Spectral Sum Rules for the Circular Aharonov-Bohm Quantum Billiard

This work presents the results of a quantitative investigation of the “sum rule method” recently proposed by the author for calculating low-lying energy levels. The system considered in detail is the circular Aharonov-Bohm quantum billiard recently introduced by Berry and Robnik. Exact, expressions are derived for the spectral zeta function at positiveinteger values as a function of the magnetic flux. Using the zeta function for fixed angular momentum, we observe a very fast convergence to the exact ground state energy (“precocious convergence”).