Unbounded Perturbations of the Bose-einstein Condensate

New, commutative probability tools for studying the problem of the Bose-Einstein condensate preservation under thermooeld and standard gauge-invariant perturbations are presented. In particular, a new result on the stability of a Bose-Einstein condensate under the thermooeld perturbations of a polynomial type is presented. The problem of the Bose-Einstein condensate preservation that do arises in the free Bose Matter under switching on any realistic two-body interactions between particles has still remained open despite of its long history and eeorts of many the-oretists. Even in the case of noninteracting Bose particles, the process of providing a complete and rigorous (from the mathematical physics point of view) proof of Einstein's heuristic indications 1] took a very long time period during which Mark Kac's fundamental contribution 2] played a crucial role. It is due to the long-time activity of Dublin's group that Mark Kac's ideas have been cleaned up and, at present, a very clear, general understanding of the condensation phenomenon in noninteracting Bose systems is available 3]. Let ((n) be a monotonous family of bounded regions in the Euclidean space R d ; d 3, such that S n n = R d and let (h n) be a family of one particle kinetic energy operators that possess purely discrete spectra (h n) = f n (k)g k with the lowest egenvalues n (1) 0 for all n. We deene also n (k) = n (k) ? n (1); for k = 2; 3; : : : Then we will say that the family ((n ; h n) is admissible ii: