Related papers: Dirac-Coulomb operators with general charge distri…
We prove a sharp Hardy-type inequality for the Dirac operator. We exploit this inequality to obtain spectral properties of the Dirac operator perturbed with Hermitian matrix-valued potentials $\mathbf V$ of Coulomb type: we characterise its…
We carry out the spectral analysis of singular matrix valued perturbations of 3-dimensional Dirac operators with variable magnetic field of constant direction. Under suitable assumptions on the magnetic field and on the perturbations, we…
This paper is devoted to the study of the two-dimensional Dirac-Coulomb operator in presence of an Aharonov-Bohm external magnetic potential. We characterize the highest intensity of the magnetic field for which a two-dimensional magnetic…
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.…
The purpose of this paper is to study the eigenvalues $\{\lambda_{\mu,i} \}_i$ for the Dirichlet Hardy-Leray operator, i.e. $$ -\Delta u+\mu|x|^{-2}u=\lambda u\ \ {\rm in}\ \, \Omega,\quad\quad u=0\ \ {\rm on}\ \ \partial\Omega,$$ where…
We generalize the Donsker-Varadhan minimax formula for the principal eigenvalue of a uniformly elliptic operator in nondivergence form to the first principal half-eigenvalue of a fully nonlinear operator which is concave (or convex) and…
The self-adjointness of $H+V$ is studied, where $H=-i\alpha\cdot\nabla +m\beta$ is the free Dirac operator in $\R^3$ and $V$ is a measure-valued potential. The potentials $V$ under consideration are given by singular measures with respect…
The lowest eigenvalue of the Schr\"odinger operator $-\Delta+\mathcal{V}$ on a compact Riemannian manifold without boundary is studied. We focus on the particularly subtle case of a sign changing potential with positive average.
In this work we prove that the eigenvalues of the $n$-dimensional massive Dirac operator $\mathscr{D}_0 + V$, $n\ge2$, perturbed by a possibly non-Hermitian potential $V$, are localized in the union of two disjoint disks of the complex…
We consider a linear symmetric operator in a Hilbert space that is neither bounded from above nor from below, admits a block decomposition corresponding to an orthogonal splitting of the Hilbert space and has a variational gap property…
In this article, we consider the Dirac operator with constant magnetic field in $\mathbb R^2$. Its spectrum consists of eigenvalues of infinite multiplicities, known as the Landau-Dirac levels. Under compactly supported perturbations, we…
The finite-rank Lieb-Thirring inequality provides an estimate on a Riesz sum of the $N$ lowest eigenvalues of a Schr\"odinger operator $-\Delta-V(x)$ in terms of an $L^p(\mathbb{R}^d)$ norm of the potential $V$. We prove here the existence…
We study the eigenvalue problem for a one-dimensional Dirac operator with a spatially varying ``mass'' term. It is well-known that when the mass function has the form of a kink, or \emph{domain wall}, transitioning between strictly positive…
The Dirac equation for a massive spin-1/2 field in a central potential V in three dimensions is studied without fixing a priori the functional form of V. The second-order equations for the radial parts of the spinor wave function are shown…
In this paper, we describe the leftmost eigenvalue of the non-selfadjoint operator $\mathcal{A}_h = -h^2\Delta+iV(x)$ with Dirichlet boundary conditions on a smooth bounded domain $\Omega\subset\mathbb{R}^n\,$, as $h\rightarrow0\,$. $V$ is…
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 this work, we use regularized determinant approach to study the discrete spectrum generated by relatively compact non-self-adjoint perturbations of the magnetic Schr\"odinger operator $(-i\nabla - \textbf{\textup{A}})^{2} - b$ in…
Quantum dynamical lower bounds for continuous and discrete one-dimensional Dirac operators are established in terms of transfer matrices. Then such results are applied to various models, including the Bernoulli-Dirac one and, in contrast to…
We prove a lower bound for the first eigenvalue of the Dirac operator on a compact Riemannian spin manifold depending on the scalar curvature as well as a chosen Codazzi tensor. The inequality generalizes the classical estimate from [2].
The present paper studies concentration phenomena of semiclassical approximation of a massive Dirac equation with general nonlinear self-coupling: \[ -i\hbar\alpha\cdot\nabla w+a\beta w+V(x)w=g(|w|)w \,. \] Compared with some existing…