Remarks on a paper of Kotani concerning generalized reflectionless Schrödinger potentials (Q1956546)

Within the class \({\mathcal R}\) of generalized reflectionless Schrödinger potentials various results are proved. Given a Schrödinger operator \[ L=-\frac{d^2}{dx^2}+q(x) \] with \(q\in{\mathcal R}\), there corresponds a regular Borel measure \(\sigma\) which is supported on a finite symmetric subinterval of \(\mathbb R\). The first goal of the paper is to study the correspondence \(q\to\sigma\) in the case when \(q\) is a generic element of a ``stationary ergodic process'' \(Q\subset{\mathcal R}\). The second goal relates the so-called Marchenko-Lundina class \({\mathcal R}\) to another class which consists, roughly speaking, of real-valued Schrödinger potentials \(q\) which are of limit point type \(\pm\infty\) in the classical Weyl terminology, and which are also of Sato-Segal-Wilson type. Finally, some facts regarding the Floquet exponent of a stationary ergodic process \(Q\subset{\mathcal R}\) are discussed.