Ground states of Bose–Einstein condensates with higher order interaction

Abstract

We analyze the ground state of a Bose-Einstein condensate in the presence of higher-order interaction (HOI), modeled by a modified Gross-Pitaevskii equation (MGPE). In fact, due to the appearance of HOI, the ground state structures become very rich and complicated. We establish the existence and non-existence results under different parameter regimes, and obtain their limiting behaviors and/or structures with different combinations of HOI and contact interactions. Both the whole space case and the bounded domain case are considered, where different structures of ground states are identified.

Computing Ground States of Bose--Einstein Condensates with Higher Order Interaction via a Regularized Density Function Formulation

“…Taking a minimizing sequence {ρ n } in W , then {ρ n } is uniformly bounded in X V with a weak limit ρ ∞ in H 1 -norm. Following a similar procedure as shown in [7] and applying Lemma 2.3, we can show that ρ ∞ is indeed the ground state density. The details are omitted here for brevity.…”

“…Mathematically speaking, the ground state of a BEC described by the MGPE (1.3) is defined as the minimizer of the energy functional (1.4) under the normalization constraint (1.5). Specifically, we have the ground state φ g = φ g (x) defined as [7,41] (1. 7) φ g := arg min φ∈SẼ (φ) ,…”

“…Specifically, we have the ground state φ g = φ g (x) defined as [7,41] (1. 7) φ g := arg min φ∈SẼ (φ) ,…”

“…under the normalization constraint φ 2 = 1, where µ is the corresponding chemical potential (or eigenvalue in mathematics) and can be computed as [7,41] µ(φ) =Ẽ(φ) + The existence and uniqueness of the ground state of the non-convex optimization problem (1.7) has been thoroughly studied in [7]. When δ = 0, the MGPE (1.3) degenerates to a GPE and the existence and uniqueness of the ground state is clear.…”

“…The formulation. When β ≥ 0 and δ ≥ 0, it is sufficient to consider the positive wave function for the ground state [7]. Reformulating the energy functional (1.4) with the density function ρ := |φ| 2 , we get…”