Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: HARDY INEQUALITIES IN STRIPS ON RULED SURFACES | HARDY INEQUALITIES IN STRIPS ON RULED SURFACES DAVID KREJCIRIK Received 17 August 2005 Accepted 8 November 2005 We consider the Dirichlet Laplacian in infinite two-dimensional strips defined as uniform tubular neighbourhoods of curves on ruled surfaces. We show that the negative Gauss curvature of the ambient surface gives rise to a Hardy inequality and we use this to prove certain stability of spectrum in the case of asymptotically straight strips about mildly perturbed geodesics. Copyright 2006 David KrejciHk. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 1. Introduction Problems linking the geometry of two-dimensional manifolds and the spectrum of associated Laplacians have been considered for more than a century. While classical motivations come from theories of elasticity and electromagnetism the same rather simple models can be also remarkably successful in describing even rather complicated phenomena in quantum heterostructures. Here an enormous amount of recent research has been undertaken on both the theoretical and experimental aspects of binding in curved striplike waveguide systems. More specifically as a result of theoretical studies it is well known now that the Dirichlet Laplacian in an infinite planar strip of uniform width always possesses eigenvalues below its essential spectrum whenever the strip is curved and asymptotically straight. We refer to 13 15 for initial proofs and to 8 19 21 for reviews with many references on the topic. The existence of the curvature-induced bound states is interesting from several respects. First of all one deals with a purely quantum effect of geometrical origin with negative consequences for the electronic transport in nanostructures. From the mathematical point of view the strips represent a class of noncompact noncomplete manifolds for which the .
