Quantum Many-Body Systems with Short-Range Interactions

In this dissertation, the central theme is evaluation of the ground energy and the first few excited energies of Bose or Fermi gas system in continuum R3 or lattice Z3 with short-range interactions. In Chapter 2 and 3, we carry out an analysis on low-dimensional behaviors of dilute Bose gas in traps. In Chapter 4 and 5, we generalize the results on the ground state energies of dilute Bose or Fermi gases in thermodynamic limit. In quantum mechanics, many-body quantum system is completely described by the Hamiltonian, which is a self-adjoint operator on a suitable Hilbert space. In proper scaling limits and parameter regimes, these three-dimensional Hamiltonians can be rigorously described by effective low-dimensional Hamiltonians or equations. In Chapter 2 we show that the Lieb-Liniger model for one-dimensional bosons with repulsive Delta function interaction can be rigorously derived from a dilute three-dimensional Bose gas with arbitrary repulsive interaction potential of finite scattering length. In Chapter 3 we prove that the two dimensional rotating Gross-Pitaevskii (GP) equation correctly describes the ground state energy and corresponding one-particle density matrix of rotating, dilute, interacting Bose gas in three dimensions in a potential that is strongly confining in one direction. Another one of the most remarkable recent developments in study of dilute Bose gases is the rigorous proofs on the leading terms of the effect of the repulsive interaction potential on the ground state energy or the free energy in the thermodynamic limit. For interacting Bose gases, in [44], Lieb and Yngvason proved the correction per volume on ground energy is 4pi aϱ2, where a is the scattering length of interaction potential and ϱ is the density. In this dissertation we generalize this result as follows, we prove that the upper bound part holds for all interaction potentials of positive scattering length, i.e., a > 0, and the lower bound part holds for some interaction potentials with shallow and/or narrow negative parts. For Fermi gases, in [32], Lieb, Seiringer and Solovej proved the correction on ground energy is 8pi aϱuϱd. Here ϱu (d) are the density of the spin up(down) particles. We extend the lower bound part of the result in [32] to the Hubbard model, i.e., the correction per volume on the ground energy of Fermi gas in Hubbard model is not less than 8pi aϱuϱd, our result completes the discussion in [15] where Giuliani proved the upper bound part. Our discussion can be easily generalized to other short-range interacting Fermi gas systems in Lattice.