Asymptotic Methods in the Spectral Analysis of Sturm-Liouville Operators

We consider the relationship between the asymptotic behavior of solutions of the singular Sturm-Liouville equation and spectral properties of the corresponding self-adjoint operators. In particular, we review the main features of the theory of subordinacy by considering two standard cases, the half-line operator on L2([0, ∞)) and the full-line operator on L2(ℝ). It is assumed that the coefficient function q is locally integrable, that 0 is a regular endpoint in the half-line case, and that Weyl’s limit point case holds at the infinite endpoints. We note some consequences of the theory for the well-known informal characterization of the spectrum in terms of bounded solutions. We also consider extensions of the theory to related differential and difference operators, and discuss its application, in conjunction with other asymptotic methods, to some typical problems in spectral analysis.