Ground states for 3D dipolar Bose-Einstein condensate involving quantum fluctuations and three-body losses

TL;DR

This paper investigates the existence and behavior of ground states in a 3D dipolar Bose-Einstein condensate model with quantum fluctuations and three-body losses, establishing conditions for stability and describing asymptotic properties as the mass diminishes.

Contribution

It introduces a local minimization framework for unbounded energy functionals and characterizes the asymptotic behavior of ground states as the condensate mass approaches zero.

Findings

Existence of stable ground states under certain parameter conditions.

Precise asymptotic description of ground states as mass tends to zero.

Identification of conditions leading to unbounded energy on the L^2 sphere.

Abstract

We consider ground states of three-dimensional dipolar Bose-Einstein condensate involving quantum fluctuations and three-body losses, which can be described equivalently by positive $L^2$-constraint critical point of the Gross-Pitaevskii energy functional \[E(u)\!=\!\frac{1}{2}\int_{{\mathbb{R}^3}} {|\nabla u|}^2dx+\frac{\lambda_{1}}{2}\int_{{\mathbb{R}^3}} {| u|}^4dx+\frac{\lambda_{2}}{2} \int_{\mathbb{R}^{3}}\left(K \star|u|^{2}\right)|u|^{2} d x+\frac{2\lambda_{3}}{p}\int_{{\mathbb{R}^3}} {|u|}^{p}dx,\] where $2<p<\frac{10}{3}$, $\lambda_{3}<0$, $\star$ is the convolution, $ K(x) \!=\! \frac{{1-3{{\cos }^2}\theta(x) }}{{{{| x |}^3}}}$, $\theta(x)$ is the angle between the dipole axis determined by $(0,0,1)$ and the vector $x$. If ${\lambda _1} \!\!<\!\! \frac{4\pi} {3} {\lambda _2}\!\leq\! 0$ or ${\lambda _1} \!\!<\!- \frac{8\pi}{3} {\lambda _2}\!\leq\! 0$, $E(u)$ is unbounded on the $L^2$-sphere $S_{c}\!:=\!\Big\{ u \!\in\! H^1({\mathbb{R}^3}): \int_{{\mathbb{R}^3}} {{|u|}^2}dx\!=\!c^2 \Big\}$, so we turn to study a local minimization problem $$ m(c,R_0)\!:=\!\inf _{u \in V^c_{R_0}} E(u)$$ for a suitable $R_0\!>\!0$ with $V^c_{R_0} \!:=\!\left\{u \!\in\! S_c : \big(\int_{{\mathbb{R}^3}} {{|\nabla u|}^2dx}\big)^{\frac{1}{2}} \!<\!R_0\right\}$. We show that $m(c,R_0)$ is achieved by some $u_c>0$, which is a stable ground state. Furthermore, by refining the upper bound of $m(c, R_0)$, we provide a precise description of the asymptotic behavior of $u_c$ as the mass $c$ vanishes, i.e. $${[\frac{{p|{\lambda _3}|}}{{2{\gamma _c}}}]^{\frac{1}{{p - 2}}}}{u_c}(\frac{{x + {y_c}}}{{\sqrt {2{\delta _p}{\gamma _c}} }}) \to {W_p}\;\;\;\;{\rm{in}}\;\;\;\;{H^1}({\mathbb{R}^3})\;\;\;\;{\rm{for some}}\;\;\;\;{y_c} \in {\mathbb{R}^3}\;\;\;\;{\rm{as}}\;\;\;\;c \to {0^ + }.$$