ar X iv : m at h / 98 05 02 5 v 1 [ m at h . SP ] 6 M ay 1 99 8 Absolute continuity and spectral concentration for slowly decaying potentials

We consider the spectral function ρ(µ) (µ ≥ 0) for the Sturm-Liouville equation y ′′ + (λ − q)y = 0 on [0, ∞) with the boundary condition y(0) = 0 and where q has slow decay O(x −α) (a > 0) as x → ∞. We develop our previous methods of locating spectral concentration for q with rapid exponential decay (this Journal 81 (1997) 333-348) to deal with the new theoretical and computational complexities which arise for slow decay.