On the Spectrum of Schr\"{o}dinger Operators Interacting at Two Distinct Scales

Schr\"{o}dinger operators of the form $\Delta - W$ 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 with distinct spatial scales: $W=W_\varepsilon= V_0+V_{1,\varepsilon}$. The second component $V_{1,\varepsilon}(|x|) = \varepsilon^2V_1(\varepsilon |x|)$ represents a scaled potential that becomes increasingly delocalized as $\varepsilon \to 0$. We will assume that both potentials $V_0(r), V_1(r)$ exhibit certain decay properties as $r \to \infty$. We show how the eigenvalue count on the positive real axis is built out of the spectra associated with the two reduced eigenvalue problems on their separate scales. The result is that the total number of eigenvalues of $\Delta - W$ is the sum of the number of positive eigenvalues of $\Delta - V_0$ and $\Delta - V_1$. Our analysis combines dynamical systems techniques with a separation of scales argument, providing a novel framework for studying spectral properties of differential operators where multiple spatial scales interact.