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 $R\sim a(ρa^3)^{-\frac{3}{4}-η}$ .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 $R\sim a(ρa^3)^{-\frac{3}{4}-η}$.

arXiv:2603.20776Mar 21, 20260 citations

Boundary problems for quantum Bose-gas

А. В. Латышев, Aleksandr Alekseevich Yushkanov

Russian Physics JournalJan 1, 20020 citations

R\sim a(ρa^3)^{-\frac{3}{4}-η}