Ground State Energy of Dense Gases of Strongly Interacting Fermions

We study the ground state energy of a gas of <i>N</i> fermions confined to a unit box in <i>d</i> dimensions. The particles interact through a two-body potential with strength scaled in an <i>N</i>-dependent way as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow><mml:msup><mml:mi>N</mml:mi> <mml:mrow><mml:mo>-</mml:mo> <mml:mi>α</mml:mi></mml:mrow> </mml:msup> <mml:mi>v</mml:mi></mml:mrow> </mml:math> , where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>α</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>R</mml:mi></mml:mrow> </mml:math> and <i>v</i> is a function of positive type satisfying a mild regularity assumption. Our focus is on the strongly interacting case <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>α</mml:mi> <mml:mo><</mml:mo> <mml:mn>1</mml:mn> <mml:mo>-</mml:mo> <mml:mfrac><mml:mn>2</mml:mn> <mml:mi>d</mml:mi></mml:mfrac> </mml:mrow> </mml:math> . We contrast our result with existing results in the weakly interacting case <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>α</mml:mi> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> <mml:mo>-</mml:mo> <mml:mfrac><mml:mn>2</mml:mn> <mml:mi>d</mml:mi></mml:mfrac> </mml:mrow> </mml:math> and the transition happening at the mean-field scaling <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>α</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>-</mml:mo> <mml:mfrac><mml:mn>2</mml:mn> <mml:mi>d</mml:mi></mml:mfrac> </mml:mrow> </mml:math> . Our proof is an adaptation of the bosonization technique used to treat the mean-field case.