On estimates for the discrete eigenvalues of two-dimensional quantum waveguides

In this paper, we give upper estimates for the number and sum of eigenvalues below the bottom of the essential spectrum counting multiplicities of quantum waveguides in two dimensions. We consider both straight and curved waveguides of constant width, and the estimates are presented in terms of norms of the potential. For curved quantum waveguide, we assume that the waveguide is not self-intersecting and its curvature is a continuous and bounded function on R. The estimates are new, particularly for the case of curved quantum waveguides and this opens a window for their extension to different configurations such as waveguides with local defamations.