On the singular continuous spectrum of self-adjoint extensions (Q1906885)

Let \(H\) be a closed symmetric operator in the Hilbert space \({\mathcal H}\) with gap \(J\) and infinite deficiency indices. Given any open subset \(J_0\) of \(J\) we shall give explicitly a selfadjoint extension \(\widetilde {H}\) of \(H\) such that up to some discrete spectrum the spectrum of \(\widetilde {H}\) within the gap \(J\) of \(H\) is purely singular continuous and the singular continuous spectrum of \(\widetilde {H}\) within \(J\) equals the closure of \(J_0\) in \(J\).