On the Spectrum of Schrödinger Operators Interacting at Two Distinct Scales

Schrödinger operators of the form $Δ- W$ on $L^2_{\text{rad}}(\mathbb{R}^3)$, the space of radially symmetric square integrable functions, are relevant in a variety of physical contexts. The potential $W$ is taken to be radially symmetric (i.e. $W(x) = W(|x|)$) and to decompose into two components w