Description of the scattering data for Sturm-Liouville operators on the half-line

Abstract

We describe the set of the scattering data for self-adjoint Sturm-Liouville operators on the half-line with potentials belonging to $L_1(\mathbb{R}+,\rho(x),\text{d} x)$, where $\rho:\mathbb{R}+\to\mathbb{R}_+$ is a monotonically nondecreasing function from some family $\mathscr{R}$. In particular, R includes the functions $\rho(x)=(1+x)^{\alpha}$ with $\alpha \geq 1$.