On estimates for the discrete eigenvalues of two-dimensional quantum waveguides
Martin Karuhanga, Catherine Ashabahebwa
TL;DR
The paper addresses upper bounds for the number and sum of eigenvalues of two-dimensional quantum waveguides that lie below the bottom of the essential spectrum, addressing both straight and curved geometries. The authors combine CLR- and LT-type spectral inequalities with Orlicz-norm techniques and a transverse-longitudinal mode decomposition to reduce to effective one-dimensional problems and to handle curvature effects. Key contributions include explicit CLR-type bounds for straight waveguides with terms depending on longitudinal moments $\beta_k$ and Orlicz-norm data, and curved-waveguide bounds that incorporate curvature through explicit constants; a Lieb-Thirring bound shows the sum of negative eigenvalues is controlled by $\|V\|^2_{L^2(\Omega)}$ with curvature-modulated constants. The results provide new, sharp-type estimates for curved quantum waveguides and offer potential extensions to impurities, magnetic fields, and more general waveguide configurations, with implications for spectral stability and bound-state formation in quasi-1D quantum systems.
Abstract
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 the 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.