The BCS energy gap at low density

Abstract

We show that the energy gap for the BCS gap equation is Ξ=μ8e-2+o(1)expπ2μa\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\varXi = \\mu \\left( 8 {\\mathrm{e}}^{-2} + o(1)\\right) \\exp \\left( \\frac{\\pi }{2\\sqrt{\\mu } a}\\right) \\end{aligned}$$\\end{document}in the low density limit mu rightarrow 0. Together with the similar result for the critical temperature by Hainzl and Seiringer (Lett Math Phys 84: 99–107, 2008), this shows that, in the low density limit, the ratio of the energy gap and critical temperature is a universal constant independent of the interaction potential V. The results hold for a class of potentials with negative scattering length a and no bound states.