arXiv:2603.20776

Propagation of Condensation via Neumann Localization in the Dilute Bose Gas

Lukas Junge

Published Mar 21, 2026• 0 citations

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 $R\sim a(ρa^3)^{-\frac{3}{4}-η}$.

DOI: 10.48550/arxiv.2603.20776

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arXiv:2603.20776

Propagation of Condensation via Neumann Localization in the Dilute Bose Gas

Lukas Junge

Published Mar 21, 2026• 0 citations

arXiv HTML PDF

Abstract

R\sim a(ρa^3)^{-\frac{3}{4}-η}

DOI: 10.48550/arxiv.2603.20776

Related papers

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R\sim a(ρa^3)^{-\frac{3}{4}-η}

4 \pi a \left( 2 \varrho ^2 - \varrho _0^2 \right)

4πa2ϱ2-ϱ02 . Here

\varrho

ϱ denotes the density of the system and

\varrho _0

ϱ0 is the expected condensate density of the ideal gas. Additionally, we show that the one-particle density matrix of any approximate minimizer of the Gibbs free energy functional is to leading order given by the one of the ideal gas. This in particular proves Bose–Einstein condensation with critical temperature given by the one of the ideal gas to leading order. One key ingredient of our proof is a novel use of the Gibbs variational principle that goes hand in hand with the c-number substitution.

{\omega}

, the temperature T, and the particle number N are related by

{N \sim (T / \omega)^{3} \to\infty}

while the scattering length is so small that the interaction energy per particle around the center of the trap is of the same order of magnitude as the spectral gap in the trap. We prove that the difference between the canonical free energy of the interacting gas and the one of the noninteracting system can be obtained by minimizing the GP energy functional. We also prove Bose–Einstein condensation in the following sense: The one-particle density matrix of any approximate minimizer of the canonical free energy functional is to leading order given by that of the noninteracting gas but with the free condensate wavefunction replaced by the GP minimizer.

y

and the condensate fraction

\frac{{N}_{0}}{N}

are studied as functions of temperature and are evaluated explicity for cubes with

\frac{L}{\overline{l}}=40 \mathrm{and} 100

, where

L

is the edge length of the enclosure and

\overline{l}

the mean interparticle distance in the system. The finite-size corrections to the condensate fraction under Dirichlet or Neumann boundary conditions turn out to be qualitatively different from those under periodic or antiperiodic boundary conditions. This difference becomes manifest when one tries to express these corrections in a form consistent with the standard scaling theory for finite-size effects.