On the convergence of Hill’s method

Abstract

Hill's method is a means to numerically approximate spectra of linear differential operators with periodic coefficients. In this paper, we address different issues related to the convergence of Hill's method. We show the method does not produce any spurious approximations, and that for self-adjoint operators, the method converges in a restricted sense. Furthermore, assuming convergence of an eigenvalue, we prove convergence of the associated eigenfunction approximation in the L 2-norm. These results are not restricted to selfadjoint operators. Finally, for certain selfadjoint operators, we prove that the rate of convergence of Hill's method to the least eigenvalue is faster than any polynomial power.