An operator-theoretical study on the BCS-Bogoliubov model of superconductivity near absolute zero temperature

In the preceding papers the present author gave another proof of the existence and uniqueness of the solution to the BCS-Bogoliubov gap equation for superconductivity from the viewpoint of operator theory, and showed that the solution is partially differentiable with respect to the temperature twice. Thanks to these results, we can indeed partially differentiate the solution and the thermodynamic potential with respect to the temperature twice so as to obtain the entropy and the specific heat at constant volume of a superconductor. In this paper we show the behavior near absolute zero temperature of the thus-obtained entropy, the specific heat, the solution and the critical magnetic field from the viewpoint of operator theory since we did not study it in the preceding papers. Here, the potential in the BCS-Bogoliubov gap equation is an arbitrary, positive continuous function and need not be a constant.

Note that there is a function that belongs to the former Banach space but not to the latter one. For example, the function x → 1/x belongs to the former Banach space but not to the latter one. Under this circumstance, unfortunately the present author does not know which norm, which metric (which distance), which ε-neighborhoods, I could use in order to prove the statement that the former Banach space continuously tends to the latter one as the cutoff goes to zero from the view point of operator theory. I therefore introduce the cutoff ε > 0 , fix it and deal with the former Banach space For a fixed temperature T, the existence and uniqueness of the solution were established and studied in 1,2,5-21 . See also Kuzemsky 22 , Chapters 26 and 29 and 23,24 . For the role of the chemical potential in the BCS-Bogoliubov model, see Anghel and Nemnes 25 and Anghel 26,27 .
In connection to this, the BCS-Bogoliubov gap equation plays a role similar to that of the Maskawa-Nakajima equation 28,29 which has attracted considerable interest in elementary particle physics. In Professor Maskawa's Nobel lecture, he stated the reason why he dealt with the Maskawa-Nakajima equation. For an operator-theoretical treatment of this equation, see the present author's paper 30 .
In the BCS-Bogoliubov model, the thermodynamic potential is given by where u 0 is the solution to the BCS-Bogoliubov gap equation (1.1), T c is the transition temperature (see 2 , Definition 1.8 for our operator-theoretical definition of T c ) and N 0 is a positive constant and denotes the density of states per unit energy at the Fermi surface. Here we consider only the contribution from the interval [−ℏω D , ℏω D ] , and omit the contribution from the other intervals. In other words, we consider only the contribution from superconductivity. For more details, see 2 , (1.5) and (1.6). As mentioned above, thanks to 1 , Theorems 2.3 and 2.4 and 2 , Theorems 2.2 and 2.10, we can indeed partially differentiate the solution with respect to the temperature T twice, and have the solution u 0 , the first order partial derivative ∂u 0 /∂T and the second order partial derivative ∂ 2 u 0 /∂T 2 . Moreover, all of them are continuous functions of both the temperature T and the energy x. Therefore, thanks to these results, we can indeed differentiate the thermodynamic potential with respect to T twice so as to obtain the entropy and the specific heat at constant volume. Note that the potential U(·, ·) in the BCS-Bogoliubov gap equation is an arbitrary, positive continuous function and need not be a constant.

Remark 1.3
If the solution u 0 is an accumulating point of the set V in 2 , Theorem 2.2 (resp. of the set W in 2 , Theorem 2.10), then we replace u 0 by a suitably chosen element of V (resp. of W) in the form (1.2) of the thermodynamic potential . This is because u 0 is an accumulating point. Note that such a suitably chosen element is partially differentiable with respect to the temperature T twice and that it is a continuous function of both the temperature T and the energy x. Therefore, once we replace the solution u 0 by a suitably chosen element in the form (1.2), we can differentiate the thermodynamic potential with respect to the temperature T twice so as to obtain the entropy and the specific heat at constant volume. (1.2) The same is true for ∂ 2 u 0 /∂T 2 . Therefore we apply the following approximation. Approximation (A) Let T 0 (> 0) be in a neighborhood of absolute zero temperature T = 0 and let . Since all of the solution u 0 , the first order partial derivative ∂u 0 /∂T and the second order partial derivative ∂ 2 u 0 /∂T 2 are continuous functions of both the temperature T and the energy x, we apply the following approximation: Here, X > 0 and n is every nonnegative integer.

Remark 2.5
As far as the present author knows, similar results are obtained in the physics literature under the restriction that the potential U(·, ·) in the BCS-Bogoliubov gap equation is a constant. But Theorem 2.2 holds true even when the potential U(·, ·) is not a constant but an arbitrary, positive continuous function.

Remark 2.6
Suppose that the potential U(·, ·) is a constant, i.e., U(·, ·) = U 0 . Here, U 0 is a positive constant. Then the solution u 0 to the BCS-Bogoliubov gap equation does not depend on the energy x and becomes a function of the temperature T only. We denote the solution by u 0 (T) . Then the forms of S(T), C V (T) and u 0 (T, x) in Theorem 2.2 are reduced to the following well-known forms, respectively: At T ∈ [0, T 0 ],

Proof of Theorem 2.2
We first give a proof for the behavior of the entropy S at T ∈ [0, Note that the sixth term on the right side of (3.1) is negligible. This is because the sixth term becomes (at T ∈ [0, T 0 ]) which is negligible compared to the seventh term. www.nature.com/scientificreports/ We next give a proof for the behavior for the specific heat C V at constant volume at T ∈ [0, T 0 ] . To this end we differentiate ∂�/∂T with respect to T again and obtain the second order partial derivative ∂ 2 �/∂T 2 . The second order partial derivative of the first term on the right side of (3.1) becomes which is approximated by 0 at T ∈ [0, T 0 ] under Approximation (A). On the other hand, the second order partial derivative of the last term on the right side of (3.1) includes which is the only term that we have at T ∈ [0, T 0 ] under Approximation (A). We deal with the other terms on the right side of (3.1) similarly.