Point Interactions in a Strip

ArticleinAnnals of Physics 252(1) · July 1996with 13 Reads 
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DOI: 10.1006/aphy.1996.0127 · Source: arXiv
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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. Comment: a LaTeX file, 38 pages, to appear in Ann. Phys.; 12 figures available at request from tater@ujf.cas.cz

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    We have introduced a controllable artificial impurity or ``antidot'' into a quantum wire and report on the novel phenomena observed as this system evolves from classical behavior at low magnetic fields to the quantum Hall regime. In the transition, conductance resonances due to magnetically bound impurity states are detected. The resonant oscillations exhibit beating and sharp period changes. A theoretical model based on an interedge state coupling mechanism and a new nonlocal effect of edge state formation at local potential energy maxima account for the principal experimental features.
  • Article
    We deduce the effects of quantum interference on the conductance of chaotic cavities by using a statistical ansatz for the S matrix. Assuming that the circular ensembles describe the S matrix, we find that the conductance fluctuation and weak-localization magnitudes are universal: they are independent of the size and shape of the cavity if the number of incoming modes, N, is large. The limit of small N is more relevant experimentally; here we calculate the full distribution of the conductance and find striking differences as N changes or a magnetic field is applied.
  • Article
    We discuss periodic Schrödinger operators for a particle on a rectangular lattice of sides l1, l2. In addition to the standard ( delta-type) coupling with continuous wave functions at lattice nodes, we introduce two other boundary conditions which generalize naturally the one-dimensional delta' interaction and its symmetrized version; both of them can be used as models for geometric scatterers. We show that the band spectrum of these models depends on number-theoretic properties of the parameters. In particular, the delta lattice has no gaps above the threshold if l2/l1 is badly approximable by rationals and the coupling constant is small enough.
  • Article
    We consider curvature-induced resonances in a planar two-dimensional Dirichlet tube of a width $ d $. It is shown that the distances of the corresponding resonance poles from the real axis are exponentially small as $ d\to 0+ $, provided the curvature of the strip axis satisfies certain analyticity and decay requirements.