We study, using the Bogolyubov approximation, the thermodynamic behaviour of a superstable Bose system whose energy operator in the second-quantized form contains a nonlinear expression in the occupation numbers operators. We prove that for all values of the chemical potential satisfying $\mu > \lambda(0)$, where $\lambda (0)\leq 0$ is the lowest energy value, the system undergoes Bose--Einstein condensation

Abstract. In the framework of the Bogolyubov approximation and using the Bogolyubov inequalities we give a simple proof of the coexistence of two nonconventional Bose-Einstein condensates in the case of some superstable Bose system whose atoms have an internal two-level structure and their energy operators in the second quantized form depend on the number operators only.

For a non-interacting many particle Bose system whose energy operator is diagonal in the number of occupation operatorsn j upper bounds on the thermal averages n j are obtained. These bounds lead to the proof of Bose-Einstein condensation for finite values of the inverse temperature β and for chemical potential µ = 0. Finally for µ < 0, in the case of a generalization of the studied model system, the property of Local Gaussian Domination for the grand canonical partition function is proved.

A quantum system of nonlinear oscillators is considered. Within the framework of Berezin's functional integral we prove the gaussian domination at finite temperature for some values of the chemical potential. Upper and lower bounds for the average number of particles with momentum p are derived.

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