For a translation invariant system of N bosons in the Gross–Pitaevskii regime, we establish a precise bound for the ground state energy <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="220_2025_5320_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="26" /> </InlineMediaObject> <EquationSource Format="TEX">$E_N$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>E</mi> <mi>N</mi> </msub> </math></EquationSource> </InlineEquation>. While the leading, order N, contribution to <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="220_2025_5320_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="26" /> </InlineMediaObject> <EquationSource Format="TEX">$E_N$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>E</mi> <mi>N</mi> </msub> </math></EquationSource> </InlineEquation> has been known since (Lieb et al. in Phys Rev A 61:043602, 2000; Lieb and Yngvason in Phys Rev Lett 80:2504–2507, 1998) and the second order corrections (of order one) have been first determined in Boccato et al. (Acta Math 222(2):219–335, 2019), our estimate also resolves the next term in the asymptotic expansion of <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="220_2025_5320_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="26" /> </InlineMediaObject> <EquationSource Format="TEX">$E_N$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>E</mi> <mi>N</mi> </msub> </math></EquationSource> </InlineEquation>, which is of the order <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="220_2025_5320_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="76" /> </InlineMediaObject> <EquationSource Format="TEX">$(\log N) / N$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mo>log</mo> <mi>N</mi> <mo stretchy="false">)</mo> <mo stretchy="false">/</mo> <mi>N</mi> </mrow> </math></EquationSource> </InlineEquation>.

Third Order Corrections to the Ground State Energy of a Bose Gas in the Gross–Pitaevskii Regime

期刊论文
出版模式：

开放获取
发表日期：

2025年6月3日

查看全文

Cristina Caraci,
Alessandro Olgiati,
Diane Saint Aubin,
Benjamin Schlein

摘要

For a translation invariant system of N bosons in the Gross–Pitaevskii regime, we establish a precise bound for the ground state energy $E_N$ $E_{N}$ . While the leading, order N, contribution to $E_N$ $E_{N}$ has been known since (Lieb et al. in Phys Rev A 61:043602, 2000; Lieb and Yngvason in Phys Rev Lett 80:2504–2507, 1998) and the second order corrections (of order one) have been first determined in Boccato et al. (Acta Math 222(2):219–335, 2019), our estimate also resolves the next term in the asymptotic expansion of $E_N$ $E_{N}$ , which is of the order $(\log N) / N$ $(log N) / N$ .