An asymptotic formula for energy eigenvalues in the bound-state Aharonov-Bohm effect

An asymptotic formula for the energy eigenvalues of a charged particle contained in a cylindrical shell is derived. The formula is used to estimate the eigenvalues when there is a cylindrically symmetric static magnetic-induction field present and the system exhibits the bound-state Aharonov-Bohm effect. It is shown that the errors involved form an l2 sequence with terms labelled by the radial quantum number. Numerical approximations obtained from the asymptotic formula are compared to those obtained from analytical solutions of Schroedinger's equation in the following two cases. (i) There is no magnetic-induction field in the shell itself but there is a field inside the inner cylinder. (ii) There is a constant magnetic-induction field in the shell as well as a (possibly different) field inside the inner cylinder. In these cases, the radial eigenfunctions are expressible in terms of Bessel functions and confluent hypergeometric functions, respectively. The numerical results are consistent with the asymptotic character of the formula for the energy eigenvalues. (author)