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 $V$ on ${\mathbf {R}}$ for which the Sturm-Liouville equations \[ - y'' + \lambda V(x)y = 0,\quad x \in \mathbf {R},\] are oscillatory at $\pm \infty$ for every real $\lambda \ne 0$, or, equivalently, for which there exists a real $\lambda \ne 0$ such that the equation has a positive solution on ${\mathbf {R}}$.

Locations

• Proceedings of the American Mathematical Society - View - PDF