# Proper condensates

September 22, 2021

In this article a novel characterization of Bose-Einstein condensates is
proposed. Instead of relying on occupation numbers of a few dominant modes,
which become macroscopic in the limit of infinite particle numbers, it focuses
on the regular excitations whose numbers stay bounded in this limit. In this
manner, subspaces of global, respectively local regular wave functions are
identified. Their orthogonal complements determine the wave functions of
particles forming proper (infinite) condensates in the limit. In contrast to
the concept of macroscopic occupation numbers, which does not sharply fix the
wave functions of condensates in the limit states, the notion of proper
condensates is unambiguously defined. It is outlined, how this concept can be
used in the analysis of condensates in models. The method is illustrated by the
example of trapped non-interacting ground states and their multifarious
thermodynamic limits, differing by the structure of condensates accompanying
the Fock vacuum. The concept of proper condensates is also compared with the
Onsager-Penrose criterion, based on the analysis of eigenvalues of one-particle
density matrices. It is shown that the concept of regular wave functions is
useful there as well for the identification of wave functions forming proper
condensates.

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