 96320 P. Exner, R. Gawlista, P. \v{S}eba, M. Tater
 Point interactions in a strip
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Jul 2, 96

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Abstract. We study the behavior of a quantum particle confined to
a hardwall 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 stateproblem is
analogous to that of point interactions in the plane: since a
twodimensional 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 pointperturbation 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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