Abstract. We prove that the critical temperature for the BCS gap equation is given by 8 Tc = µ π eγ−2 « + o(1) e π/(2 √ µa) in the low density limit µ → 0. The formula holds for a suitable class of interaction potentials with negative scattering length a in the absence of bound states.
Abstract We show that the energy gap for the BCS gap equation is Ξ=μ(8e−2+o(1))exp(2μaπ) Ξ = μ 8 e - 2 + o ( 1 ) exp π 2 μ a in the low density limit μ→0 μ → 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.
We derive upper and lower bounds on the critical temperature Tc and the energy gap $\ensuremath{\Xi}$ (at zero temperature) for the BCS gap equation, describing spin-21 fermions interacting via a local two-body interaction potential $\ensuremath{\lambda}V(x)$. At weak coupling $\ensuremath{\lambda}⪡1$ and under appropriate assumptions on V(x), our bounds show that ${T}_{c}\ensuremath{\sim}A\phantom{\rule{0.2em}{0ex}}\mathrm{exp}(\ensuremath{-}B∕\ensuremath{\lambda})$ and $\ensuremath{\Xi}\ensuremath{\sim}C\phantom{\rule{0.2em}{0ex}}\mathrm{exp}(\ensuremath{-}B∕\ensuremath{\lambda})$ for some explicit coefficients A, B, and C depending on the interaction V(x) and the chemical potential $\ensuremath{\mu}$. The ratio A∕C turns out to be a universal constant, independent of both V(x) and $\ensuremath{\mu}$. Our analysis is valid for any $\ensuremath{\mu}$; for small $\ensuremath{\mu}$, or low density, our formulas reduce to well-known expressions involving the scattering length of V(x).
Abstract We study the BCS energy gap Ξ Ξ in the high–density limit and derive an asymptotic formula, which strongly depends on the strength of the interaction potential V on the Fermi surface. In combination with the recent result by one of us (Math. Phys. Anal. Geom. 25, 3, 2022) on the critical temperature Tc T c at high densities, we prove the universality of the ratio of the energy gap and the critical temperature.
Abstract. We present a rigorous analysis of the Bardeen-Cooper-Schrieffer (BCS) model for general pair interaction potentials. For both zero and positive temperature, we show that the existence of a non-trivial solution of the nonlinear BCS gap equation is equivalent to the existence of a negative eigenvalue of a certain effective linear operator. From this we conclude the existence of a critical temperature below which the BCS pairing wave function does not vanish identically. For attractivepotentials, we prove that the critical temperature is non-zero and exponentially small in the strength of the potential. 1.
Abstract. The Bardeen-Cooper-Schrieffer (BCS) functional has recently received renewed attention as a description of fermionic gases interacting with local pairwise interactions. We present here a rigorous analysis of the BCS functional for general pair interaction potentials. For both zero and positive temperature, we show that the existence of a non-trivial solution of the nonlinear BCS gap equation is equivalent to the existence of a negative eigenvalue of a certain linear operator. From this we conclude the existence of a critical temperature below which the BCS pairing wave function does not vanish identically. For attractive potentials, we prove that the critical temperature is non-zero and exponentially small in the strength of the potential. 1.
The Bardeen-Cooper-Schrieffer (BCS) functional has recently received renewed attention as a description of fermionic gases interacting with local pairwise interactions. We present here a rigorous analysis of the BCS functional for general pair interaction potentials. For both zero and positive temperature, we show that the existence of a non-trivial solution of the nonlinear BCS gap equation is equivalent to the existence of a negative eigenvalue of a certain linear operator. From this we conclude the existence of a critical temperature below which the BCS pairing wave function does not vanish identically. For attractive potentials, we prove that the critical temperature is non-zero and exponentially small in the strength of the potential.