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

For a translation invariant system of <i>N</i> bosons in the Gross-Pitaevskii regime, we establish a precise bound for the ground state energy <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>E</mml:mi> <mml:mi>N</mml:mi></mml:msub> </mml:math> . While the leading, order <i>N</i>, contribution to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>E</mml:mi> <mml:mi>N</mml:mi></mml:msub> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>E</mml:mi> <mml:mi>N</mml:mi></mml:msub> </mml:math> , which is of the order <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo> <mml:mo>log</mml:mo> <mml:mi>N</mml:mi> <mml:mo>)</mml:mo> <mml:mo>/</mml:mo> <mml:mi>N</mml:mi></mml:mrow> </mml:math> .