The article surveys the main topics, techniques, and results of the theory of periodic
operators arising in mathematical physics and other areas. Close attention is paid to …

In this paper, we prove the existence of a bound state in a waveguide that consists of two
semi-infinite periodic structures separated by an interface. The two periodic structures are …

In this paper, we consider the propagation of waves in an open waveguide in ℝ 2 where the
index of refraction is a local perturbation of a function which is periodic along the axis of the …

This work deals with one-dimensional infinite perturbation---namely, line defects---in
periodic media. In optics, such defects are created to construct an (open) waveguide that …

We study the transport properties of Schr\"{o} dinger operators on $\mathbb {R}^ d $ and
$\mathbb {Z}^ d $ with potentials that are periodic in some directions and compactly …

The recent development of subwavelength photonic and phononic crystals shows the
possibility of controlling wave propagation at deep subwavelength scales. Subwavelength …

We present a new numerical scheme to solve the Helmholtz equation in a wave-guide. We
consider a medium that is bounded in the x_2-direction, unbounded in the x_1-direction …

The Helmholtz equation is considered in an unbounded wave guide, a bounded domain.
The coefficient a is strictly elliptic and either periodic in the unbounded direction or periodic …

The structure of band edges of periodic Schrödinger operators is an interesting and wide
open question of mathematical physics. For example, suppose that a band function k↦! E (k) …

We analyze the propagation of waves in unbounded photonic crystals. Waves are described
by a Helmholtz equation with x-dependent coefficients, the scattering problem must be …