On the Emergence of Discrete Spectrum for Weakly Disordered Schrödinger Operators

We investigate the spectral properties of the Anderson operator perturbed by a localized negative potential, $-V$. Specifically, we analyze the random Schrödinger operator defined by $H=-\mathrm{\Delta }+\text{ve}{\sum }_n{\omega }_n{\chi }_n-V$, where the unperturbed operator exhibits a disordered energy landscape. Our primary focus is to establish precise estimates on the number of negative eigenvalues (bound states) induced by the attractive perturbation. By analyzing the competition between Anderson localization and the binding capacity of the potential, we provide quantitative bounds on the discrete spectrum. These results offer new insights into how randomness enhances the eigenvalue bounds.