96-320 P. Exner, R. Gawlista, P. \v{S}eba, M. Tater
Point interactions in a strip (116K, LaTeX) Jul 2, 96
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Abstract. 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 \$\,N\,\$ of point perturbations. Constructing the resolvent of the corresponding Hamiltonian by means of Krein'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 \$\,m\,\$ discrete eigenvalues, \$\,1\le m\le 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 infinite heigth, 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, the spectrum may exhibit any finite number of gaps. Finally, we study numerically conductance fluctuations in case of random point perturbations.

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