Numerical computation of eigenvalues in spectral gaps of Sturm–Liouville operators

We consider two different approaches for the numerical calculation of eigenvalues of a singular Sturm–Liouville problem -y″+Q(x)y=λy-y″+Q(x)y=λy, x∈R+x∈R+, where the potential Q   is a decaying L1L1 perturbation of a periodic function and the essential spectrum consequently has a band-gap structure. Both the approaches which we propose are spectrally exact: they are capable of generating approximations to eigenvalues in any gap of the essential spectrum, and do not generate any spurious eigenvalues.We also prove (Theorem 2.4) that even the most careless of regularizations of the problem can generate at most one spurious eigenvalue in each spectral gap, a result which does not seem to have been known hitherto.