Point interactions in a strip

We study the behavior of a quantum particle confined to a hard-wall strip of a constant width in which there is a finite number {ital N} of point perturbations. Constructing the resolvent of the corresponding Hamiltonian by means of Krein{close_quote}s formula, we analyze its spectral and scattering properties. The bound state problem is analogous to that of point interactions in the plane: since a two-dimensional point interaction is never repulsive, there are {ital m} discrete eigenvalues, 1{le}{ital m}{le}{ital N}, the lowest of which is nondegenerate. On the other hand, due to the presence of the boundary the point interactions give rise to infinite series of resonances; if the coupling is weak they approach the thresholds of higher transverse modes. We derive also spectral and scattering properties for point perturbations in several related models: a cylindrical surface, both of a finite and of an infinite height, threaded by a magnetic flux, and a straight strip which supports a potential independent of the transverse coordinate. As for strips with an infinite number of point perturbations, we restrict ourselves to the situation when the latter are arranged periodically; we show that in distinction to the case of a point-perturbation array in the plane,more » the spectrum may exhibit any finite number of gaps. Finally, we study numerically conductance fluctuations in the case of random point perturbations. Copyright {copyright} 1996 Academic Press, Inc.« less