Related papers: Maxwell operator in a cylinder. Separation of vari…
In the paper we consider the Maxwell operator in a three-dimensional cylinder with coefficients periodic along the axis of a cylinder. It is proved that for cylinders with circular and rectangular cross-section the spectrum of the Maxwell…
As a first step to deriving effective dynamics and ray optics, we prove that the perturbed periodic Maxwell operator in d = 3 can be seen as a pseudodifferential operator. This necessitates a better understanding of the periodic Maxwell…
The purpose of this paper is to prove that the spectrum of an isotropic Maxwell operator with electric permittivity and magnetic permeability that are periodic along certain directions and tending to a constant super-exponentially fast in…
This paper provides a view of Maxwell's equations from the perspective of complex variables. The study is made through complex differential forms and the Hodge star operator in $\mathbb{C}^2$ with respect to the Euclidean and the Minkowski…
In this paper, we establish a condition on the coefficients of differential operators generated in the space of square-integrable functions on the entire real line by an ordinary differential expression with periodic, complex-valued…
We consider the algebra of mixed multidimensional integral operators. In particular, Fredholm integral operators of the first and second kind belongs to this algebra. For the piecewise constant kernels we provide an explicit representation…
A geometric theory of the irreducible tensor operators of quantum spin systems. It is based upon the Maxwell-Sylvester geometric representation of the multipolar electrostatic potential. In the latter, an order-$\ell$ multipolar potential…
In the paper the general case of a normal discrete Hausdorff operators in $L^2(\mathbb{R}^d)$ is considered. The main result states that under some natural arithmetic condition the spectrum of such an operator is rotationally invariant.…
Recently a new formalism for perturbations of Maxwell's equations on the background of the Kerr-NUT-(A)dS spacetime was proposed, with which the equations are reduced to a equation of motion of a scalar field that can be solved by…
We consider the third order operator with small 1-periodic coefficients on the real line. The spectrum of the operator is absolutely continuous and covers all real line. Under the minimal conditions on the coefficients we show that there…
For an elliptic complex of first order differential operators on a smooth manifold, we define a system of two equations which can be thought of as abstract Maxwell equations. The formal theory of this system proves to be very similar to…
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…
We introduce an analog of the Maxwell operator on a q-Minkowski space algebra (treated as a particular case of the so-called Reflection Equation Algebra) and on certain of its quotients. We treat the space of "quantum differential forms" as…
Using tools from functional analysis we show that for bounded and convex domains in three dimensions, the Maxwell constants are bounded from below and above by Friedrichs' and Poincare's constants.
Representations of polynomial covariance commutation relations by pairs of linear integral and differential operators are constructed in the space of infinitely continuously differentiable functions. Representations of polynomial covariance…
The free Maxwell theory in D<>4 dimensions provides a physical example of a unitary, scale invariant theory which is NOT conformally invariant. The easiest way to see this is that the field strength operator F_mn is neither a primary nor a…
Using systematic calculations in spinor language, we obtain simple descriptions of the second order symmetry operators for the conformal wave equation, the Dirac-Weyl equation and the Maxwell equation on a curved four dimensional Lorentzian…
Coherent-state representations are a standard tool to deal with continuous-variable systems, as they allow one to efficiently visualize quantum states in phase space. Here, we work out an alternative basis consisting of monomials on the…
Integration over curved manifolds with higher codimension and, separately, discrete variants of continuous operators, have been two important, yet separate themes in harmonic analysis, discrete geometry and analytic number theory research.…
The Bethe-Sommerfeld conjecture states that the spectrum of the stationary Schrodinger operator with a periodic potential in dimensions higher than 1 has only finitely many gaps. After work done by many authors, it has been proven by now in…