David Krejcirik
Quantum strips on surfaces
(54K, LaTeX)

ABSTRACT.  Motivated by the theory of quantum waveguides, 
we investigate the spectrum of the Laplacian, 
subject to Dirichlet boundary conditions, 
in a curved strip of constant width 
that is defined as a tubular neighbourhood 
of an infinite curve in a two-dimensional Riemannian manifold. 
Under the assumption that the strip is asymptotically 
straight in a suitable sense, we localise the essential 
spectrum and find sufficient conditions which 
guarantee the existence of geometrically induced 
bound states. In particular, the discrete spectrum exists 
for non-negatively curved strips which are studied 
in detail. The general results are used to recover 
and revisit the known facts about quantum strips in the plane. 
As an example of non-positively curved 
quantum strips, we consider strips on ruled surfaces.