On the spectrum of a second-order periodic differential equation (Q1880837)

Consider the equation \[ y^{\prime\prime}(t)+(\lambda - q(t))y(t)=0,\tag{E} \] where \(\lambda\) is a real parameter and \(q\) is a real-valued \(\pi\)-periodic function. For some \(N\geq 2\) assume that \(q^{(N-1)}(t)\) exists and is integrable on \([0,\pi]\). Two types of boundary conditions are associated with (E): periodic boundary conditions (i.e., \(y(0)=y(\pi), \;y'(0)=y'(\pi)\)), and semi-periodic boundary conditions (i.e., \(y(0)=-y(\pi), \;y'(0)=-y'(\pi)\)). Asymptotic estimates on both the periodic and semi-periodic eigenvalues are derived, with an error term of order \(O(n^{-(N+1)})\). Examples, including the Mathieu equation, are carried out by MAPLE.