Related papers: Absence of zero resonances of massless Dirac opera…

Let $H$ be a real Hilbert space and $\Phi:H\to H$ be a $C^1$ operator with Lipschitzian derivative and closed range. We prove that $\Phi^{-1}(0)\neq \emptyset$ if and only if, for each $\epsilon>0$, there exist a convex set $X\subset H$ and…

Discussed are $\pm m$ modes and $\pm m$ resonances of Dirac operators with vector potentials $H_{\!A}= \alpha \cdot (D - A(x)) + m \beta$. Asymptotic limits of $\pm m$ modes at infinity are derived when $|A(x)| \le C<x>^{-\rho}$, $\rho >…

We compute the spectral density of the (Hermitean) Dirac operator in Quantum Chromodynamics with two light degenerate quarks near the origin. We use CLS/ALPHA lattices generated with two flavours of O(a)-improved Wilson fermions…

The absolute continuity of the spectrum for the periodic Dirac operator $$ \hat D=\sum_{j=1}^n(-i\frac {\partial}{\partial x_j}-A_j)\hat \alpha_j + \hat V^{(0)}+\hat V^{(1)}, x\in R^n, n\geq 3, $$ is proved given that either $A\in…

In this article, we have introduced a $\mathcal{PT}$ symmetric non-Hermitian Hamiltonian model which is given as $\hat{\mathcal{H}}=\omega (\hat{b}^{\dag}\hat{b}+1/2)+ \alpha (\hat{b}^{2}-(\hat{b}^{\dag})^{2})$ where $\omega$ and $\alpha$…

This paper is mainly devoted to study time decay estimates of the higher-order Schr\"{o}dinger type operator $H=(-\Delta)^{m}+V(x)$ in $\mathbf{R}^{n}$ for $n>2m$ and $m\in\mathbf{N}$. For certain decay potentials $V(x)$, we first derive…

We present some results obtained in collaboration with prof. Piero D'Ancona concerning global existence for the 3D cubic non linear massless Dirac equation with a potential for small initial data in $H^1$ with slight additional assumptions.…

We show that a simplified version of the Dirac interaction operator given by $\hat H_\mathrm{I} \propto \int d\mathbf{k}d\mathbf{p}(\hat a(\mathbf{k}) + \hat a^\dagger(-\mathbf{k})) \hat b^\dagger(\mathbf{p} + \mathbf{k}) \hat…

The Dirac equation has been studied in which the Dirac matrices $\hat{\boldmath$\alpha$}, \hat\beta$ have space factors, respectively $f$ and $f_1$, dependent on the particle's space coordinates. The $f$ function deforms Heisenberg algebra…

We consider the Schr{\"o}dinger operator --$\Delta$ + V on the Euclidean space with potential in the Lorentz space L^{n/2,1} and we find necessary and sufficient conditions for zero to be a resonance or an eigenvalue. We consider functions…

Let $\Delta_M$ be the Laplace operator on a compact $n$-dimensional Riemannian manifold without boundary. We study the zero sets of its eigenfunctions $u:\Delta u + \lambda u =0$. In dimension $n=2$ we refine the Donnelly-Fefferman estimate…

We study Kohn-Dirac operators $D_\theta$ on strictly pseudoconvex CR manifolds with ${\rm spin}^{\mathbb C}$ structure of weight $\ell\in{\mathbb Z}$. Certain components of $D_\theta$ are CR invariants. We also derive CR invariant twistor…

The zeta and eta-functions associated with massless and massive Dirac operators, in a D-dimensional (D odd or even) manifold without boundary, are rigorously constructed. Several mathematical subtleties involved in this process are…

We consider an elliptic self-adjoint first order differential operator L acting on pairs (2-columns) of complex-valued half-densities over a connected compact 3-dimensional manifold without boundary. The principal symbol of the operator L…

We prove that the massless Dirac operator in $\mathbb{R^3}$ with long-range potential has an a.c. spectrum which fills the whole real line. The Dirac operators with matrix-valued potentials are considered as well.

In this article, we show that if $A$ is a maximal monotone operator on a Hilbert space $H$ with $0$ in the range $\textrm{Rg}(A)$ of $A$, then for every $0<s<1$, the Dirichlet problem associated with the Bessel-type equation $$…

The importance of the spectral density of the Dirac operator in studying spontaneous chiral symmetry breaking and anomalous U(1) axial symmetry breaking are reviewed. It is shown that both types of symmetry breaking can be traced to effects…

By the Aharonov-Casher theorem, the Pauli operator $P$ has no zero eigenvalue when the normalized magnetic flux $\alpha$ satisfies $|\alpha|<1$, but it does have a zero energy resonance. We prove that in this case a Lieb-Thirring inequality…

Let ${\mathscr A}(D)$ be an algebra of functions continuous in the disk $D=\{z\in{\mathbb C}\,|\,\,\,|z|\leqslant 1\}$ and {\it holomorphic} into $D$. The well-known fact is that the set ${\mathscr M}$ of its characters (homomorphisms…

We introduce a family of metric-deformed Heisenberg algebras $M_1$ and $M_2$, where the commutation relations are expressed directly in terms of the components of a diagonal Lorentzian metric. We show that these algebras unify several known…