Propagation of Condensation via Neumann Localization in the Dilute Bose Gas

Abstract

We prove a Neumann localization inequality for the Laplacian that includes a spectral gap. This result is obtained by partitioning a cube into overlapping families of subcubes and analysing the associated projection operators. The resulting operator inequality goes through a discrete Neumann Laplacian on the lattice of boxes and yields a quantitative spectral gap estimate. As an application, we consider the dilute Bose gas with Neumann boundary conditions. Combining the localization method with recently established free-energy lower bounds, we propagate strong condensation estimates from the Gross Pitaevskii scale to larger boxes of side length $R\sim a(\rho a^3{)}^{-\frac{3}{4}-\eta }$.