## Dilute, trapped Bose gases and Bose-Einstein condensation.(English)Zbl 1103.82028

Dereziński, Jan (ed.) et al., Large Coulomb systems. Lecture notes on mathematical aspects of QED. Selected papers based on the presentations at the summer school on large quantum systems – QED, Nordfjordeid, Norway, August 11–18, 2003, and the summer school ”Quantum field theory – from Hamiltonian point of view”, Sandbjerg Manor, Denmark, August 2–9, 2000. Berlin: Springer (ISBN 3-540-32578-6/hbk). Lecture Notes in Physics 695, 249-274 (2006).
The author considers $$N$$-particle Hamiltonian of the form $H = \sum_{i=1}^{N} (- \Delta_i + V(x_i)) + \sum_{i < j} v(x_i - x_j)$ in the Hilbert space $$\otimes_i L^2(R^3, dx^i)$$ where the external potential $$V$$ is a locally bounded function with $$\lim _{| x| \rightarrow \infty} V(x) = \infty$$ and the interaction potential $$v$$ is positive, radial and has a compact support. He is interested in the ground state energy of $$H$$ for large number $$N$$ of particle and a small scattering length $$a = (4 \pi)^{-1} N^{-1}g$$, where $$g$$ is a positive constant. The main result is that the ground state energy and one particle density matrix have limits when the number of particles $$N$$ goes to infinity.
For the entire collection see [Zbl 1094.81005].

### MSC:

 82D05 Statistical mechanics of gases 81-06 Proceedings, conferences, collections, etc. pertaining to quantum theory 82-06 Proceedings, conferences, collections, etc. pertaining to statistical mechanics 81T13 Yang-Mills and other gauge theories in quantum field theory