Operadores efetivos para guia de ondas com condição de Robin no plano e espaço

Abstract

First we briefly discuss the spectral properties of the Robin Laplacian in a straight and infinitely long planar strip with Robin coupling parameter changing signs. Assuming that such parameter is a constant outside a compact set, we find the essential spectrum, and also derive a sufficient condition that guarantees the existence of bound states. Next, we have studied the Laplacian in some thin curved domains in the plane and space, with a particular type of Robin boundary conditions and cross-sections; such conditions are nonhomogeneous, in the sense that the Robin coupling parameter depends on the curvature of the reference curve. We derive, when the diameter of the cross sections tends to zero, effective operators by means of a kind of convergence of the resolvents. With such modeling framework, the main novelties are that the curvature contributes with a repulsive potential whereas the torsion (in the spatial case) plays no role to effective operators.