Smoothness and monotone decreasingness of the solution to the BCS-Bogoliubov gap equation for superconductivity

We show the temperature dependence such as smoothness and monotone decreasingness with respect to the temperature of the solution to the BCS-Bogoliubov gap equation for superconductivity. Here the temperature belongs to the closed interval $[0,\, \tau]$ with $\tau>0$ nearly equal to half of the transition temperature. We show that the solution is continuous with respect to both the temperature and the energy, and that the solution is Lipschitz continuous and monotone decreasing with respect to the temperature. Moreover, we show that the solution is partially differentiable with respect to the temperature twice and the second-order partial derivative is continuous with respect to both the temperature and the energy, or that the solution is approximated by such a smooth function.


Introduction and main result
In this paper we show the temperature dependence such as smoothness and monotone decreasingness with respect to the temperature of the solution to the BCS-Bogoliubov gap equation [2,4] for superconductivity: where the solution u is a function of the absolute temperature T ≥ 0 and the energy x (0 ≤ x ≤ ℏω D ). The solution u corresponds to the energy gap between the superconducting ground state and the superconducting first excited state, and so the value of the solution is nonnegative, i.e., u(T, x) ≥ 0. The constant ω D > 0 stands for the Debye angular frequency, and the potential U satisfies U(x, ξ) > 0 at all (x, ξ) ∈ [0, ℏω D ] 2 . In (1.1) we consider the solution u as a function of the absolute temperature T and the energy x. Accordingly, we deal with the integral with respect to the energy ξ. Sometimes one considers the solution u as a function of the absolute temperature and the wave vector of an electron. Accordingly, instead of the integral with respect to the energy ξ in (1.1), one deals with the integral with respect to the wave vector over the three dimensional Euclidean space R 3 . The existence and uniqueness of the solution to the BCS-Bogoliubov gap equation were established in previous papers [11,3,12,5,6,7] for each fixed temperature. So the temperature dependence such as smoothness and monotone decreasingness with respect to the temperature of the solution is not covered except for the paper [1]. In [1] the gap equation in the Hubbard model for a constant potential was studied, and its solution was shown to be strictly decreasing with respect to the temperature. In this connection, for interdisciplinary reviews of the BCS-Bogoliubov model of superconductivity, see [8,9].
As is well known, studying the temperature dependence of the solution to the BCS-Bogoliubov gap equation is very important in condensed matter physics. This is because studying the temperature dependence of the solution, by dealing with the thermodynamic potential, leads to a mathematical proof of the statement that the transition to the superconducting state is a second-order phase transition in the BCS-Bogoliubov model of superconductivity. In order to give its proof, we have to differentiate the thermodynamic potential, and hence the solution with respect to the temperature twice, and we have to study some properties of the second-order partial derivative of the solution. So it is highly desirable to study the temperature dependence such as smoothness and monotone decreasingness with respect to the temperature of the solution to the BCS-Bogoliubov gap equation (1.1).
We now define a nonlinear integral operator A by Here the right side of this equality is exactly the right side of the BCS-Bogoliubov gap equation ( Here the temperature τ 1 > 0 is defined by (see [2]) See also Niwa [10] and Ziman [17]. As is well known in the BCS-Bogoliubov model, physicists and engineers studying superconductivity always assume that there is a unique nonnegative solution ∆ 1 to the simple gap equation (1.3), that the solution ∆ 1 is continuous and strictly decreasing with respect to the temperature T , and that the solution ∆ 1 is of class C 2 with respect to the temperature T , and so on. But, as far as the present authors know, there is no mathematical proof for these assumptions of the BCS-Bogoliubov model. Then, applying the implicit function theorem to the simple gap equation (1.3), one of the present authors obtained the following proposition that indeed gives a mathematical proof for these assumptions mentioned just above: Then there is a unique nonnegative solution ∆ 1 : [ 0, τ 1 ] → [0, ∞) to the simple gap equation (1.3) such that the solution ∆ 1 is continuous and strictly decreasing with respect to the temperature T on the closed interval [ 0, τ 1 ]: Moreover, the solution ∆ 1 is of class C 2 with respect to the temperature T on the interval [ 0, τ 1 ) and satisfies We introduce another positive constant Here, τ 2 > 0 is defined by We again set ∆ 2 (T ) = 0 for T > τ 2 . See figure 1. The function ∆ 2 has properties similar to those of the function ∆ 1 . We now deal with the BCS-Bogoliubov gap equation (1.1), where the potential U is not a constant but a function. We assume the following condition on U: Let 0 ≤ T ≤ τ 2 and fix T . We now consider the Banach space C[0, ℏω D ] consisting of continuous functions of the energy x only, and deal with the following temperature dependent subset V T : Consequently, the solution u 0 (T, ·) with T fixed is continuous with respect to the energy x and varies with the temperature as follows: Superconductivity is observed when the temperature T satisfies T < T c . Here, T c is the transition temperature (critical temperature) and divides superconductivity (T < T c ) and normal conductivity (T > T c ). The existence and uniqueness of the transition temperature T c were pointed out in previous papers [5,6,7,12]. In our case, we can define it as follows: Definition 1.6. Let u 0 (T, ·) be as in Theorem 1.5. Then the transition temperature T c is defined by Note that the transition temperature T c satisfies τ 1 ≤ T c ≤ τ 2 . Let u 0 (T, ·) be as in Theorem 1.5. A straightforward calculation gives that if there is a point Remark 1.7. Theorem 1.5 tells us nothing about continuity of the solution u 0 with respect to the temperature T . Applying the Banach fixed-point theorem, we then showed in [14, Theorem 1.2] that the solution u 0 is indeed continuous both with respect to the temperature T and with respect to the energy x under the restriction that the temperature T is sufficiently small. See also [15].
When the potential U(·, ·) is not a constant but a function, one of the present authors [16] studied the temperature dependence such as smoothness and monotone decreasingness of the solution to the BCS-Bogoliubov gap equation (1.1) with respect to the temperature near the transition temperature T c , and gave the behavior of the solution near the transition temperature T c . Then, dealing with the thermodynamic potential, it was shown that the transition to the superconducting state is a second-order phase transition from the viewpoint of operator theory [16]. Moreover, the exact and explicit expression for the gap in the specific heat at constant volume at the transition temperature T c was also obtained in [16].
Remark 1.8. Observed values in many experiments by using superconductors imply the temperature τ 0 is nearly equal to T c /2.
Let 0 < τ < τ 0 and fix τ . We then deal with the following subset V of the Banach Here, γ > 0 is defined by (   2 Proof of Theorem 1.10 We prove Theorem 1.10 in a sequence of lemmas.

Proof. A straightforward calculation gives
Note that √ X ≥ ∆ 1 (τ 0 ) and that z 0 is nearly equal to 2.07. The result thus follows.
A straightforward calculation gives the following.
Lemma 2.3. The subset V is bounded, closed, convex and nonempty.
Proof. Since u(T, x) ≤ ∆ 2 (T ), it follows that Therefore (1.4) gives Similarly we can show the rest. Proof.
Step 1. We first show Au(T, x) − Au(T ′ , x) ≥ 0. where where T < T ′′ < T ′ . Thus, by (2.2), Proof. Let T < T ′ . Then Since U(·, ·) is uniformly continuous, for an arbitrary ε > 0, there is a δ 1 > 0 such that Note that the δ 1 > 0 depends neither on x, nor on x ′ , nor on ξ, nor on u ∈ V . Hence the second term on the right of (2.3) becomes On the other hand, the first term on the right of (2.3) becomes by the preceding lemma. Here, T ′ − T < ε/(2γ). Thus Note that the δ > 0 depends neither on x, nor on x ′ , nor on ξ, nor on u ∈ V , nor on T , nor on T ′ .
A straightforward calculation gives the following.
Lemma 2.7. Let u ∈ V . Then Au is partially differentiable with respect to T twice (0 ≤ T ≤ τ ), and ∂Au ∂T , The preceding lemmas imply the following.
Lemma 2.9. The set AV is relatively compact.
Proof. Let u ∈ V . Lemma 2.4 then implies So the set AV is uniformly bounded. As mentioned in the proof of Lemma 2.6, the δ does not depend on u ∈ V . Hence the set AV is equicontinuous. The result thus follows from the Ascoli-Arzelà theorem. Proof. Let u, v ∈ V . Then combining a similar discussion to that in the proof of Lemma 2.5 with (1.3) gives Here, d is between u(T, ξ) and v(T, ξ), and · denotes the norm of the Banach space . The result thus follows.
We now extend the domain V of our operator A to the closure V . Let u ∈ V . Then there is a sequence {u n } ∞ n=1 ⊂ V satisfying u − u n → 0 as n → ∞. A similar discussion to that in the proof of Lemma 2.10 gives {Au n } ∞ n=1 ⊂ V is a Cauchy sequence, and hence there is an Au ∈ V satisfying Au − Au n → 0 as n → ∞. Note that Au ∈ V does not depend on how to choose the sequence {u n } ∞ n=1 ⊂ V . We thus have the following.
It is not obvious that Au (u ∈ V ) is expressed as (1.2). The next lemma shows this is the case. A similar discussion to that in the proof of Lemma 2.10 gives the following.
Proof. For u ∈ V , set and let {u n } ∞ n=1 ⊂ V be a sequence satisfying u − u n → 0 as n → ∞. Note that the function (T, x) → I(T, x) just above is well-defined and continuous. Then A similar discussion to that in the proof of Lemma 2.10 gives the second term becomes |Au n (T, x) − I(T, x)| ≤ U 2 U 1 u n − u → 0 (n → ∞).
The result thus follows.
Similar discussions to those in Lemmas 2.4 and 2.5 give the following. Lemma 2.13 implies Au(T, x) ≤ ∆ 2 (0) for u ∈ V since the function ∆ 2 is strictly decreasing with respect to the temperature T . Hence the set AV is uniformly bounded. Similar discussions to those in the proofs of Lemmas 2.6 and 2.9 give the following. Lemma 2.14. Let u ∈ V . Then Au ∈ C([0, τ ] × [0, ℏω D ]). Moreover, the set AV is equicontinuous, and hence the set AV is relatively compact.
By Lemma 2.12, a smilar discussion to that in the proof of Lemma 2.10 gives the following.    The uniqueness of the nonzero fixed point of A : V → V was pointed out in Theorem 1.5. Our proof of Theorem 1.10 is now complete.