论文标题
涉及量子波动和三体损失的3D偶极玻色菌冷凝物的基态
Ground states for 3D dipolar Bose-Einstein condensate involving quantum fluctuations and three-body losses
论文作者
论文摘要
我们考虑三维偶极玻色 - 因斯坦冷凝水的基态，涉及量子波动和三体损失，可以用正$ l^2 $ -Sontaint-rossaint-Pittaint临界点等效地描述。 \ [e（u）\！= \！\ frac {1} {2} {2} \ int _ {{\ MathBb {r}^3}}} {| \ nabla u |}^2dx^2dx^2dx+\ \\\\\\\\\\\ frac {λ_{λ_{1}}}}} {2} {2} {2} {2} {2} {2 {| u |}^4dx+\ frac {λ_{2}}} {2} {2} \ int _ {\ mathbb {\ mathbb {r}^{3}}} \ left（k \ star | U |^{2} {2} {2} \ right） x+\ frac {2λ_{3}} {p} \ int _ {{\ mathbb {r}^3}}}} {| u |}^{p}^{p} dx，\ \ \]，其中$ 2 <p <p <p <\ frac {10}}} {3} {3} {3} {3} {3} $，$ c $，$λ_= 3} $ k. \！= \！ \ frac {{1-3 {{\ cos}^2}θ（x）}}} {{{{{| x |}^3}}}} $，$θ（x）$是由$（0,0,1）$和向量$ x $确定的偶极轴之间的角度。如果$ {λ_1} \！\！<\！\！ \ frac {4π} {3} {λ_2} \！\ leq \！ 0 $或$ {λ_1} \！\！<\！ - \ frac {8π} {3} {λ_2} \！\ leq \！ 0 $，$ e（u）$在$ l^2 $ -sphere $ s_ {c} \！：= \！\ big \ {u \！\ in \！ h^1（{\ Mathbb {r}^3}）：\ int _ {{\ Mathbb {r}^3}}} {{| U |}^2} dx^2} dX \！= \！ \ in V^c_ {r_0}} e（u）$$对于合适的$ r_0 \！> \！0 $ with $ v^c_ {r_0} \！：= \！\！\！\！\！ s_c：\ big（\ int _ {{\ mathbb {r}^3}}} {{| \ nabla u |}^2dx}^2dx} \ big）^{\ frac {1} {1} {2}}}}}}}}}}} \ \！ 我们证明$ m（c，r_0）$是通过某些$ u_c> 0 $实现的，这是稳定的基态状态。此外，通过完善$ m（c，r_0）$的上限，我们提供了$ u_c $的渐近行为的精确描述，即质量$ c $ c $ nishes，即。 $$ {[\ frac {{{p | {λ_3} |}}} {2 {γ_c}}] {2 {Δ_P} {γ_C}}}}}} \ to {w_p} \; \; \; \ {\ rm {in}}} \; \; \; \; \; {h^1}（{\ Mathbb {r}^3}^3}） {\ mathbb {r}^ 3} \; \; \; \; {\ rm {as}}}} \; \; \; \; \; c \ to {0^ +}。
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 \[E(u)\!=\!\frac{1}{2}\int_{{\mathbb{R}^3}} {|\nabla u|}^2dx+\frac{λ_{1}}{2}\int_{{\mathbb{R}^3}} {| u|}^4dx+\frac{λ_{2}}{2} \int_{\mathbb{R}^{3}}\left(K \star|u|^{2}\right)|u|^{2} d x+\frac{2λ_{3}}{p}\int_{{\mathbb{R}^3}} {|u|}^{p}dx,\] where $2<p<\frac{10}{3}$, $λ_{3}<0$, $\star$ is the convolution, $ K(x) \!=\! \frac{{1-3{{\cos }^2}θ(x) }}{{{{| x |}^3}}}$, $θ(x)$ is the angle between the dipole axis determined by $(0,0,1)$ and the vector $x$. If ${λ_1} \!\!<\!\! \frac{4π} {3} {λ_2}\!\leq\! 0$ or ${λ_1} \!\!<\!- \frac{8π}{3} {λ_2}\!\leq\! 0$, $E(u)$ 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. $${[\frac{{p|{λ_3}|}}{2{γ_c}}]^{\frac{1}{p - 2}}}{u_c}(\frac{x + {y_c}}{{\sqrt {2{δ_p}{γ_c}} }}) \to {W_p}\;\;\;\;{\rm{in}}\;\;\;\;{H^1}({\mathbb{R}^3})\;\;\;\;{\rm{for some}}\;\;\;\;{y_c} \in {\mathbb{R}^3}\;\;\;\;{\rm{as}}\;\;\;\;c \to {0^ + }.$$
