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Propagation of Condensation via Neumann Localization in the Dilute Bose Gas

Lukas Junge

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 $R\sim a(ρa^3)^{-\frac{3}{4}-η}$.

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 .
Paper Structure (3 sections, 4 theorems, 53 equations)

This paper contains 3 sections, 4 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. The Neumann localisation
  3. Application to the dilute Bose gas

Key Result

Theorem 1

For $\ell^{-1}\in \mathbb{N}$, we consider the two partitions where Then for all $\ell^{-1}\in \mathbb{N}$, the following operator inequality holds on $L^2([0,1]^3)$: Note that the sets $B_{\ell,k}^{shift}$ are not all cubes: intervals adjacent to the boundary at $1$ have length $\frac{\ell}{2}$, while those adjacent to the boundary at $0$ have length $\frac{3\ell}{2}$.

Theorems & Definitions (10)

  • Theorem 1
  • Remark 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • Remark 5
  • Corollary 6
  • Remark 7
  • proof