The Second-Order Correction to the Ground State Energy of the Dilute Bose Gas

Abstract

We establish the Lee–Huang–Yang formula for the ground state energy of a dilute Bose gas for a broad class of repulsive pair-interactions in 3D as a lower bound. Our result is valid in an appropriate parameter regime of soft potentials and confirms that the Bogoliubov approximation captures the right second-order correction to the ground state energy.

Appendix A: Taylor Expansions for the Bogoliubov Integral

The quadratic Hamiltonian and thus the integral $\int h_0(k) \, {\mathrm {d}}k$ (see Lemma 4.4) plays an important role in this article. In this appendix, we will estimate this integral using Taylor approximations to various order on different regions. The resulting estimates are given in a general form as powers in n. This allows us in several lemmas to obtain bounds on the energy from bounds on n, respectively, to obtain bounds on n for states which have sufficiently low energy.

Throughout this appendix we will assume that $n\ge 1$ and that the condition on $n|{B} |^{-1}$ in Lemma 4.6 is satisfied, so that by Lemma 4.4 we have the lower bounds

$$\begin{aligned} H_{\mathrm{Quad}}\ge \frac{1}{2}(2\pi )^{-3}\int _{{\mathbb {R}}^3} h_{0}(k) \, {\mathrm {d}}k -Cn_+a\min \{R^{-3},|B|^{-1}\}\max \chi _B^2 \end{aligned}$$

and

$$\begin{aligned} h_0(k)\ge&\!-\!\left( \!n^{-1}\!\tau _B(k^2)\!+\!|B|^{-1}\!\widehat{W} (k) \!-\!\sqrt{\!n^{-2}\tau _B(k^2)^2\! +\!2n^{-1}\!|B|^{-1}\tau _B(k^2)\widehat{W}(k)\!}\right) \!n_0 \int \chi _B^2. \end{aligned}$$

(A.1)

1.1 A.1. Bounds for the Quadratic Part of $H_B$

We estimate the operator-valued integral $\frac{1}{2}(2\pi )^{-3}\int h_0(k)\, {\mathrm {d}}k$ using the following facts.

Around $x=0$ we can write $\sqrt{1+x}=1+\frac{1}{2}x-\frac{1}{8}x^2+\frac{1}{16}x^3+O(x^4)$ yielding the bounds

$$\begin{aligned}&1+\frac{1}{2}x-Cx^2\le \sqrt{1+x}\le 1+\frac{1}{2}x-\frac{1}{8}x^2+C|{x} |^3 \end{aligned}$$

(A.2)

$$\begin{aligned}&\quad \quad \quad \quad \quad \,\, \quad \quad \sqrt{1+x}\ge 1+\frac{1}{2}x-\frac{1}{8}x^2\qquad (\text {if and only if }x\ge 0)\end{aligned}$$

(A.3)

$$\begin{aligned}&1+\frac{1}{2}x-\frac{1}{8}x^2-C|{x} |^3\le \sqrt{1+x}\le 1+\frac{1}{2}x. \end{aligned}$$

(A.4)

If B is either a small or a large box, then, for all $k\in {\mathbb {R}}^3$, the parentheses in (A.1) is positive.

This is easy to see. If B is a small box and $|{k} |\le (ds\ell )^{-1}$, then $\tau _B(k^2)=0$ and $\widehat{W}(k)>0$ by (104) since $(ds\ell )^{-1}<R^{-1}$. For $|{k} |>(ds\ell )^{-1}$ we have $\tau _B(k^2)>0$ and apply (A.4). For the large box the claim is proven analogously. We may therefore replace $n_0$ by n in (A.1) when bounding $h_0$ from below such that

$$\begin{aligned} h_0(k)\ge&\!-\!\left( \!n^{-1}\tau _B(k^2)\!+\!|B|^{-1}\widehat{W}(k)\!-\!\sqrt{\!n^{-2}\tau _B(k^2)^2\! +\!2n^{-1}|B|^{-1}\tau _B(k^2)\widehat{W}(k)\!}\right) \!n \int \chi _B^2. \end{aligned}$$

(A.5)

Provided $\tau _B(k^2)>0$, we write

$$\begin{aligned} h_0(k)\ge&-\!\left( \tau _B(k^2)+n|B|^{-1}\widehat{W} (k) -\tau _B(k^2)\sqrt{1 +\frac{2n\widehat{W}(k)}{|B|\tau _B(k^2)}}\right) \int \chi _B^2. \end{aligned}$$

(A.6)

1.1.1 A.1.1. Estimates on the Small Box

$\tau _B(k^2)=0$ if $|{k} |<(ds\ell )^{-1}$ while $\widehat{W}(k)>0$ if $|{k} |<R^{-1}$. Since $\sqrt{1+x}\ge 1$ if $x\ge 0$, we have

$$\begin{aligned} \overset{{}}{\underset{{|{k} |<2(ds\ell )^{-1}}}{\int }} h_0(k)\, {\mathrm {d}}k \ge&-\overset{{}}{\underset{{|{k} |<2(ds\ell )^{-1}}}{\int }}\frac{n_0}{|{B} |}\widehat{W}(k)\, {\mathrm {d}}k\int \chi _B^2 \ge -C\frac{n}{|{B} |}a(ds\ell )^{-3}\int \chi _B^2. \end{aligned}$$

(A.7)

Using (A.3) for $2(ds\ell )^{-1}<|{k} |<R^{-1}$ and (A.4) for $|{k} |>R^{-1}$ gives

$$\begin{aligned} \frac{1}{2}(2\pi )^{-3}\overset{{}}{\underset{{|{k} |>2(ds\ell )^{-1}}}{\int }} h_0(k)\, {\mathrm {d}}k \ge&\frac{1}{2}(2\pi )^{-3}\overset{{}}{\underset{{|{k} |>2(ds\ell )^{-1}}}{\int }}-\frac{1}{2}\frac{n^2}{|{B} |^2}\frac{\widehat{W}(k)^2}{\tau _B(k^2)}\, {\mathrm {d}}k \int \chi _B^2\nonumber \\&-C\overset{{}}{\underset{{|{k} |>R^{-1}}}{\int }}\,\frac{n^3}{|{B} |^3}\frac{|{\widehat{W}(k)} |^3}{\tau _B(k^2)^2}\, {\mathrm {d}}k \int \chi _B^2. \end{aligned}$$

(A.8)

Since $\tau _B=(1-\varepsilon _0)[|{k} |-(ds\ell )^{-1}]_+^2\ge C|{k} |^2$ if $|{k} |\ge 2(ds\ell )^{-1}$ and we have $|{\widehat{W}(k)} |\le \widehat{W}(0)\le Ca$, it is easy to estimate the last term in (A.8)

$$\begin{aligned} \,\overset{{}}{\underset{{|{k} |>R^{-1}}}{\int }}\frac{n^3}{|{B} |^3}\frac{|{\widehat{W}(k)} |^3}{\tau _B(k^2)^2}\, {\mathrm {d}}k\int \chi _B^2\le C\overset{{}}{\underset{{|{k} |>R^{-1}}}{\int }}\frac{n^3}{|{B} |^3}\frac{a^3}{|{k} |^4}\, {\mathrm {d}}k \int \chi _B^2 \le C\frac{n^3}{|{B} |^3}a^3\!R \int \chi _B^2. \end{aligned}$$

(A.9)

The integral $\int _{{|{k} |>2(ds\ell )^{-1}}}^{{}}\frac{\widehat{W}(k)^2}{\tau _B(k^2)}\, {\mathrm {d}}k$ is related to the second Born term, and we have to estimate it carefully. Recall that on the small box we have by (31) that $v_R(x)\le W(x)\le v_R(x)(1+C(\frac{R}{d\ell })^2)$. Thus we have

$$\begin{aligned} \vert \vert {v_R-W} \vert \vert _{\frac{6}{5}}\le C({\textstyle \frac{R}{d\ell }})^2\vert \vert {v_R} \vert \vert _{\frac{6}{5}}\qquad \text {and}\qquad \vert \vert {v_R} \vert \vert _{\frac{6}{5}}= R^{-\frac{1}{2}}\vert \vert {v_{1}} \vert \vert _{\frac{6}{5}}. \end{aligned}$$

(A.10)

Since $v_R(x)=\frac{1}{R^3} v_{1}(\frac{x}{R})$, we have $\widehat{v_R}(k)=\widehat{v_{1}}(Rk)$ and that

$$\begin{aligned} \int _{{|{k} |\ge 2(ds\ell )^{-1}}}^{{}}\frac{\widehat{v_R}(k)^2}{|{k} |^2}\, {\mathrm {d}}k =\frac{1}{R}\int _{{|{k} |\ge 2(ds\ell )^{-1}R}}^{{}}\frac{\widehat{v}_{1}(k)^2}{|{k} |^2}\, {\mathrm {d}}k. \end{aligned}$$

(A.11)

For $f\in L^{\frac{6}{5}}({\mathbb {R}}^{3})$ a real space representation (see [21] Cor 5.10) followed by an application of the Hardy–Littlewood–Sobolev inequality gives

$$\begin{aligned} \int |{\widehat{f}(k)} |^2|{k} |^{-2}\, {\mathrm {d}}k\le C \vert \vert {f} \vert \vert _{\frac{6}{5}}^2. \end{aligned}$$

(A.12)

On the small box, we have for $|{k} |>2(ds\ell )^{-1}$

$$\begin{aligned} \tau _B(k^2)^{-1}\le (1+C\varepsilon _0)|{k} |^{-2}+C(ds\ell )^{-1}|{k} |^{-3}. \end{aligned}$$

(A.13)

We use a Cauchy–Schwarz inequality, (A.10) and (A.12) to obtain the estimate

$$\begin{aligned} \overset{{}}{\underset{{|{k} |>2(ds\ell )^{-1}}}{\int }}\frac{\widehat{W}(k)^2}{|{k} |^2}\, {\mathrm {d}}k \le {}&\overset{{}}{\underset{{|{k} |>2(ds\ell )^{-1}}}{\int }}\frac{\widehat{v}_R(k)^2}{|{k} |^2}\, {\mathrm {d}}k\nonumber \\&+2 \bigg (\overset{{}}{\underset{{|{k} |>2(ds\ell )^{-1}}}{\int }}\frac{|{\widehat{W}(k)-\widehat{v}_R(k)} |^2}{|{k} |^2}\, {\mathrm {d}}k\bigg )^{\frac{1}{2}}\bigg (\overset{{}}{\underset{{|{k} |>2(ds\ell )^{-1}}}{\int }}\frac{\widehat{v}_R(k)^2}{|{k} |^2}\, {\mathrm {d}}k\bigg )^{\frac{1}{2}}\nonumber \\&+ \overset{{}}{\underset{{|{k} |>2(ds\ell )^{-1}}}{\int }}\frac{|{\widehat{W}(k)-\widehat{v}_R(k)} |^2}{|{k} |^2}\, {\mathrm {d}}k\nonumber \\ \le {}&\frac{1}{R}\overset{{}}{\underset{{|{k} |>2(ds\ell )^{-1}R}}{\int }}\frac{\widehat{v}_{1}(k)^2}{|{k} |^2}\, {\mathrm {d}}k+C\vert \vert {W-v_R} \vert \vert _{\frac{6}{5}}\vert \vert {v_R} \vert \vert _\frac{6}{5}+ C\vert \vert {W-v_R} \vert \vert _{\frac{6}{5}}^2\nonumber \\ \le {}&\frac{1}{R}\overset{{}}{\underset{{|{k} |>2(ds\ell )^{-1}R}}{\int }}\frac{\widehat{v}_{1}(k)^2}{|{k} |^2}\, {\mathrm {d}}k+ C\left( {\frac{R}{d\ell }}\right) ^2\frac{a^2}{R}. \end{aligned}$$

(A.14)

Using $|{\widehat{W}(k)} |\le Ca$, gives

$$\begin{aligned} \overset{{}}{\underset{{|{k} |>2(ds\ell )^{-1}}}{\int }}\widehat{W}\left( {k}\right) ^2|{k} |^{-3} \, {\mathrm {d}}k ={}&\overset{{}}{\underset{{2(ds\ell )^{-1}< |{k} |< R^{-1}}}{\int }}\widehat{W}(k)^2|{k} |^{-3}\, {\mathrm {d}}k+ \overset{{}}{\underset{{ |{k} |>R^{-1}}}{\int }}\widehat{W}\left( {k}\right) ^2|{k} |^{-3} \, {\mathrm {d}}k \nonumber \\ \le {}&C\overset{{}}{\underset{{2(ds\ell )^{-1}R< |{k} | < 1 }}{\int }}\widehat{W}(0)^2|{k} |^{-3}\, {\mathrm {d}}k+ R\overset{{}}{\underset{{|{k} |>2(ds\ell )^{-1}}}{\int }}\widehat{W}\left( {k}\right) ^2|{k} |^{-2} \, {\mathrm {d}}k \nonumber \\ \le {}&Ca^2\ln \Big ( \frac{ds\ell }{R}\Big ). \end{aligned}$$

(A.15)

Combining (A.13), (A.14) and (A.15), we arrive at

$$\begin{aligned} \overset{{}}{\underset{{|{k} |>2(ds\ell )^{-1}}}{\int }}\frac{\widehat{W}(k)^2}{\tau _B(k^2)}\, {\mathrm {d}}k \le {}&(1+C\varepsilon _0)\overset{{}}{\underset{{|{k} |>2(ds\ell )^{-1}}}{\int }}\frac{\widehat{W}(k)^2}{|{k} |^2}\, {\mathrm {d}}k+C(ds\ell )^{-1}\overset{{}}{\underset{{|{k} |>2(ds\ell )^{-1}}}{\int }}\frac{\widehat{W}\left( {k}\right) ^2}{|{k} |^{3}} \, {\mathrm {d}}k \nonumber \\ \le {}&(1+C\varepsilon _0)\frac{1}{R}\overset{{}}{\underset{{}}{\int }}\frac{\widehat{v}_{1}(k)^2}{|{k} |^2}\, {\mathrm {d}}k +Ca\frac{a}{ds\ell }\ln \Big (\frac{ds\ell }{R}\Big ). \end{aligned}$$

(A.16)

In the last inequality, we used that the second term in (A.14) and the positive term $(1+C\varepsilon _0)\frac{1}{R}\int _{{|{k} |<2(ds\ell )^{-1}R}}^{{}}\frac{\widehat{v}_{1}(k)^2}{|{k} |^2}\, {\mathrm {d}}k$ by Condition 1 are bounded by the last term in (A.16).

1.1.2 A.1.2. Estimates on the Large Box

Recall that on the large box, we have ${\int \chi _B^2=|{B} |}$ and that

$$\begin{aligned} {\tau _B(k^2)=(1-\varepsilon _0)(1-\varepsilon _{T})\left[ {{|k|-{\textstyle \frac{1}{2}} (s\ell )^{-1}}}\right] _+^2+(1-\varepsilon _0)\varepsilon _{T}\,\left[ {{|k|-{\textstyle \frac{1}{2}} (ds\ell )^{-1}}}\right] _+^2}. \end{aligned}$$

Hence, as in (A.7),

$$\begin{aligned} \overset{{}}{\underset{{|{k} |<(s\ell )^{-1}}}{\int }} h_0(k)\, {\mathrm {d}}k \ge&-\overset{{}}{\underset{{|{k} |<(s\ell )^{-1}}}{\int }}\frac{n_0}{|{B} |}\widehat{W}(k)\, {\mathrm {d}}k\int \chi _B^2 \ge -C\frac{n}{|{B} |}a(s\ell )^{-3}|{B} |. \end{aligned}$$

(A.17)

Analogously to (A.8), we have

$$\begin{aligned} \frac{1}{2}(2\pi )^{-3}\overset{{}}{\underset{{|{k} |>(s\ell )^{-1}}}{\int }} h_0(k)\, {\mathrm {d}}k \ge \frac{1}{2}(2\pi )^{-3}\overset{{}}{\underset{{|{k} |>(s\ell )^{-1}}}{\int }}-\frac{1}{2}\frac{n^2}{|{B} |^2}\frac{\widehat{W}(k)^2}{\tau _B(k^2)}\, {\mathrm {d}}k |{B} |-C\overset{{}}{\underset{{|{k} |>R^{-1}}}{\int }}\frac{n^3}{|{B} |^3}\frac{|{\widehat{W}(k)} |^3}{\tau _B(k^2)^2}\, {\mathrm {d}}k |{B} |. \end{aligned}$$

(A.18)

The last term in (A.18) is estimated in the same way as the last term in (A.8)

$$\begin{aligned}&\overset{{}}{\underset{{|{k} |>R^{-1}}}{\int }}\frac{n^3}{|{B} |^3}\frac{|{\widehat{W}(k)} |^3}{\tau _B(k^2)^2}\, {\mathrm {d}}k\int \chi _B^2\le C\overset{{}}{\underset{{|{k} |>R^{-1}}}{\int }}\frac{n^3}{|{B} |^3}\frac{a^3}{|{k} |^4}\, {\mathrm {d}}k |{B} |\le C\frac{n^3}{|{B} |^3}a^3R |{B} |. \end{aligned}$$

(A.19)

By (30), we have $v_R(x)\!\le \! W(x)\!\le \! v_R(x)(1\!+\!C(\!\frac{R}{\ell })^2)\!$. Thus, similarly to (A.14),

$$\begin{aligned} \overset{{}}{\underset{{|{k} |>(s\ell )^{-1}}}{\int }}\frac{\widehat{W}(k)^2}{|{k} |^2}\, {\mathrm {d}}k \le&\overset{{}}{\underset{{|{k} |>(s\ell )^{-1}}}{\int }}\frac{\widehat{v}_{R}(k)^2}{|{k} |^2}\, {\mathrm {d}}k+ C\frac{a^2R}{\ell ^2}. \end{aligned}$$

(A.20)

Similarly to (A.15), we have

$$\begin{aligned} \overset{{}}{\underset{{|{k} |>(s\ell )^{-1}}}{\int }}\widehat{W}\left( {k}\right) ^2|{k} |^{-3} \, {\mathrm {d}}k \le Ca^2\ln \left( { \frac{s\ell }{R}}\right) . \end{aligned}$$

(A.21)

Since

$$\begin{aligned} \tau _B(k^2)^{-1}\!\le \!{\left\{ \begin{array}{ll} \!\left( {1\!+\!C\varepsilon _0\!+\!C\varepsilon _T\!}\right) \!|{k} |^{{-}2}\!+\!C(s\ell )^{-1}|{k} |^{{-}3} &{} \text {for }(s\ell )^{-1}\!<\!|{k} |\!<\!(ds\ell )^{-1}\\ \!\left( {1\!+\!C\varepsilon _0\!}\right) \!|{k} |^{{-}2}\!+\!C\!\left( {(s\ell )^{-1}\!+\!\varepsilon _T(ds\ell )^{-1}\!}\right) \!|{k} |^{{-}3}\! &{} \text {for }|{k} |\ge (ds\ell )^{-1}, \end{array}\right. } \end{aligned}$$

(A.22)

we have, similar to (A.16),

$$\begin{aligned} \,\overset{{}}{\underset{{|{k} |>(s\ell )^{-1}}}{\int }}\frac{\widehat{W}(k)^2}{\tau _B(k^2)}\, {\mathrm {d}}k \le&\,(1+C\varepsilon _0+C\varepsilon _T)\overset{{}}{\underset{{(s\ell )^{-1}<|{k} |<(ds\ell )^{-1}}}{\int }}\frac{\widehat{W}(k)^2}{|{k} |^2}+C(s\ell )^{-1}\frac{\widehat{W}(k)^2}{|{k} |^3}\! \, {\mathrm {d}}k\nonumber \\&+\!\left( {1\!+\!C\varepsilon _0}\right) \overset{{}}{\underset{{|{k} |>(ds\ell )^{-1}}}{\int }}\frac{\widehat{W}(k)^2}{|{k} |^2}+C\!\left( {(s\ell )^{-1}+\!\varepsilon _T(ds\ell )^{-1}}\right) \! \frac{\widehat{W}(k)^2}{|{k} |^3}\!\, {\mathrm {d}}k\nonumber \\ \le {}&\overset{{}}{\underset{{|{k} |>(s\ell )^{-1}}}{\int }}\frac{\widehat{v_R}(k)^2}{|{k} |^2}\, {\mathrm {d}}k +C\varepsilon _0a\frac{a}{R}+ C\frac{a^2R}{\ell ^2}+C\varepsilon _Ta^2(ds\ell )^{-1}\nonumber \\&+C(s\ell )^{-1}a^2\ln \left( {\frac{s\ell }{R}}\right) +C\varepsilon _T(ds\ell )^{-1}a^2\ln \left( {\frac{ds\ell }{R}}\right) \nonumber \\ \le {}&\int _{{}}^{{}}\frac{\widehat{v_R}(k)^2}{|{k} |^2}\, {\mathrm {d}}k +C\varepsilon _0a\frac{a}{R} +C(s\ell )^{-1}a^2\ln \left( {\frac{s\ell }{R}}\right) \nonumber \\&+C\varepsilon _T(ds\ell )^{-1}a^2\ln \left( {\frac{ds\ell }{R}}\right) . \end{aligned}$$

(A.23)

In the last inequality, we used that by Condition 1 the terms $C\frac{aR}{\ell ^2}a$ and $\int _{{|{k} |<(s\ell )^{-1}}}^{{}}\,\frac{\widehat{v}_{R}(k)^2}{|{k} |^2}\, {\mathrm {d}}k$ are bounded by the second to last term in (A.23).