Critical Coupling Constants and Eigenvalue Asymptotics of Perturbed Periodic Sturm–Liouville Operators

Abstract:

Perturbations of asymptotic decay c/r ² arise in the partial-wave analysis of rotationally symmetric partial differential operators. We show that for each end-point λ₀ of the spectral bands of a perturbed periodic Sturm–Liouville operator, there is a critical coupling constant c _crit such that eigenvalues in the spectral gap accumulate at λ₀ if and only if c/c _crit>1. The oscillation theoretic method used in the proof also yields the asymptotic distribution of the eigenvalues near λ₀.