On the speed at which solutions of the Sturm-Liouville equation tend to zero (Q2257089)

For a locally integrable, non-negative potential \(q(x)\) on \(\mathbb R\), sufficient and necessary conditions on \(q(x)\) have been found by the authors themselves [the authors, Methods Appl. Anal. 5, No. 3, 259--272 (1998; Zbl 0924.34012)] so that for each \(f\in L_p=L_p({\mathbb R})\), where \(p\in [1,\infty]\), the Sturm-Liouville equation \[ -y''(x) + q(x) y(x) =f(x), \qquad x\in \mathbb R \] admits a unique solution \(y\in L_p\). In particular, these solutions \(y(x)\) decay to zero as \(| x|\) tends to infinity. In the present paper, the authors have gone further. For any \(p\in[1,\infty]\) and \(q\) verifying the conditions, they obtain the optimal decay speed of these solutions at \(x=\infty\).