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The full diagonal model of a Bose gas. (English) Zbl 0799.58088
The Hamiltonian of bosons interacting in the volume V in \(\mathbb{R}^ d\) through a pair potential can be written as \[ H = T + {1\over 2V} \sum_{q,k,k'}v(q) a^*_{k + q} a^*_{k'-q} a_ ka_{k'}, \] where \(T\) is a kinetic energy. The authors investigate the full diagonal approximation of this model (by omitting the terms \(q \neq 0\) and \(q \neq k - k'\)): \[ H^{FD} = T + {1\over 2V} \sum_{k,k'} v(k - k') n_ k n_{k'} + {a\over 2V} \biggl\{N^ 2 - \sum_{k} n^ 2_ k\biggr\},\tag{*} \] where \(n_ k\) are occupation numbers, \(N = \sum_ k n_ k\) and \(a = v(0) > 0\). In this model they find in the thermodynamic limit the variational formula for the pressure \[ p^{FD}(\mu) = -\inf {\mathcal E}^ \mu[m],\quad \mu\text{ the chemical potential,} \] where \(m\) is a bounded positive measure on \(\mathbb{R}^ d\). They show that there are two regions: one in which the kinetic term \(T\) dominates and one in which the last term in \((*)\) dominates. This, so- called Huang-Yang-Luttinger term is responsible for the Bose-Einstein condensation. For a large class of potentials there is a critical temperature below which there is a condensation.

MSC:
58Z05 Applications of global analysis to the sciences
82B10 Quantum equilibrium statistical mechanics (general)
82B21 Continuum models (systems of particles, etc.) arising in equilibrium statistical mechanics
82D05 Statistical mechanical studies of gases
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