Denis Borisov

Bound States in Weakly Deformed Strips and Layers

Abstract. We consider Dirichlet Laplacians on straight strips in

\(\mathcal {PT}\) -Symmetric Waveguides

{\Bbb R}^2

Geometric coupling thresholds in a two-dimensional strip

or layers in

On a waveguide with frequently alternating boundary conditions: homogenized Neumann condition

{\Bbb R}^3

Homogenization of the planar waveguide with frequently alternating boundary conditions

with a weak local deformation. First we generalize a result of Bulla et al. to the three-dimensional situation showing that weakly coupled bound states exist if the volume change induced by the deformation is positive;we also derive the leading order of the weak-coupling asymptotics. With the knowledge of the eigenvalue analytic properties, we demonstrate then an alternative method which makes it possible to evaluate the next term in the asymptotic expansion for both the strips and layers. It gives,in particular, a criterion for the bound-state existence in the critical case when the added volume is zero.

Spectrum of the Magnetic Schrodinger Operator in a Waveguide with Combined Boundary Conditions

Abstract.We introduce a planar waveguide of constant width with non-Hermitian

Quantum waveguides with small periodic perturbations: gaps and edges of Brillouin zones

\mathcal {PT}

Planar waveguide with “twisted” boundary conditions: Discrete spectrum

-symmetric Robin boundary conditions. We study the spectrum of this system in the regime when the boundary coupling function is a compactly supported perturbation of a homogeneous coupling. We prove that the essential spectrum is positive and independent of such perturbation, and that the residual spectrum is empty. Assuming that the perturbation is small in the supremum norm, we show that it gives rise to real weakly-coupled eigenvalues converging to the threshold of the essential spectrum. We derive sufficient conditions for these eigenvalues to exist or to be absent. Moreover, we construct the leading terms of the asymptotic expansions of these eigenvalues and the associated eigenfunctions.