Bogoliubov Theory for Trapped Bosons in the Gross–Pitaevskii Regime

Abstract

We consider systems of N bosons in ${\mathbb {R}}^3$, trapped by an external potential. The interaction is repulsive and has a scattering length of the order $N^{-1}$ (Gross–Pitaevskii regime). We determine the ground state energy and the low-energy excitation spectrum up to errors that vanish in the limit $N\rightarrow \infty$.

Properties of the Gross–Pitaevskii Functional

In this appendix, we collect several well-known results about the Gross–Pitaevskii functional ${\mathcal {E}}_\mathrm{GP}$, defined in equation (1.4). Let us recall that ${\mathcal {E}}_\mathrm{GP}:{\mathcal {D}}_\mathrm{GP}\rightarrow {\mathbb {R}}$ is given by

$$\begin{aligned} {\mathcal {E}}_\mathrm{GP}(\varphi ) = \int _{{\mathbb {R}}^3} \left( \vert \nabla \varphi _0 (x) \vert ^2 + V_\text {ext}(x) \vert \varphi _0 (x)\vert ^2 + 4\pi \mathfrak {a}_0 \vert \varphi _0 (x)\vert ^4 \right) \hbox {d}x \end{aligned}$$

with domain

$$\begin{aligned} {\mathcal {D}}_\mathrm{GP} =\big \{ \varphi \in H^1({\mathbb {R}}^3)\cap L^4({\mathbb {R}}^3): \ V_\text {ext}\vert \varphi \vert ^2\in L^1({\mathbb {R}}^3)\big \}. \end{aligned}$$

Recall, moreover, assumption (2) in Eq. (1.7) on the external potential $V_\text {ext}$. The following was proved in [20, Theorems 2.1, 2.5 & Lemma A.6].

Lemma A.1

There exists a minimizer $\varphi _0 \in {\mathcal {D}}_\mathrm{GP}$ with $\Vert \varphi _0 \Vert _2=1$ such that

$$\begin{aligned} \inf _{\psi \in {\mathcal {D}}_\mathrm{GP} \ : \Vert \psi \Vert _2 =1} {\mathcal {E}}_\mathrm{GP}(\psi ) = {\mathcal {E}}_\mathrm{GP}(\varphi _0 ). \end{aligned}$$

The minimizer $\varphi _0$ is unique up to a complex phase, which can be chosen so that $\varphi _0$ is strictly positive. Furthermore, the minimizer $\varphi _0$ solves the Gross–Pitaevskii equation

$$\begin{aligned} -\Delta \varphi _0 + V_\text {ext} \varphi _0 + 8\pi \mathfrak {a}_0 \vert \varphi _0 \vert ^2 \varphi _0 = \varepsilon _\mathrm{GP}\varphi _0 , \end{aligned}$$

with $\mu$ given by

$$\begin{aligned} \varepsilon _\mathrm{GP}= {\mathcal {E}}_\mathrm{GP}(\varphi _0 ) + 4\pi \mathfrak {a}_0 \Vert \varphi _0 \Vert _4^4. \end{aligned}$$

Moreover, $\varphi _0 \in L^\infty ({\mathbb {R}}^3)\cap C^1({\mathbb {R}}^3)$ and for every $\nu >0$ there exists $C_\nu$ (which only depends on $\nu$ and ${\mathfrak {a}}_0$) such that for all $x\in {\mathbb {R}}^3$ it holds true that

$$\begin{aligned} \vert \varphi _0 (x)\vert \le C_\nu e^{-\nu \vert x \vert }. \end{aligned}$$

(A.1)

As in the main text, $\varphi _0$ denotes the unique, strictly positive minimizer of ${\mathcal {E}}_\mathrm{GP}$, subject to the constraint $\Vert \varphi _0 \Vert _2=1$. In addition to Lemma A.1, we collect some additional facts about the regularity of $\varphi _0$. The following was shown in [11, Theorem A.2].

Lemma A.2

Let $V_\text {ext}$ satisfy the assumptions in (1.7). Then $\varphi _0 \in H^2({\mathbb {R}}^3)\cap C^2({\mathbb {R}}^3)$ and for every $\nu >0$ there exists $C_\nu >0$ such that for every $x\in {\mathbb {R}}^3$ we have

$$\begin{aligned} \vert \nabla \varphi _0 (x) \vert , \ \vert \Delta \varphi _0 (x) \vert , | \nabla \Delta \varphi _0 (x)| \le C_\nu e^{-\nu \vert x \vert }. \end{aligned}$$

(A.2)

Furthermore, if ${\widehat{\varphi }}_0$ denotes the Fourier transform of $\varphi _0$, we have for all $p\in {\mathbb {R}}^3$ that

$$\begin{aligned} | {\widehat{\varphi }}_0 (p)|, | \widehat{\varphi _0 ^2} (p)| \le \frac{C}{(1+ | p|)^3}. \end{aligned}$$

(A.3)