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The Bardeen–Cooper–Schrieffer functional of superconductivity and its mathematical properties
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45. The complex conjugation of is denoted by . In an abstract Hilbert space, this simply means for an anti-linear involution J. In the concrete setting of ℋ = L2(ℝd) below, it will always mean that .
46. Here and the following, we shall use the notation C for generic constants, possibly having a different value in each appearance.
47. This is, in fact, the convention used in Ref. 13.
48. The results in Ref. 13 were stated for D > 0, but the proof is equally valid for D ≤ 0.
49. To see that (4.72) follows from Ref. 13 [Lemma 4], simply note that we can assume that otherwise the right side of (4.72) is negative. Since we already know that and thus also by Sobolev’s inequality, the desired result follows in a straightforward way.
50. As already mentioned in Section II, complex conjugation in an abstract Hilbert space corresponds to the choice of an anti-linear involution. The reason for its necessity is the antilinearity of the annihilation operator. Alternatively, one could define the annihilation operator to accept as its argument an element of instead of (i.e., one replaces a by , where with being the conjugate linear map such that (Jψ)(ϕ) = 〈ψ|ϕ〉), in which case the antilinearity would be naturally absorbed in J. This is the approach followed in Ref. 40.
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We review recent results concerning the mathematical properties of the Bardeen–Cooper–Schrieffer (BCS) functional of superconductivity, which were obtained in a series of papers, partly in collaboration with R. Frank, E. Hamza, S. Naboko, and J. P. Solovej. Our discussion includes, in particular, an investigation of the critical temperature for a general class of interaction potentials, as well as a study of its dependence on external fields. We shall explain how the Ginzburg–Landau model can be derived from the BCS theory in a suitable parameter regime.
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