On the spectrum of a second-order periodic differential equation

In this paper we derive asymptotic approximations for the periodic and semi-periodic eigenvalues for a second-order periodic differential equation known as Hill's equation. Our results are sharper than the existing results in the literature in that they give sharper error bounds whilst relaxing the smoothness assumptions. For some particular potentials, including that of Mathieu equation, we provide estimates for the corresponding eigenvalues using the symbolic manipulator package, Maple.