Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Arbitrary harmonic functions as Bose--Einstein condensates

Michiel De Wilde, Robert Seiringer

TL;DR

The work addresses what condensate structures can arise in an ideal Bose gas in the thermodynamic limit when the Laplacian boundary conditions are allowed to vary. It introduces a family of Laplacians $-\Delta_{\Phi}^{\Omega}$ parametrized by a harmonic sequence $\Phi=\{\phi_k\}$ and proves strong resolvent convergence to the standard Laplacian as domains expand, enabling an explicit expression for the thermodynamic-limit two-point function. The main result shows that the limit comprises the usual Bose-term $\int_{\mathbb{R}^d} \frac{ \overline{\hat f(p)} \hat g(p)}{e^{\beta |p|^2}-1} dp$ plus a condensate contribution $\frac{1}{\beta} \sum_k (\int \overline f \phi_k)(\int \overline{\phi_k} g)$, i.e., arbitrary condensates described by harmonic functions $\phi_k$ can be engineered via boundary control. This provides a constructive link between boundary-parameterized Laplacians, harmonic-function condensates, and KMS-type states in the thermodynamic limit, with potential implications for the design of condensate patterns in bosonic systems.

Abstract

We show that a suitable choice of boundary conditions for the Laplacian allows for the appearance of an an arbitrary number of condensates, described by arbitrary harmonic functions, in the thermodynamic limit of an ideal Bose gas.

Arbitrary harmonic functions as Bose--Einstein condensates

TL;DR

The work addresses what condensate structures can arise in an ideal Bose gas in the thermodynamic limit when the Laplacian boundary conditions are allowed to vary. It introduces a family of Laplacians parametrized by a harmonic sequence and proves strong resolvent convergence to the standard Laplacian as domains expand, enabling an explicit expression for the thermodynamic-limit two-point function. The main result shows that the limit comprises the usual Bose-term plus a condensate contribution , i.e., arbitrary condensates described by harmonic functions can be engineered via boundary control. This provides a constructive link between boundary-parameterized Laplacians, harmonic-function condensates, and KMS-type states in the thermodynamic limit, with potential implications for the design of condensate patterns in bosonic systems.

Abstract

We show that a suitable choice of boundary conditions for the Laplacian allows for the appearance of an an arbitrary number of condensates, described by arbitrary harmonic functions, in the thermodynamic limit of an ideal Bose gas.
Paper Structure (4 sections, 3 theorems, 18 equations)

This paper contains 4 sections, 3 theorems, 18 equations.

Table of Contents

  1. Introduction and Main Result
  2. Construction of the Laplacians
  3. Strong resolvent convergence
  4. Proof of Theorem \ref{['thm:main']}

Key Result

Theorem 1

Consider a sequence of open, bounded domains $\Omega_j \subset \mathbb{R}^d$, with smooth boundary $\partial \Omega_j$, converging to $\mathbb{R}^d$ as $j\to \infty$, in the sense that any bounded set is contained in $\Omega_j$ for $j$ large enough. Given a (possibly infinite) sequence $\Phi = \{\ph for any $\beta>0$ and any $f,g \in L^2(\mathbb{R}^d)$ with compact support. In dimensions $d\in\{1,

Theorems & Definitions (6)

  • Theorem 1
  • Lemma 1
  • proof
  • Example 1
  • Theorem 2
  • proof