Related papers: On gap properties for the linearized 1D Dirac--Sol…
In the planar limit, in the deconfined phase, the Euclidean Dirac operator has a spectral gap around zero. We show that functions of eigenvalues close to the spectral edge, which are independent of common rescalings and shifts gauge…
A model in which a Dirac particle in $\mathbb{R}^{3}$ is bound by $N\geqslant1$ spatially distributed zero-range potentials is presented. Interactions between the particle and the potentials are modeled by subjecting a particle's bispinor…
We present new results on the block-diagonalization of Dirac operators on three-dimensional Euclidean space with unbounded potentials. Classes of admissible potentials include electromagnetic potentials with strong Coulomb singularities and…
In the context of some deformed canonical commutation relations leading to isotropic nonzero minimal uncertainties in the position coordinates, a Dirac equation is exactly solved for the first time, namely that corresponding to the Dirac…
We investigate $L^1\to L^\infty$ dispersive estimates for the Dirac equation with a potential in four spatial dimensions. We classify the structure of the obstructions at the thresholds as being composed of an at most two dimensional space…
In this survey we gather recent results on Dirac operators coupled with $\delta$-shell interactions. We start by discussing recent advances regarding the question of self-adjointness for these operators. Afterward we switch to an…
One dimensional Dirac operators $$ L_{bc}(v) y = i 1 & 0 0 & -1 \frac{dy}{dx} + v(x) y, \quad y = y_1 y_2, \quad x\in[0,\pi]$$, considered with $L^2$-potentials $ v(x) = 0 & P(x) Q(x) & 0$ and subject to regular boundary conditions ($bc$),…
We consider the nonlinear Dirac equation, also known as the Soler model: $i\p\sb t\psi=-i\alpha \cdot \nabla \psi+m \beta \psi-f(\psi\sp\ast \beta \psi) \beta \psi$, $\psi(x,t)\in\mathbb{C}^{N}$, $x\in\mathbb{R}^n$, $n\le 3$, $f\in C\sp…
We consider a nonlinear eigenvalue problem for some elliptic equations governed by general operators including the $p$-Laplacian. The natural framework in which we consider such equations is that of Orlicz-Sobolev spaces. we exhibit two…
We consider the Schr\"odinger operator $H$ on the half-line with a periodic potential $p$ plus a compactly supported potential $q$. For generic $p$, its essential spectrum has an infinite sequence of open gaps. We determine the asymptotics…
We characterise regions in the complex plane that contain all non-embedded eigenvalues of a perturbed periodic Dirac operator on the real line with real-valued periodic potential and a generally non-symmetric matrix-valued perturbation V .…
We investigate spectral features of the Dirac operator with infinite mass boundary conditions in a smooth bounded domain of $\mathbb{R}^2$. Motivated by spectral geometric inequalities, we prove a non-linear variational formulation to…
We consider an elliptic self-adjoint first order pseudodifferential operator acting on columns of m complex-valued half-densities over a connected compact n-dimensional manifold without boundary. The eigenvalues of the principal symbol are…
We consider the nonlinear Dirac equation with Soler-type nonlinearity concentrated at one point and present a detailed study of the spectrum of linearization at solitary waves. We then consider two different perturbations of the…
We consider a Dirac operator in three space dimensions, with an electrostatic (i.e. real-valued) potential $V(x)$, having a strong Coulomb-type singularity at the origin. This operator is not always essentially self-adjoint but admits a…
Let $\Omega\subset\mathbb{R}^3$ be an open set, we study the spectral properties of the free Dirac operator $\mathcal{H}$ coupled with the singular potential $V_\kappa=(\epsilon I_4 +\mu\beta+\eta(\alpha\cdot N))\delta_{\partial\Omega}$.…
We review the energy spectrum and transport properties of several types of one- dimensional superlattices (SLs) on single-layer and bilayer graphene. In single-layer graphene, for certain SL parameters an electron beam incident on a SL is…
We study the spectrum of one dimensional integral operators in bounded real intervals of length $2L$, for value of $L$ large. The integral operators are obtained by linearizing a non local evolution equation for a non conserved order…
The aim of this work is to explore the discrete spectrum generated by complex perturbations in $L^{2}(\mathbb{R}^3,\mathbb{C}^4)$ of the $3d$ Dirac operator $\alpha \cdot (-i\nabla - \textbf{A}) + m \beta$ with variable magnetic field.…
We consider the two-dimensional Dirac operator with Lorentz-scalar $\delta$-shell interactions on each edge of a star-graph. An orthogonal decomposition is performed which shows such an operator is unitarily equivalent to an orthogonal sum…