Spectroscopy of annular drums and quantum rings: perturbative and nonperturbative results

Abstract

We obtain systematic approximations to the states (energies and wave functions) of quantum rings (annular drums) of arbitrary shape by conformally mapping the annular domain to a simply connected domain. Extending the general results of Ref.\cite{Amore09} we obtain an analytical formula for the spectrum of quantum ring of arbirtrary shape: for the cases of a circular annulus and of a Robnik ring considered here this formula is remarkably simple and precise. We also obtain precise variational bounds for the ground state of different quantum rings. Finally we extend the Conformal Collocation Method of \cite{Amore08,Amore09} to the class of problems considered here and calculate precise numerical solutions for a large number of states ($\approx 2000$).

12 pages, 12 figures, 2 tables

References in corpus (3)

• Solving the Helmholtz equation for membranes of arbitrary shape
• Spectroscopy of drums and quantum billiards: perturbative and non-perturbative results
• Can one hear the density of a drum? Weyl's law for inhomogeneous media

Cited by in corpus (5)

• Breaking the resilience of a two-dimensional Bose-Einstein condensate to fragmentation
• Exact sum rules for inhomogeneous drums
• A perturbative approach to the spectral zeta functions of strings, drums and quantum billiards
• Spatially partitioned many-body vortices
• Exact sum rules for quantum billiards of arbitrary shape