Note on the absolutely continuous spectrum of Sturm-Liouville operators (Q749730)

Consider the differential expression \(\ell (y)=-y''+q(x)y\) on an interval (a,b) where \(\ell\) is in the limit-point case at b. For some \(c\in (a,b)\) let \(L_ b\) be a selfadjoint realization of \(\ell\) in \({\mathcal L}^ 2[c,b)\). Denoting by \(m_ b(\lambda)\) the Titchmarsh-Weyl coefficient of \(L_ b\) and by \(J\subseteq {\mathbb{R}}\) any open interval the principal result of this note states: If for some constants \(\alpha,\beta,\epsilon >0\) we have \(\alpha\leq Im m_ b(\mu +i\nu)\leq \beta\) for \(\mu\in J\), \(0<\nu <\epsilon\) then each selfadjoint realization L of \(\ell\) in \({\mathcal L}^ 2(a,b)\) has purely absolutely continuous spectrum in J. The condition on \(m_ b\) can be replaced by a similar condition on the derivative of the spectral function of \(L_ b\) in J. This theorem implies that essentially the absolutely continuous spectrum of L is only affected by the behaviour of the potential q at one singular endpoint.