Trapped States in Bent Quantum Wires

Abstract

In this chapter the basic results on trapped states in bent quantum wires are presented. L-shaped structures have been studied in acoustics by Lippert (1954, 1955), in flat electromagnetic waveguides by Carini and coworkers (1992) following the theoretical work of Exner, Seba and Stovicek (1989), and in quantum wires by Schult, Ravenhall and Wyld (1989). Curvature induced bound states were discussed by Goldstone and Jaffe (1992), which results were later improved by Renger and Bulla (1995) and Duclos and Exner (1995). We review below also the case of quantum wires with bumps, which has been treated by a number of researchers, as well as the case of waveguides with a boundary window. The next topic of interest to the experimentalists is curvature induced resonances. The mathematical physics of resonances has been addressed by Duclos, Exner and coworkers. Quantum wire systems with discrete states embedded in the continuous spectrum are presented.