Searcharxiv⌕ Search

arXiv subjects

Edgardo Stockmeyer

Publications and source records attributed to Edgardo Stockmeyer.

At least 19 recordsLinked to original sources
arXiv2025

On gap properties for the linearized 1D Dirac--Soler model

We study spectral properties of the Dirac operator $L_0$ arising as the upper-right off-diagonal block in the linearization around standing wave solutions of the one-dimensional Soler model with power nonlinearity $f(s)=s|s|^{p-1}$, $p>0$. Our main results concern the so-called gap property: we show that if $p \geq 1$, then the only eigenvalues of $L_0$ are its ground state energies, $-2\omega$ and $0$. In contrast, for $p<1$, additional eigenvalues appear from the thresholds of the essential spectrum. Furthermore, we prove that the thresholds never admit eigenvalues and that they have at most one resonance.

math-ph↗
arXiv2025

Green's functions for magnetic Dirac operators and bulk-edge correspondence

We study the resolvent kernel (Green's function) of magnetic Dirac operators on a half-plane with boundary conditions interpolating between infinite mass and zigzag cases, excluding the latter. We show that these kernels have all the required properties so that the proof of bulk-edge correspondence from the infinite mass case shown recently can be repeated ad litteram. The zigzag case exhibits qualitatively different behavior and will be addressed in future work.

math-ph↗
arXiv2024

Magnetic Dirac systems: Violation of bulk-edge correspondence in the zigzag limit

We consider a Dirac operator with constant magnetic field defined on a half-plane with boundary conditions that interpolate between infinite mass and zigzag. By a detailed study of the energy dispersion curves we show that the infinite mass case generically captures the profile of these curves, which undergoes a continuous pointwise deformation into the topologically different zigzag profile. Moreover, these results are applied to the bulk-edge correspondence. In particular, by means of a counterexample, we show that this correspondence does not always hold true in the zigzag case.

math-ph↗
arXiv2023

Tunneling estimates for two-dimensional perturbed magnetic Dirac systems

We prove tunneling estimates for two-dimensional Dirac systems which are localized in space due to the presence of a magnetic field. The Hamiltonian driving the motion admits the decomposition $H = H_0 + W$, where $H_0 $ is a rotationally symmetric magnetic Dirac operator and $W$ is a position-dependent matrix-valued potential satisfying certain smoothness condition in the angular variable. A consequence of our results are upper bounds for the growth in time of the expected size of the system and its total angular momentum.

math-ph↗
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↗
arXiv2021

On the Dirac bag model in strong magnetic fields

In this work we study Dirac operators on two-dimensional domains coupled to a magnetic field perpendicular to the plane. We focus on the infinite-mass boundary condition (also called MIT bag condition). In the case of bounded domains, we establish the asymptotic behavior of the low-lying (positive and negative) energies in the limit of strong magnetic field. Moreover, for a constant magnetic field $B$, we study the problem on the half-plane and find that the Dirac operator has continuous spectrum except for a gap of size $a\_0\sqrt{B}$, where $a\_0\in (0,\sqrt{2})$ is a universal constant. Remarkably, this constant characterizes certain energies of the system in a bounded domain as well. We discuss how these findings, together with our previous work, give a fairly complete description of the eigenvalue asymptotics of magnetic two-dimensional Dirac operators under general boundary conditions.

math.AP↗
arXiv2020

On the two-dimensional quantum confined Stark effect in strong electric fields

We consider a Stark Hamiltonian on a two-dimensional bounded domain with Dirichlet boundary conditions. In the strong electric field limit we derive, under certain local convexity conditions, a three-term asymptotic expansion of the low-lying eigenvalues. This shows that the excitation frequencies are proportional to the square root of the boundary curvature at a certain point determined by the direction of the electric field.

math-ph↗
arXiv2020

On the asymptotic dynamics of 2-D magnetic quantum systems

In this work we provide results on the long time localisation in space (dynamical localisation) of certain two-dimensional magnetic quantum systems. The underlying Hamiltonian may have the form $H=H_0+W$, where $H_0$ is rotationally symmetric, has dense point spectrum, and $W$ is a perturbation that breaks the rotational symmetry. In the latter case, we also give estimates for the growth of the angular momentum operator in time.

math-ph↗
arXiv2020

On the semiclassical spectrum of the Dirichlet-Pauli operator

This paper is devoted to semiclassical estimates of the eigenvalues of the Pauli operator on a bounded open set whose boundary carries Dirichlet conditions. Assuming that the magnetic field is positive and a few generic conditions, we establish the simplicity of the eigenvalues and provide accurate asymptotic estimates involving Segal-Bargmann and Hardy spaces associated with the magnetic field.

math.SP↗
arXiv2018

Resolvent convergence to Dirac operators on planar domains

Consider a Dirac operator defined on the whole plane with a mass term of size m supported outside a domain Omega. We give a simple proof for the norm resolvent convergence, as m goes to infinity, of this operator to a Dirac operator defined on Omega with infinite mass boundary conditions. The result is valid for bounded and unbounded domains and gives estimates on the speed of convergence. Moreover, the method easily extends when adding external matrix-valued potentials.

math-ph↗
arXiv2017

Spectral gaps in graphene antidot lattices

We consider the gap creation problem in an antidot graphene lattice, i.e. a sheet of graphene with periodically distributed obstacles. We prove several spectral results concerning the size of the gap and its dependence on different natural parameters related to the antidot lattice.

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↗
arXiv2016

Infinite mass boundary conditions for Dirac operators

We study a self-adjoint realization of a massless Dirac operator on a bounded connected domain $Ω\subset \mathbb{R}^2$ which is frequently used to model graphene quantum dots. In particular, we show that this operator is the limit, as $M\to \infty$, of a Dirac operator defined on the whole plane, with a mass term of size $M$ supported outside $Ω$.

math-ph↗
arXiv2014

Ballistic dynamics of Dirac particles in electro-magnetic fields

Investigating properties of two-dimensional Dirac operators coupled to an electric and a magnetic field (perpendicular to the plane) requires in general unbounded (vector-) potentials. If the system has a certain symmetry, the fields can be described by one-dimensional potentials $V$ and $A$. Assuming that $|A|<|V|$ outside some arbitrary large ball, we show that absolutely continuous states of the effective Dirac operators spread ballistically. These results are based on well-known methods in spectral dynamics together with certain new Hilbert-Schmidt bounds. We use Lorentz boosts to derive these new estimates.

math-ph↗
arXiv2014

Confinement-deconfinement transitions for two-dimensional Dirac particles

We consider a two-dimensional massless Dirac operator coupled to a magnetic field $B$ and an electric potential $V$ growing at infinity. We find a characterization of the spectrum of the resulting operator $H$ in terms of the relation between $B$ and $V$ at infinity. In particular, we give a sharp condition for the discreteness of the spectrum of $H$ beyond which we find dense pure point spectrum.

math-ph↗