On the spectrum of the Laplacian on a strip with various boundary conditions

Abstract

In this paper, the spectrum of the Laplace operator on a strip with constant width subject to four different boundary conditions is investigated. In all the four situations, we prove that its spectrum starts from the first eigenvalue of the one-dimensional Laplacian considered along the width of the strip. Unlike the other cases, we demonstrate that in the case of Robin boundary conditions, the negative part of the spectrum is not necessarily empty and establish sufficient conditions for this to happen.