Bose–Einstein Condensation Beyond the Gross–Pitaevskii Regime
Arka Adhikari, Christian Brennecke, Benjamin Schlein
Abstract
Abstract We consider N bosons in a box with volume one, interacting through a two-body potential with scattering length of the order $$N^{-1+\kappa }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>N</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mi>κ</mml:mi> </mml:mrow> </mml:msup> </mml:math> , for $$\kappa >0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>κ</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> . Assuming that $$\kappa \in (0;1/43)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>κ</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>;</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>43</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , we show that low-energy states exhibit Bose–Einstein condensation and we provide bounds on the expectation and on higher moments of the number of excitations.