Hierarchical Wentzel-Kramers-Brillouin quantization of nonseparable variables: Magnetic-flux effects in a varying width geometry

The energy spectrum of a problem described by nonseparable variables is derived by an approximation procedure based upon the existence of small geometrical parameters. The method is exemplified for the states of an electron confined to a doubly connected, varying-width geometry in the presence of a magnetic field. It is shown that the spectrum includes extended states as well as states that are exponentially decaying over macroscopic parts of the sample, the latter yielding exponential quenching of the Aharonov-Bohm effect. The modifications introduced by a homogeneous magnetic field are analyzed; the Aharonov-Bohm periodicities of the states extending between the sample boundaries and the magnetic edge states are derived. The relevance to mesoscopic systems of varying dimensions is discussed.