Related papers: On gap properties for the linearized 1D Dirac--Sol…

We study the spectral stability of solitary wave solutions to the nonlinear Dirac equation in one dimension. We focus on the Dirac equation with cubic nonlinearity, known as the Soler model in (1+1) dimensions and also as the massive…

Gap opening at the Dirac point of the single-layer graphene with periodic scalar and vector potentials has been theoretically investigated under the continuum model. The symmetry analysis indicates that the two-fold degeneracy at the Dirac…

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…

On the half line $[0,\infty)$ we study first order differential operators of the form $B 1/i d/(dx) + Q(x)$, where $B:=\mat{B_1}{0}{0}{-B_2}$, $B_1,B_2\in M(n,\C)$ are self--adjoint positive definite matrices and $Q:\R_+\to M(2n,\C)$,…

We consider the two-dimensional Dirac operator with infinite mass boundary conditions posed in a tubular neighborhood of a $C^4$-planar curve. Under generic assumptions on its curvature $\kappa$, we prove that in the thin-width regime the…

At nonzero density the eigenvalues of the Dirac operator move into the complex plane, while its singular values remain real and nonnegative. In QCD-like theories, the singular-value spectrum carries information on the diquark (or pionic)…

Analyzing the point spectrum, i.e. bound state energy eigenvalue, of the Dirac delta function in two and three dimensions is notoriously difficult without recourse to regularization or renormalization, typically both. The reason for this in…

We consider half-line Dirac operators with operator data of Wigner-von Neumann type. If the data is a finite linear combination of Wigner-von Neumann functions, we show absence of singular continuous spectrum and provide an explicit set…

We provide quantitative estimates on the location of eigenvalues of one-dimensional discrete Dirac operators with complex $\ell^p$-potentials for $1\leq p\leq\infty$. As a corollary, subsets of the essential spectrum free of embedded…

A generalized two-dimensional periodic Dirac operator is considered, with $L^{\infty}$-matrix-valued coefficients of the first order derivatives and with complex matrix-valued potential. It is proved that if the matrix-valued potential has…

This review is devoted to the study of stationary solutions of linear and nonlinear equations from relativistic quantum mechanics, involving the Dirac operator. The solutions are found as critical points of an energy functional. Contrary to…

We consider the nonlinear Dirac equation in one dimension, also known as the Soler model in (1+1) dimensions, or the massive Gross-Neveu model: $i\partial_t\psi=-i\alpha\partial_x\psi+m\beta\psi-f(\psi^\ast\beta\psi)\beta\psi$,…

We study perturbed Dirac operators of the form $ D_s= D + s\A :\Gamma(E^0)\rightarrow \Gamma(E^1)$ over a compact Riemannian manifold $(X, g)$ with symbol $c$ and special bundle maps $\A : E^0\rightarrow E^1$ for $s>>0$. Under a simple…

We derive the spectrum of the Dirac operator for the linear sigma-model with quarks in the large N_c approximation using renormalization group flow equations. For small eigenvalues, the Banks-Casher relation and the vanishing linear term…

We analyze the low-lying spectrum and eigenmodes of lattice Dirac operators with a twisted mass term. The twist term expels the eigenvalues from a strip in the complex plane and all eigenmodes obtain a non-vanishing matrix element with…

This is the last paper in a series of three in which we have studied the Peierls substitution in the case of a weak magnetic field. Here we deal with two $2d$ Bloch eigenvalues which have a conical crossing. It turns out that in the…

In this paper we present a complete spectral analysis of Dirac operators with non-Hermitian matrix potentials of the form $i\operatorname{sgn}(x)+V(x)$ where $V\in L^1$. For $V=0$ we compute explicitly the matrix Green function. This allows…

We consider a positive and power-bounded linear operator $T$ on $L^p$ over a finite measure space and prove that, if $TL^p \subseteq L^q$ for some $q > p$, then the essential spectral radius of $T$ is strictly smaller than $1$. As a special…

The spectrum of periodic differential operators typically exhibits a band-gap structure. In this paper, we will consider perturbations to periodic differential operators and investigate the spectrum the perturbation induces in the gaps.…

We study the $L^p$-spectrum of the Dirac operator on complete manifolds. One of the main questions in this context is whether this spectrum depends on $p$. As a first example where $p$-independence fails we compute explicitly the…