Mathematical Analysis of Dilute, Trapped Bose Gases

The phenomenon of Bose-Einstein condensation, where a macroscopic number of atomic particles coherently occupies a single quantum state, was predicted by Einstein in 1925. An experimental realization had to wait for 70 years, but was finally accomplished by groups at MIT and the University of Boulder in 1995. This feat, that earned the principal researchers the 1999 Nobel prize for physics, triggered great interest in the quantum phenomena exhibited by dilute Bose gases at low temperatures and this subject is now studied both experimentally and theoretically by many groups worldwide. Einstein`s original work dealt only with ideal particles where interactions are ignored, but most of the present research is concerned with the effects of the unavoidable atomic interactions. The present project is concerned with several fundamental aspects of this subject from the point of view of mathematical physics. The general idea is to derive basic properties of the ground state of a dilute Bose gas, confined in a magnetic or an optical trap, starting from the full quantum mechanical many-body description and reducing it by means of rigorous mathematical theorems to simpler, effective descriptions. Specific goals are: 1) To improve existing estimates on the ground state energy and its localization and use them to demonstrate Bose- Einstein condensation from first principles also in the case of realistic interactions. 2) Determine the parameter domains for rotating Bose gases where instability of vortices and rotational symmetry breaking occurs. 3) Prove that a rotaing Bose gas has an effective desrciption in terms of a nonlinar Schrödinger equation (Gross-Pitaevskii equation) also in the symmetry breaking case.

The phenomenon of Bose-Einstein condensation, where a macroscopic number of atomic particles coherently occupies a single quantum state, was predicted by Einstein in 1925. An experimental realization had to wait for 70 years, but was finally accomplished by groups at MIT and the University of Boulder in 1995. This feat, that earned the principal researchers the 1999 Nobel prize for physics, triggered great interest in the quantum phenomena exhibited by dilute Bose gases at low temperatures and this subject is now studied both experimentally and theoretically by many groups worldwide. Einstein`s original work dealt only with ideal particles where interactions are ignored, but most of the present research is concerned with the effects of the unavoidable atomic interactions. The present project is concerned with several fundamental aspects of this subject from the point of view of mathematical physics. The general idea is to derive basic properties of the ground state of a dilute Bose gas, confined in a magnetic or an optical trap, starting from the full quantum mechanical many-body description and reducing it by means of rigorous mathematical theorems to simpler, effective descriptions. Specific goals are: 1. To improve existing estimates on the ground state energy and its localization and use them to demonstrate Bose-Einstein condensation from first principles also in the case of realistic interactions. 2. Determine the parameter domains for rotating Bose gases where instability of vortices and rotational symmetry breaking occurs. 3. Prove that a rotaing Bose gas has an effective desrciption in terms of a nonlinar Schrödinger equation (Gross-Pitaevskii equation) also in the symmetry breaking case.