On the continuous analog of Rakhmanov's theorem for orthogonal polynomials. (Q1868689)

Consider the Sturm-Liouville operator in \(L^2(0, \infty)\) defined by \[ H_\alpha f=-f'' +q f, \quad f(0)=\alpha f'(0),\; \alpha\in {\mathbb R}\cup \{\infty\}, \] where \(q\) is a uniformly locally bounded real-valued function. The main result of the paper says that, if the essential spectrum and the absolutely continuous component of the spectral measure fill \((0, \infty)\), then it holds \[ \lim_{x\to \infty}\int_{x-\delta}^{x+\delta}q(t)\,dt\to 0 \] for all \(\delta>0\). To prove this result, the author defines a continuous analog of orthogonal polynomials on the unit circle (associated with a measure), and, for this new polynomials, he establishes a variant of Rakhmanov' s theorem.