Related papers: A priori estimates for the Hill and Dirac operator…
We consider Schr\"odinger operators in $L^2(\mathbb{R}^3)$ with a singular interaction supported by a finite curve $\Gamma$. We present a proper definition of the operators and study their properties, in particular, we show that the…
We consider the discrete spectrum of the two-dimensional Hamiltonian $H=H_0+V$, where $H_0$ is a Schr\"odinger operator with a non-constant magnetic field $B$ that depends only on one of the spatial variables, and $V$ is an electric…
In this survey we discuss spectral and quantum dynamical properties of discrete one-dimensional Schr\"odinger operators whose potentials are obtained by real-valued sampling along the orbits of an ergodic invertible transformation. After an…
Let M_n be the collection of n x n complex matrices equipped with operator norm. Suppose U, V \in M_n are two unitary matrices, each possessing a gap larger than \Delta in their spectrum, which satisfy ||UV-VU|| \le \epsilon. Then it is…
We consider one-dimensional difference Schroedinger equations on the discrete line with a potential generated by evaluating a real-analytic potential function V(x) on the one-dimensional torus along an orbit of the shift x-->x+nw. If the…
We continue the study of the A-amplitude associated to a half-line Schrodinger operator, -d^2/dx^2+ q in L^2 ((0,b)), b <= infinity. A is related to the Weyl-Titchmarsh m-function via m(-\kappa^2) =-\kappa - \int_0^a A(\alpha)…
The present paper deals with the estimate of the differences of certain positive linear operators and their derivatives. Our approach involves operators defined on bounded intervals, as Bernstein operators, Kantorovich operators, genuine…
Using representation theory, we compute the spectrum of the Dirac operator on the universal covering group of $SL_2(\mathbb R)$, exhibiting it as the generator of $KK^1(\mathbb C, \mathfrak A)$, where $\mathfrak A$ is the reduced…
This paper addresses the problem of computing the eigenvalues lying in the gaps of the essential spectrum of a periodic Schrodinger operator perturbed by a fast decreasing potential. We use a recently developed technique, the so called…
It is shown that the eigenvalue problem for the hermitian Wilson-Dirac operator of for a uniform magnetic field in two dimensions can be reduced to one-dimensional problem described by a relativistic analog of the Harper equation. An…
In this article, we first prove quantitative estimates associated to the unique continuation theorems for operators with partially analytic coefficients of Tataru, Robbiano-Zuily and H\"ormander. We provide local stability estimates that…
These classical inequalities allow one to estimate the number of negative eigenvalues and the sums $S_{\gamma}=\sum |\lambda_i|^{\gamma}$ for a wide class of Schr\"{o}dinger operators. We provide a detailed proof of these inequalities for…
We establish a quantitative version of strong almost reducibility result for $\mathrm{sl}(2,\mathbb{R})$ quasi-periodic cocycle close to a constant in Gevrey class. We prove that, for the quasi-periodic Schr\"odinger operators with small…
A new proof of the conformal covariance of the powers of the flat Dirac operator is obtained. The proof uses their relation with the Knapp-Stein intertwining operators for the spinorial principal series. We also treat the compact picture,…
Let $L$ be a linear, closed, densely defined in a Hilbert space operator, not necessarily selfadjoint. Consider the corresponding wave equations &(1) \quad \ddot{w}+ Lw=0, \quad w(0)=0,\quad \dot{w}(0)=f, \quad \dot{w}=\frac{dw}{dt}, \quad…
In this paper, we study the spectrum of the complex Hill operator $L=\frac{d^2}{dx^2}+q(x;\tau)$ in $L^2(\mathbb{R},\mathbb{C})$ with the Darboux-Treibich-Verdier potential \[q(x;\tau):=-\sum_{k=0}^{3}n_{k}(n_{k}+1)\wp \left(…
In this paper the two-dimensional Dirac operator with a general hermitian $\delta$-shell interaction supported on a straight line is introduced as a self-adjoint operator and its spectral properties are investigated in detail. In…
We study the large mass asymptotics of the Dirac operator with a nondegenerate mass matrix m={diag}(m_1,m_2,m_3) in the presence of scalar and pseudoscalar background fields taking values in the Lie algebra of the U(3) group. The…
We provide a representation of the $C^*$-algebra generated by multidimensional integral operators with piecewise constant kernels and discrete ergodic operators. This representation allows us to find the spectrum and to construct the…
Assume that A is a bounded selfadjoint operator in a Hilbert space H. Then, the variational principle is obtained for some functional. As an application of this principle, a variational principle for the electrical capacitance of a…