly) regular self-adjoint operators. By combining the general theory of boundary value problems for Dirac operators as in [BB12] and pointwise considerations, for local smooth boundary conditions the que...

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 some Gagliardo–Nirenberg–Sobolev interpolation inequalities. Domain, self...

We study the linearized Vlasov-Poisson equation in the gravitational case around steady states that are decreasing and continuous functions of the energy. We identify the absolutely continuous spectrum and give criteria for the existence of oscillating modes and estimate their number. Our method allows us to take into account an attractive external...

We study the spectral stability of the nonlinear Dirac operator in dimension \(1+1\), restricting our attention to nonlinearities of the form \(f(\left\langle \psi ,\beta \psi \right\rangle _{\mathbb {C}^2}) \beta \). We obtain bounds on eigenvalues for the linearized operator around standing wave solutions of the form \(e^{-i\omega t} \phi _0\). F...

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 t...

We estimate the lowest eigenvalue in the gap 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 Schr\"odinger operator, which are equivalent to Gagliardo-Nirenberg-Sobolev interpolation inequalities. Domain, self-adjointness, optimality and...

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 ⟨time⟩ ⁻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...

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 $\langle \mathrm{time} \rangle^{-1}$.

We investigate the self-adjointness of the two-dimensional Dirac operator D , with quantum - dot and Lorentz - scalar \delta - 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 form...

We study stability properties of kinks for the (1+1)-dimensional nonlinear scalar field theory models $$\begin{aligned} \partial _t^2\phi -\partial _x^2\phi + W'(\phi ) = 0, \quad (t,x)\in \mathbb {R}\times \mathbb {R}. \end{aligned}$$The orbital stability of kinks under general assumptions on the potential W is a consequence of energy arguments. O...

We study the spectral stability of the nonlinear Dirac operator in dimension 1+1, restricting our attention to nonlinearities of the form $f(\langle\psi,\sigma_3\psi\rangle_{\mathbb{C}^2}) \sigma_3$. We obtain bounds on eigenvalues for the linearized operator around standing wave solutions of the form $e^{-i\omega t} \phi_0$. For the case of power...

We study stability properties of kinks for the (1+1)-dimensional nonlinear scalar field theory models \begin{equation*} \partial_t^2\phi -\partial_x^2\phi + W'(\phi) = 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 m...

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 Rd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage...

We study the self-adjointenss of the two-dimensional Dirac operator with Quantum-dot and Lorentz-scalar $\delta$-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 domai...

We prove that in Müller theory, a nucleus of charge Z can bind at most Z+C electrons for a constant C independent of Z.

In this paper we study the existence and non-existence of minimizers for a type of (critical) Poincar\'{e}-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}^...

We prove that in M\"uller theory, a nucleus of charge $Z$ can bind at most $Z+C$ electrons for a constant $C$ independent of $Z$.

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 conv...

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 conv...

We prove that in Müller theory, a nucleus of charge Z can bind at most Z + C electrons for a constant C independent of Z.

We consider Dirac operators defined on planar domains. For a large class of boundary conditions, we give a direct proof of their self-adjointness in the Sobolev space $H^1$.

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 \({\Omega } \subset...

We consider Dirac operators defined on planar domains. For a large class of boundary conditions, we give a direct proof of their self-adjointness in the Sobolev space \(H^1\).

We study the ionization problem in the Thomas-Fermi-Dirac-von Weizs\"acker 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 potent...

We consider an atom described by M\"uller 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...

We consider an atom described by M\"uller 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...

We prove that in Thomas-Fermi-Dirac-von Weizs\"acker 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...

We prove that in Thomas-Fermi-Dirac-von Weizs\"acker 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...

We study the ionization problem in the Thomas-Fermi-Dirac-von Weizs\"acker 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 potent...

The two-dimensional Dirac operator describes low-energy excitations in graphene. Different boundary conditions correspond to different cuts of graphene samples, the most prominent being the so-called zigzag, armchair, and infinite mass conditions. We prove a lower bound to the spectral gap around zero, proportional to $|\Omega|^{-1/2}$, for Dirac o...

We consider the Pauli operator in $\mathbb R^3$ for magnetic fields in $L^{3/2}$ that decay at infinity as $|x|^{-2-\beta}$ with $\beta > 0$. In this case we are able to prove that the existence of a zero mode for this operator is equivalent to a quantity $\delta(\mathbf B)$, defined below, being equal to zero. Complementing a result from [Balinsky...

Probabilistic Cellular Automata (PCA) are simple models used to study dynamical phase transitions. There exist mean field approximations to PCA that can be shown to exhibit a phase transition. We introduce a model interpolating between a class of PCA, called majority voters, and their corresponding mean field models. Using graphical methods, we pro...