Local Density Approximation and Other Limit Regimes for a Homogeneous Bose Gas with Repulsive Three-Body Interactions in Low-Dimensional Space
In the whole space of low spatial dimensions, namely $d \leq 2$, we study the minimizers of an energy functional with an attractive cubic nonlinearity and a repulsive quintic nonlinearity, which describes a quantum Bose gas with a two-body attraction and a three-body repulsion. We prove the existence of minimizers at fixed effective statistics parameter. In the limit of a large effective statistics parameter of the mass-critical nonlinearity (with respect to the kinetic energy), we derive an effective Thomas-Fermi-like model for the homogeneous Bose gas. We also consider other limit regimes depending on the mass-critical nonlinearity.
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