Absolute continuity and spectral concentration for slowly decaying potentials

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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 (JCAM 81 (1997) 333-348) to deal with the new theoretical and computational complexities which arise for slow decay.