On the oscillation of almost-periodic Sturm-Liouville operators with an arbitrary coupling constant

Abstract

In this paper we characterize those (Bohr) almost periodic functions V V on R {\mathbf {R}} for which the Sturm-Liouville equations \[ − y + λ V ( x ) y = 0 , x ∈ R , - y + \lambda V(x)y = 0,\quad x \in \mathbf {R}, \] are oscillatory at ± ∞ \pm \infty for every real λ ≠ 0 \lambda \ne 0 , or, equivalently, for which there exists a real λ ≠ 0 \lambda \ne 0 such that the equation has a positive solution on R {\mathbf {R}} .