Absolutely continuous spectrum for the isotropic Maxwell operator with coefficients that are periodic in some directions and decay in others

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 the remaining directions is purely absolutely continuous. The basic technical tools is a new ``operatorial'' identity relating the Maxwell operator to a vector-valued Schrodinger operator. The analysis of the spectrum of that operator is then handled using ideas developed by the same authors in a previous paper

Spectral asymptotics of Pauli operators and orthogonal polynomials in complex domains

We consider the spectrum of a two-dimensional Pauli operator with a compactly supported electric potential and a variable magnetic field with a positive mean value. The rate of accumulation of eigenvalues to zero is described in terms of the logarithmic capacity of the support of the electric potential. A connection between these eigenvalues and orthogonal polynomials in complex domains is established.Comment: 16 page

On the P\'olya conjecture for the Neumann problem in planar convex domains

Denote by $N_{\cal N} (\Omega,\lambda)$ the counting function of the spectrum of the Neumann problem in the domain $\Omega$ on the plane. G. P\'olya conjectured that $N_{\cal N} (\Omega,\lambda) \ge (4\pi)^{-1} |\Omega| \lambda$. We prove that for convex domains $N_{\cal N} (\Omega,\lambda) \ge (2 \sqrt 3 \,j_0^2)^{-1} |\Omega| \lambda$. Here $j_0$ is the first zero of the Bessel function $J_0$

The Maxwell operator with periodic coefficients in a cylinder

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 operator is absolutely continuous