Bose-Einstein condensation on a Reimannian manifolds with nonnegative Ricci curvature

Abstract

In this thesis, the properties of the Bose-Einstein condensation in at spaces are reviewed. Three dimensional weakly interacting Bose systems are examined by using the Bogoliubov approximation. The heat kernel of the Laplace operator is introduced. The Bose-Einstein condensation for an ideal gas and for a weakly interacting gas in the nonrelativistic limit on a Riemannian manifold with nonnegative Ricci curvature is studied using the heat kernel and eigenvalue estimates of the Laplace operator. The Bose gas is assumed to obey the Neumann boundary conditions. Behaviour of the chemical potential at low temperatures is described. Bounds for the depletion of the condensate and for the critical temperature of an ideal Bose system are derived in the thermodynamic limit. We observed that the condensation does not take place in two dimensions, however it is formed in three dimenions which is consistent with the at space results. In the case of dilute gases on a compact Riemannian manifold, Bogoliubov theory is applied. The ground state of a dilute Bose system is analyzed using the heat kernel methods. Specifically, the depletion of the condensate is estimated at absolute zero temperature. For finite volumes, we concluded that the condensate exists in two dimensions for weakly interacting gases. We also analyzed the depletion of the condensate at finite temperatures. Inconsistency of the Bogoliubov approximation in the thermodynamic limit is shown using heat kernel methods. Justification of the c-number substitution on a manifold is given.