Spectra of soft ring graphs

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.

LaTeX 2e, 17 pages, with 10 ps figures

References in corpus (5)

• Asymptotics of eigenvalues of the Schroedinger operator with a strong delta-interaction on a loop
• Band gap of the Schroedinger operator with a strong delta-interaction on a periodic curve
• Persistent currents for 2D Schrödinger operator with a strong $\delta$-interaction on a loop
• Magnetic transport in a straight parabolic channel
• Persistent currents due to point obstacles

Cited by in corpus (14)

• Quantum Graphs: Applications to Quantum Chaos and Universal Spectral Statistics
• Leaky quantum graphs: approximations by point interaction Hamiltonians
• Schrödinger operators with δ- and δ'-interactions on Lipschitz surfaces and chromatic numbers of associated partitions
• Leaky Quantum Graphs: A Review
• Strong coupling asymptotics for a singular Schroedinger operator with an interaction supported by an open arc
• Strong coupling asymptotics for Schrödinger operators with an interaction supported by an open arc in three dimensions
• Generalized interactions supported on hypersurfaces
• Schroedinger operators with singular interactions: a model of tunneling resonances
• On absence of bound states for weakly attractive ${\delta }^{\prime }$-interactions supported on non-closed curves in ${\mathbb{R}}^2$
• Optimization of the lowest eigenvalue of a soft quantum ring
• Approximation by point potentials in a magnetic field
• Spectral asymptotics for $\delta$-interactions on sharp cones
• On Schrödinger Operators Modified by $\delta$ Interactions
• Rank One Perturbations Supported by Hybrid Geometries and Their Deformations