The Minimization Problems Of Quantum Systems With Attractive Interactions

This dissertation is devoted to two kinds of quantum systems which possess attractive interactions: one is Bose-Einstein condensation and another is Boson star. The former is a non-relativistic quantum system, which can be described by the Gross-Pitaevskii energy functional, while the latter is a pseudo-relativistic quantum system and can be described by the pseudo-relativistic Hartree energy functional. We focus on two minimization problems that correspond to a Gross-Pitaevskii energy on R2 and a pseudo-relativistic Hartree energy on R3 respectively.Firstly, we give a generalization to the results of Guo and Seiringer [28], which guarantee the existence and mass concentration for Bose-Einstein condensate with attractive interactions when the trapping potential V(x) satisfies the condition lim|x|??V(x) = +?. We extend the results to the situation that the potential is periodic or of Coulomb-type, which are important in experiments.When the potential is periodic, by virtue of concentration-compactness principle we prove that minimizers of the Gross-Pitaevskii energy exist when the interaction strength a satisfies a* < a < a*= = ?Q?22 for some constant a* ? 0, where Q(x) is the unique positive radial solution of the equation -?u + u - u3 = 0, and there is no minimizer for a > a*. Using concentration-compactness arguments again we obtain an optimal energy estimate as a approaches a*. Moreover, we analyze the mass concentration as a tends to a*, and prove that the mass concentrates at the bottom of one well of the periodic potential.When the potential is of Coulomb-type just like -1/|x|1-? for 0 ??< 1 and x ? R2, we study a two dimensional Bose-Einstein Condensate with inhomoge-neous attractive interaction. After obtaining some delicate estimates on the Gross-Pitaevskii energy, we present the existence theorem for the minimizers of the Gross-Pitaevskii energy. In particular, for 0 < ? < 1, we find that the critical value a* is no longer a jumping point of the Gross-Pitaevskii energy (denoted by e(a)), it behaves as: e(a) is continuous and decreasing with respect to a, such that e(a) = -? for all a > a*. Moreover, when 0 < ? < 1, we analyze the asymptotic property of the minimizers of Gross-Pitaevskii energy as a tends to the critical value a*, it is proved that all the mass concentrates at a singular point of negative singular potentials.Secondly, we investigate a minimization problem on the pseudo-relativistic Hartree equation for Boson Stars. We fill a gap for the results in [66]. In [66], the authors establish the existence of boosted ground states for the pseudo-relativistic Hartree equation on R3 when the number of particles N is less then the thresh-old Nc(v). We obtain an optimal estimate for the boosted ground state energy as the number of particles N tends to the threshold Nc(v), and then we present the asymptotic property of the boosted ground states as N approaches Nc(v)...