On discrete ground states of rotating Bose-Einstein condensates

Abstract

The ground states of Bose-Einstein condensates in a rotating frame can be described as constrained minimizers of the Gross-Pitaevskii energy functional with an angular momentum term. In this paper we consider the corresponding discrete minimization problem in Lagrange finite element spaces of arbitrary polynomial order and we investigate the approximation properties of discrete ground states. In particular, we prove a priori error estimates of optimal order in the $L^2$- and $H^1$-norm, as well as for the ground state energy and the corresponding chemical potential. A central issue in the analysis of the problem is the missing uniqueness of ground states, which is mainly caused by the invariance of the energy functional under complex phase shifts. Our error analysis is therefore based on an Euler-Lagrange functional that we restrict to certain tangent spaces in which we have local uniqueness of ground states. This gives rise to an error decomposition that is ultimately used to derive the desired a priori error estimates. We also present numerical experiments to illustrate various aspects of the problem structure.

Figures (3)

• Figure 1: Surface plot of reference ground state density function $|u_{gs}|^{2}$ in 2D for model 1 (top-left) and model 2 (bottom-left). The table shows the first 15 eigenvalues of second Frèchet derivative $E"(u)$ for both the models.
• Figure 2: Errors $\|u-u_{h}\|_{H^1(\mathcal{D})}$, $\|u-u_{h}\|_{L^2(\mathcal{D})}$, $| \lambda -\lambda_h|$ and $|E(u)-E(u_h)|$ for model problem 1.
• Figure 3: Errors $\|u-u_{h}\|_{H^1(\mathcal{D})}$, $\|u-u_{h}\|_{L^2(\mathcal{D})}$, $| \lambda -\lambda_h|$ and $|E(u)-E(u_h)|$ for model problem 2.

Theorems & Definitions (38)

• Corollary 2.1: Existence of ground states
• Definition 2.2: Quasi-isolated ground state
• Remark 2.3: Ground state eigenvalue
• Proposition 2.4
• Lemma 2.5: $H^{2}$-regularity of ground states
• proof
• Remark 3.1: Same phase
• Theorem 3.2
• Theorem 3.3
• Remark 3.4: Validity of the regularity assumptions in Theorem \ref{['order_theorem']}