The equiconvergence problem for a one-dimensional Schrödinger operator with a uniformly locally integrable potential (Q5930962)

The Schrödinger operator \(l(u)=-u''+q(x)u\) on \(\mathbb R\) is considered, where \(q\) is uniformly locally integrable, i.e., \(\sup_{y\in \mathbb R} \int_y^{y+1}|q(x)|dx<\infty \). The estimate \(\|P_{[\lambda _0,\lambda ]}-P^0_{[0,\lambda ]}\|_{2,\infty } =O(\lambda^{-1/4})\) is stated, where \(P_{[\lambda _0,\lambda ]}\) and \(P^0_{[0,\lambda ]}\) are the spectral projections associated corresponding to \(l\) and \(l\) with \(q=0\), respectively.