The higher-dimensional localization conjecture for disordered Bose–Einstein condensates

Let $d>1$, let $u_{\text{\tiny\rm gs}}$ be the ground state of the Gross–Pitaevskii eigenvalue problem (GPEVP), and let $V_\varepsilon$ be a disorder potential satisfying assumptions (A1)–(A4). Let $\varepsilon\ll 1$ and let $\kappa$ denote the interaction parameter. Higher-dimensional localization conjecture. The ground state of the GPEVP is localized if

$$\kappa\lesssim \varepsilon^{d-2}.$$

The conjecture is motivated by the preceding proposed $L^\infty$ bound, which would make the nonlinear contribution a perturbation of order at most $\mathcal O(\varepsilon^{-2})$ in the stated range of $\kappa$. The source gives no resolution, so the localization claim remains open.