Absolute continuity and spectral concentration for slowly decaying potentials

We consider the spectral function $\rho (\mu )$ $(\mu \ge 0)$ for the Sturm-Liouville equation $y^{{}^{\mathrm{\prime }\mathrm{\prime }}}+(\lambda -q)y=0$ on $[0,\mathrm{\infty })$ with the boundary condition $y(0)=0$ and where $q$ has slow decay $O(x^{-\alpha })$ $(a>0)$ as $x\to \mathrm{\infty }$. 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.