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 = λ y , x ∈ R + , where the potential Q is a decaying L 1 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. o 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.