Pavel Exner and Milos Tater
Spectra of soft ring graphs
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ABSTRACT.  We discuss of a ring-shaped soft quantum wire 
modeled by $\delta$ interaction supported by the ring of a 
generally nonconstant coupling strength. We derive condition which 
determines the discrete spectrum of such systems, and analyze the 
dependence of eigenvalues and eigenfunctions on the coupling and 
ring geometry. In particular, we illustrate that a random 
component in the coupling leads to a localization. The discrete 
spectrum is investigated also in the situation when the ring is 
placed into a homogeneous magnetic field or threaded by an 
Aharonov-Bohm flux and the system exhibits persistent currents