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The BCS functional for general pair interactions
 Commun. Math. Phys
"... Abstract. The BardeenCooperSchrieffer (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 posi ..."
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Cited by 26 (12 self)
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Abstract. The BardeenCooperSchrieffer (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 nontrivial 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 nonzero and exponentially small in the strength of the potential. 1.
MICROSCOPIC DERIVATION OF GINZBURGLANDAU THEORY
"... Abstract. We give the first rigorous derivation of the celebrated GinzburgLandau (GL) theory, starting from the microscopic BardeenCooperSchrieffer (BCS) model. Close to the critical temperature, GL arises as an effective theory on the macroscopic scale. The relevant scaling limit is semiclassica ..."
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Cited by 13 (8 self)
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Abstract. We give the first rigorous derivation of the celebrated GinzburgLandau (GL) theory, starting from the microscopic BardeenCooperSchrieffer (BCS) model. Close to the critical temperature, GL arises as an effective theory on the macroscopic scale. The relevant scaling limit is semiclassical in nature, and semiclassical analysis, with minimal regularity assumptions, plays an important part in our proof.
Critical temperature and energy gap for the BCS equation
"... Abstract. We derive upper and lower bounds on the critical temperature Tc and the energy gap Ξ (at zero temperature) for the BCS gap equation, describing spin 1/2 fermions interacting via a local twobody interaction potential λV (x). At weak coupling λ ≪ 1 and under appropriate assumptions on V (x) ..."
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Cited by 13 (6 self)
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Abstract. We derive upper and lower bounds on the critical temperature Tc and the energy gap Ξ (at zero temperature) for the BCS gap equation, describing spin 1/2 fermions interacting via a local twobody interaction potential λV (x). At weak coupling λ ≪ 1 and under appropriate assumptions on V (x), our bounds show that Tc ∼ Aexp(−B/λ) and Ξ ∼ C exp(−B/λ) for some explicit coefficients A, B and C depending on the interaction V (x) and the chemical potential µ. The ratio A/C turns out to be a universal constant, independent of both V (x) and µ. Our analysis is valid for any µ; for small µ, or low density, our formulas reduce to wellknown expressions involving the scattering length of V (x). 1.
The BCS critical temperature for potentials with negative scattering length
 Lett. Math. Phys
, 2008
"... 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. ..."
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Cited by 10 (9 self)
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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.
Spectral properties of the BCS gap equation of superfluidity
 in: Mathematical Results in Quantum Mechanics, proceedings of QMath10
"... Abstract. We present a review of recent work on the mathematical aspects of the BCS gap equation, covering our results of [9] as well our recent joint work with Hamza and Solovej [8] and with Frank and Naboko [6], respectively. In addition, we mention some related new results. 1. ..."
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Cited by 9 (6 self)
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Abstract. We present a review of recent work on the mathematical aspects of the BCS gap equation, covering our results of [9] as well our recent joint work with Hamza and Solovej [8] and with Frank and Naboko [6], respectively. In addition, we mention some related new results. 1.
Low Density Limit of BCS Theory and BoseEinstein Condensation of Fermion Pairs
, 2012
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A Numerical Perspective on HartreeFockBogoliubov Theory
, 2012
"... The method ofchoice fordescribing attractive quantum systems isHartreeFockBogoliubov (HFB) theory. This is a nonlinear model which allows for the description of pairing effects, the main explanation for the superconductivity of certain materials at very low temperature. This paper is the first stu ..."
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The method ofchoice fordescribing attractive quantum systems isHartreeFockBogoliubov (HFB) theory. This is a nonlinear model which allows for the description of pairing effects, the main explanation for the superconductivity of certain materials at very low temperature. This paper is the first study of HartreeFockBogoliubov theory from the point of view of numerical analysis. We start by discussing its proper discretization and then analyze the convergence of the simple fixed point (Roothaan) algorithm. Following works by Cancès, Le Bris and Levitt for electrons in atoms and molecules, we show that this algorithm either converges to a solution of the equation, or oscillates between two states, none of them being a solution to the HFB equations. We also adapt the Optimal Damping Algorithm of Cancès and Le Bris to the HFB setting and we analyze it. The last part of the paper is devoted to numerical experiments. We consider a purely gravitational system and numerically discover that pairing always occurs. We then examine a simplified model for nucleons, with an effective interaction similar to what is often used in nuclear physics. In both cases we discuss the importance of using a damping algorithm. c ○ 2012 by the authors. This paper may be reproduced, in its entirety, for noncommercial purposes. Contents 1
Solovej, Microscopic Derivation of GinzburgLandau Theory
 J. Amer. Math. Soc
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ASYMPTOTIC BEHAVIOR OF EIGENVALUES OF SCHRÖDINGER TYPE OPERATORS WITH DEGENERATE KINETIC ENERGY
, 2009
"... Abstract. We study the eigenvalues of Schrödinger type operators T + λV and their asymptotic behavior in the small coupling limit λ → 0, in the case where the symbol of the kinetic energy, T(p), strongly degenerates on a nontrivial manifold of codimension one. 1. ..."
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Abstract. We study the eigenvalues of Schrödinger type operators T + λV and their asymptotic behavior in the small coupling limit λ → 0, in the case where the symbol of the kinetic energy, T(p), strongly degenerates on a nontrivial manifold of codimension one. 1.