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Order Form Model Order Form Model Is So Famous, But Why?

In the physics literature, one differentiates the band-aid to the BCS-Bogoliubov gap equation, the thermodynamic abeyant and the analytical alluring acreage with account to the temperature after assuming that they are differentiable with account to the temperature. So we charge to appearance that they are differentiable with account to the temperature, as mentioned in the above-mentioned section.

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We acquaint the blow ε > 0 and accept that the abeyant (U(,cdot ,,cdot ,)) satisfies (1.4) throughout this paper. We denote by z0 > 0 a different band-aid to the blueprint (frac{2}{z}=,tanh ,z) (z > 0). The amount of z0 is about according to 2.07, and the asperity (frac{2}{z}le ,tanh ,z) holds for z ≥ z0. Let τ0(>0) satisfy

$${Delta }_{1}({tau }_{0})=2{z}_{0}{tau }_{0}.$$

(2.1)

Let 0 < τ3 < τ0 and fix τ3. Here, τ3 > 0 is baby enough. Let γ be as in (3.2) below. We accord with the afterward subset V of the Banach amplitude (C([0,,{tau }_{3}]times [varepsilon ,,hslash {omega }_{D}])):

$$begin{array}{ccc}V & = & {uin C([0,,{tau }_{3}]times [varepsilon ,hslash {omega }_{D}]),:,0le u(T,x)-u({T}^{{rm{{prime} }}},x)le gamma ({T}^{{rm{{prime} }}}-T),(T < {T}^{{rm{{prime} }}}),\ & & {Delta }_{1}(T)le u(T,,x)le {Delta }_{2}(T),,u,{rm{i}}{rm{s}},{rm{p}}{rm{a}}{rm{r}}{rm{t}}{rm{i}}{rm{a}}{rm{l}}{rm{l}}{rm{y}},{rm{d}}{rm{i}}{rm{f}}{rm{f}}{rm{e}}{rm{r}}{rm{e}}{rm{n}}{rm{t}}{rm{i}}{rm{a}}{rm{b}}{rm{l}}{rm{e}},{rm{w}}{rm{i}}{rm{t}}{rm{h}},{rm{r}}{rm{e}}{rm{s}}{rm{p}}{rm{e}}{rm{c}}{rm{t}},{rm{t}}{rm{o}},T,{rm{t}}{rm{w}}{rm{i}}{rm{c}}{rm{e}},\ & & frac{{rm{partial }}u}{{rm{partial }}T},,frac{{{rm{partial }}}^{2}u}{{rm{partial }}{T}^{2}}in C([0,,{tau }_{3}]times [varepsilon ,,hslash {omega }_{D}]),,frac{{rm{partial }}u}{{rm{partial }}T}(0,x)=frac{{{rm{partial }}}^{2}u}{{rm{partial }}{T}^{2}}(0,x)=0,{rm{a}}{rm{t}},{rm{a}}{rm{l}}{rm{l}},xin [varepsilon ,,hslash {omega }_{D}].end{array}$$

The altitude in the analogue of subset V

$$frac{{rm{partial }}u}{{rm{partial }}T}(0,,x)=frac{{{rm{partial }}}^{2}u}{{rm{partial }}{T}^{2}}(0,,x)=0,{rm{a}}{rm{t}},{rm{a}}{rm{l}}{rm{l}},xin [varepsilon ,,hslash {omega }_{D}]$$

are not imposed in 24 [Theorem 1.10]. These altitude are capital for our affidavit of Theorem 2.19 below. The added altitude in the analogue of V and in Theorem 2.2 beneath are the aforementioned as the altitude in 24 [Theorem 1.10].

We afresh ascertain our abettor A (see (1.1)) on V:

$$Au(T,x)={int }_{varepsilon }^{hslash {omega }_{D}},frac{U(x,xi )u(T,xi )}{sqrt{{xi }^{2} u{(T,xi )}^{2}}},tanh ,frac{sqrt{{xi }^{2} u{(T,xi )}^{2}}}{2T}dxi ,,uin V.$$

(2.2)

We denote by (bar{V}) the cease of the subset V with account to the barometer of the Banach amplitude (C([0,{tau }_{3}]times [varepsilon ,hslash {omega }_{D}])).

The afterward is one of our capital results.

Let us acquaint the blow ε > 0 and accept (1.4). Let (bar{V}) be as above. Afresh our abettor (A:bar{V}to bar{V}) has a different anchored point ({u}_{0}in bar{V}), and so there is a different nonnegative band-aid ({u}_{0}in bar{V}) to the BCS-Bogoliubov gap equation (1.1):

$${u}_{0}(T,x)={int }_{varepsilon }^{hslash {omega }_{D}},frac{U(x,xi ){u}_{0}(T,xi )}{sqrt{{xi }^{2} {u}_{0}{(T,xi )}^{2}}},tanh ,frac{sqrt{{xi }^{2} {u}_{0}{(T,xi )}^{2}}}{2T}dxi ,,(T,x)in [0,{tau }_{3}]times [varepsilon ,hslash {omega }_{D}].$$

Consequently, the band-aid u0 is connected on ([0,{tau }_{3}]times [varepsilon ,hslash {omega }_{D}]). Moreover, u0 is banausic abbreviating and Lipschitz connected with account to T, and satisfies ({Delta }_{1}(T)le {u}_{0}(T,x)le {Delta }_{2}(T)) at all ((T,x)in [0,{tau }_{3}]times [varepsilon ,hslash {omega }_{D}]). Furthermore, if u0 ∈ V, afresh u0 is partially differentiable with account to T twice, and the first-order and second-order fractional derivatives of u0 are both connected on ([0,{tau }_{3}]times [varepsilon ,hslash {omega }_{D}]). And, at all (xin [varepsilon ,hslash {omega }_{D}]),

$$frac{partial {u}_{0}}{partial T}(0,x)=frac{{partial }^{2}{u}_{0}}{partial {T}^{2}}(0,x)=0.$$

On the added hand, if ({u}_{0}in bar{V}{rm{setminus }}V), afresh u0 is approximated by such a action of V with account to the barometer of the Banach amplitude (C([0,{tau }_{3}]times [varepsilon ,hslash {omega }_{D}])).

Let u0 be the band-aid of Theorem 2.2. Since ({u}_{0}in bar{V}), we accept u0 ∈ V or ({u}_{0}in bar{V}{rm{setminus }}V). If u0 ∈ V, afresh the band-aid in the thermodynamic abeyant Ψ(T) (see (1.6)) is annihilation but this u0 ∈ V, and appropriately the band-aid in Ψ(T) is partially differentiable with account to the temperature T twice. So we can differentiate the thermodynamic abeyant Ψ(T) with account to the temperature T twice. On the added hand, if ({u}_{0}in bar{V}{rm{setminus }}V), afresh ({u}_{0}in bar{V}{rm{setminus }}V) is approximated by a appropriately called aspect u1 ∈ V. In such a case, we alter the band-aid in Ψ(T) by this aspect u1 ∈ V. Let us admonish actuality that the aspect u1 ∈ V is partially differentiable with account to the temperature T twice. Once we alter the band-aid in Ψ(T) by this aspect u1 ∈ V, we can afresh differentiate the thermodynamic abeyant Ψ(T) with account to the temperature T twice. In this way, in both cases, we can differentiate the thermodynamic abeyant Ψ(T), and appropriately Ω(T) with account to the temperature T twice.

The behavior of the band-aid u0 accustomed by Theorem 2.2 is in acceptable acceding with the beginning data.

The function

$$(T,x)mapsto {int }_{varepsilon }^{hslash {omega }_{D}},frac{U(x,xi )}{sqrt{{xi }^{2} {Delta }_{2}{(T)}^{2},},},tanh ,frac{sqrt{{xi }^{2} {Delta }_{2}{(T)}^{2}}}{2T}dxi $$

is continuous, and it follows from (1.3) that

$${int }_{varepsilon }^{hslash {omega }_{D}},frac{U(x,xi )}{sqrt{{xi }^{2} {Delta }_{2}{(T)}^{2}}},tanh ,frac{,sqrt{{xi }^{2} {Delta }_{2}{(T)}^{2}}}{2T}dxi < 1$$

since U(x, ξ) < U2 (see (1.4)). Agenda that the function

$$(T,x)mapsto frac{{Delta }_{2}{(tau )}^{2}}{2{varepsilon }^{2}}{int }_{varepsilon }^{hslash {omega }_{D}},frac{U(x,xi )}{xi },tanh ,frac{xi }{2T}dxi $$

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is additionally continuous. Here, 0 < τ < Tc. We afresh accede the sum of the two connected functions above:

$${int }_{varepsilon }^{hslash {omega }_{D}}frac{U(x,,xi )}{sqrt{{xi }^{2} {Delta }_{2}{(T)}^{2}}},tanh ,frac{sqrt{{xi }^{2} {Delta }_{2}{(T)}^{2}}}{2T}dxi frac{{Delta }_{2}{(tau )}^{2}}{2{varepsilon }^{2}}{int }_{varepsilon }^{hslash {omega }_{D}}frac{U(x,xi )}{xi },tanh ,frac{xi }{,2T}dxi mathrm{}.$$

Note that the additional appellation aloof aloft tends to aught as Δ2(τ)/ε goes to zero. Let τ be actual abutting to Tc and let Δ2(τ)/ε be actual baby so that the inequality

$${int }_{varepsilon }^{hslash {omega }_{D}},frac{U(x,xi )}{sqrt{{xi }^{2} {Delta }_{2}{(T)}^{2}}},tanh ,frac{sqrt{{xi }^{2} {Delta }_{2}{(T)}^{2}}}{2T}dxi frac{{Delta }_{2}{(tau )}^{2}}{2{varepsilon }^{2}}{int }_{varepsilon }^{hslash {omega }_{D}},frac{U(x,xi )}{xi },tanh ,frac{xi }{,2T}dxi < 1$$

holds true.

We afresh fix τ and ε, and we accord with the set ([tau ,{T}_{c}]times [varepsilon ,hslash {omega }_{D}]in {{mathbb{R}}}^{2}). Agenda that the larboard ancillary of the asperity aloof aloft is a connected action of ((T,x)in [tau ,{T}_{c}]times [varepsilon ,hslash {omega }_{D}]). We set

$$begin{array}{ccc}alpha & = & mathop{max}limits_{(T,,x)in [tau ,,{T}_{c}]times [varepsilon ,,hslash {omega }_{D}]}[{int }_{varepsilon }^{hslash {omega }_{D}}frac{U(x,,xi )}{,sqrt{,{xi }^{2} {Delta }_{2}(T{)}^{2},},},tanh frac{,sqrt{,{xi }^{2} {Delta }_{2}(T{)}^{2},},}{2T},dxi \ & & ,,,,, frac{,{Delta }_{2}(tau {)}^{2},}{2,{varepsilon }^{2}}{int }_{varepsilon }^{hslash {omega }_{D}}frac{,U(x,,xi ),}{xi },tanh frac{xi }{,2T,},dxi ].end{array}$$

Therefore,

We let τ be actual abutting to Tc, and we let Δ2(τ)/ε be actual baby so that (2.3) holds true.

Let us accede the afterward condition.

Condition (C). Let τ and ε be as above. An aspect (uin C([tau ,{T}_{c}]times [varepsilon ,hslash {omega }_{D}])) is partially differentiable with account to the temperature T ∈ [τ, Tc) twice, and the fractional derivatives (∂u/∂T) and (∂2u/∂T2) both accord to (C([tau ,{T}_{c})times [varepsilon ,hslash {omega }_{D}])). Moreover, for the u above, there are a different (vin C[varepsilon ,hslash {omega }_{D}]) and a different (win C[varepsilon ,hslash {omega }_{D}]) acceptable the following:

(C1) v(x) > 0 at all (xin [varepsilon ,hslash {omega }_{D}]).

(C2) For an approximate ε1 > 0, there is a δ > 0 such that |Tc − T| < δ implies

$$|v(x)-frac{,u{(T,x)}^{2},}{{T}_{c}-T}| < {T}_{c},{varepsilon }_{1},.$$

Here, δ does not depend on (xin [varepsilon ,hslash {omega }_{D}]).

(C3) For an approximate ε1 > 0, there is a δ > 0 such that |Tc − T| < δ implies

$$|w(x)-frac{,-v(x)-frac{{rm{partial }}}{,{rm{partial }}T,}{u{(T,x)}^{2}},}{{T}_{c}-T}| < {varepsilon }_{1},.$$

Here, δ does not depend on (xin [varepsilon ,hslash {omega }_{D}]).

(C4) For an arbitrarily ample R > 0, there is a δ > 0 such that |Tc − T| < δ implies

$$-frac{partial u}{partial T}(T,x) > R,$$

Here, δ does not depend on (xin [varepsilon ,hslash {omega }_{D}]).

If (uin C([tau ,{T}_{c}]times [varepsilon ,hslash {omega }_{D}])) satisfies Action (C2), afresh u(Tc, x) = 0 at all (xin [varepsilon ,hslash {omega }_{D}]).

Condition (C2) implies the action (frac{partial {u}^{2}}{partial T}(T,x)) converges to −v(x) (<0) analogously with account to x as T ↑ Tc.

Condition (C3) implies the action (frac{{partial }^{2}{u}^{2}}{partial {T}^{2}}(T,x)) converges to w(x) analogously with account to x as T ↑ Tc.

Condition (C4) implies (frac{partial u}{partial T}(T,x)to -infty ) as T ↑ Tc. Action (C4) is not imposed in 1 [Theorem 2.3]. The added altitude in the analogue of the subset W beneath and in Theorem 2.10 beneath are the aforementioned as the altitude in 1 [Theorem 2.3].

We denote by W the afterward subset of the Banach amplitude (C([tau ,{T}_{c})times [varepsilon ,hslash {omega }_{D}])):

$$begin{array}{ccc}W & = & {uin C([tau ,{T}_{c}]times [varepsilon ,hslash {omega }_{D}]):u(T,x)ge u({T}^{{rm{{prime} }}},x),(T < {T}^{{rm{{prime} }}}),\ & & {Delta }_{1}(T)le u(T,x)le {Delta }_{2}(T),{rm{a}}{rm{t}},(T,x),({T}^{{rm{{prime} }}},x)in [tau ,{T}_{c}]times [varepsilon ,hslash {omega }_{D}],\ & & u,{rm{s}}{rm{a}}{rm{t}}{rm{i}}{rm{s}}{rm{f}}{rm{i}}{rm{e}}{rm{s}},{rm{C}}{rm{o}}{rm{n}}{rm{d}}{rm{i}}{rm{t}}{rm{i}}{rm{o}}{rm{n}},({rm{C}}),{rm{a}}{rm{b}}{rm{o}}{rm{v}}{rm{e}}},end{array}$$

and we ascertain our abettor A (see (1.1)) on W:

$$Au(T,x)={int }_{varepsilon }^{hslash {omega }_{D}}frac{U(x,xi )u(T,xi )}{sqrt{{xi }^{2} u{(T,xi )}^{2}}},tanh ,frac{sqrt{{xi }^{2} u{(T,xi )}^{2}}}{2T}dxi ,,uin W.$$

(2.4)

We denote by (bar{W}) the cease of the subset W with account to the barometer of the Banach amplitude (C([tau ,{T}_{c})times [varepsilon ,hslash {omega }_{D}])).

The afterward is one of our capital results.

Let us acquaint the blow ε > 0 and accept (1.4). Let (tau > 0) be actual abutting to Tc and let Δ2(τ)/ε > 0 be actual baby so that (2.3) holds true. Afresh our abettor (A:bar{W}to bar{W}) is a abbreviating operator. Consequently, there is a different anchored point ({u}_{0}in bar{W}) of our abettor (A:bar{W}to bar{W}), and so there is a different nonnegative band-aid ({u}_{0}in bar{W}) to the BCS-Bogoliubov gap blueprint (1.1):

$${u}_{0}(T,x)={int }_{varepsilon }^{hslash {omega }_{D}},frac{U(x,xi ){u}_{0}(T,xi )}{sqrt{{xi }^{2} {u}_{0}{(T,xi )}^{2}}},tanh ,frac{,sqrt{{xi }^{2} {u}_{0}{(T,xi )}^{2}}}{2T}dxi ,,(T,x)in [tau ,{T}_{c}]times [varepsilon ,hslash {omega }_{D}].$$

The band-aid u0 is connected on ([tau ,{T}_{c}]times [varepsilon ,hslash {omega }_{D}]), and is banausic abbreviating with account to the temperature T. Moreover, u0 satisfies that ({Delta }_{1}(T)le u(T,x)le {Delta }_{2}(T)) at all ((T,x)in [tau ,{T}_{c}]times [varepsilon ,hslash {omega }_{D}]), and that u0(Tc, x) = 0 at all (xin [varepsilon ,hslash {omega }_{D}]). If u0 ∈ W, afresh u0 satisfies Action (C). On the added hand, if ({u}_{0}in overline{W}backslash W), afresh u0 is approximated by such a action of W with account to the barometer of the Banach amplitude (C([tau ,{T}_{c})times [varepsilon ,hslash {omega }_{D}])).

Let u0 be the band-aid of Theorem 2.10. Suppose u0 ∈ W. First, Action (C2) implies (frac{partial {u}_{0}^{2}}{partial T}(T,x)) converges to −v(x) (<0) analogously with account to x as T ↑ Tc. Second, Action (C3) implies (frac{{partial }^{2}{u}_{0}^{2}}{partial {T}^{2}}(T,x)) converges to w(x) analogously with account to x as T ↑ Tc. Finally, Action (C4) implies (frac{partial {u}_{0}}{partial T}(T,x)to -infty ) as T ↑ Tc. Actuality both of −v and w depend on u0. This behavior of the band-aid u0 is in acceptable acceding with the beginning data.

Let u0 be the band-aid of Theorem 2.10. Since ({u}_{0}in bar{W}), we accept u0 ∈ W or ({u}_{0}in bar{W}{rm{setminus }}W). If u0 ∈ W, afresh the band-aid in the thermodynamic abeyant Ψ(T) (see (1.6)) is annihilation but this u0 ∈ W, and appropriately the band-aid in Ψ(T) satisfies Action (C). So we can differentiate the thermodynamic abeyant Ψ(T) with account to the temperature T twice. On the added hand, if ({u}_{0}in bar{W}{rm{setminus }}W), afresh ({u}_{0}in bar{W}{rm{setminus }}W) is approximated by a appropriately called aspect u1 ∈ W. In such a case, we alter the band-aid in Ψ(T) by this aspect u1 ∈ W. Let us admonish actuality that the aspect u1 ∈ W satisfies Action (C). Once we alter the band-aid in Ψ(T) by this aspect u1 ∈ W, we can afresh differentiate the thermodynamic abeyant Ψ(T) with account to the temperature T twice. In this way, in both cases, we can differentiate the thermodynamic abeyant Ψ(T), and appropriately Ω(T) with account to the temperature T twice.

Let (g:[0,infty )to {mathbb{R}}) be accustomed by

$$g(eta )={begin{array}{cc}-frac{1}{,{eta }^{2},}left(frac{,tanh eta ,}{eta },-,frac{1}{,{cosh }^{2}eta ,}right), & (eta > 0),\ -frac{,2,}{,3,} & (eta =0).end{array}$$

(25)

Note that g(η) < 0. As mentioned before, if the band-aid u0 to the BCS-Bogoliubov gap blueprint (1.1) is partially differentiable with account to the temperature T twice, afresh the thermodynamic abeyant Ω is differentiable with account to T twice, and the specific calefaction at connected aggregate at T is accustomed by

$${C}_{V}(T)=-,Tfrac{{partial }^{2}Omega }{partial {T}^{2}}(T).$$

Therefore the gap ΔCV in the specific calefaction at connected aggregate at the alteration temperature Tc is accustomed by (see Remark 1.9)

$$Delta {C}_{V}=-,{T}_{c}frac{{partial }^{2}Psi }{partial {T}^{2}}({T}_{c}).$$

In the physics literature, one differentiates the thermodynamic abeyant to access the specific calefaction at connected aggregate after assuming that the thermodynamic abeyant is differentiable with account to T. Agenda that the thermodynamic abeyant has the band-aid to the BCS-Bogoliubov gap blueprint (1.1) in its form. In added words, one differentiates the thermodynamic abeyant with account to T after assuming that the band-aid is differentiable with account to T. But Combining Theorems 2.2 and 2.10 with Remarks 2.3 and 2.12 implies that we can differentiate the band-aid u0, and appropriately the thermodynamic abeyant Ω with account to T twice.

Let u0 be the band-aid to the BCS-Bogoliubov gap blueprint (1.1) accustomed by Theorem 2.10. Let ΔCV be the gap in the specific calefaction at connected aggregate at T = Tc, and let ({C}_{V}^{N}({T}_{c})) be the specific calefaction at connected aggregate at T = Tc agnate to accustomed conductivity, i.e., ({C}_{V}^{N}({T}_{c})=-,{T}_{c}({partial }^{2}{Omega }_{N}/partial {T}^{2})({T}_{c})). Afresh (Delta {C}_{V}/{C}_{V}^{N}({T}_{c})) is absolutely and absolutely accustomed by the expression

$$frac{Delta {C}_{V}}{{C}_{V}^{N}({T}_{c}),}=-frac{{N}_{0}}{32{({T}_{c})}^{2},J}{int }_{varepsilon /(2,{T}_{c})}^{hslash {omega }_{D}/(2,{T}_{c})},v{(2{T}_{c}eta )}^{2}g(eta )deta ,( > 0),$$

where

$$begin{array}{c}J=2{int }_{varepsilon /(2{T}_{c})}^{hslash {omega }_{D}/(2{T}_{c})},frac{{N}_{0}{eta }^{2}}{{cosh }^{2}eta }deta {int }_{-mu /(2{T}_{c})}^{-hslash {omega }_{D}/(2{T}_{c})},frac{N(2{T}_{c}eta ){eta }^{2}}{{cosh }^{2}eta }deta \ , {int }_{hslash {omega }_{D}/(2{T}_{c})}^{infty },frac{N(2{T}_{c}eta ){eta }^{2}}{{cosh }^{2}eta }deta ,end{array}$$

and v(·) is that in Action (C).

The action v(·) of Theorem 2.14 corresponds to the band-aid u0 to the BCS-Bogoliubov gap blueprint (1.1) accustomed by Theorem 2.10.

gives the absolute and exact announcement for (Delta {C}_{V}/{C}_{V}^{N}({T}_{c})). Agenda that the amount U(x, ξ) is about according to a connected at all ((x,xi )in {[varepsilon ,hslash {omega }_{D}]}^{2}) in some superconductors. Moreover, agenda that the amount (hslash {omega }_{D}/(2{T}_{c})) is actual ample in abounding superconductors. The afterward afresh gives that the announcement aloof aloft does not depend on superconductors and is a accepted constant.

Assume U(x, ξ) = U0 at all ((x,xi )in {[varepsilon ,hslash {omega }_{D}]}^{2}), area U0 > 0 is a constant. If (hslash {omega }_{D}/(2{T}_{c})simeq infty ) and (varepsilon /(2{T}_{c})simeq 0), then

$$frac{Delta {C}_{V}}{{C}_{V}^{N}({T}_{c})}simeq frac{{pi }^{2}}{4{int }_{0}^{infty },frac{{eta }^{2}}{{cosh }^{2}eta }deta ,{int }_{0}^{infty },{-g(eta )}deta },$$

which does not depend on superconductors and is a accepted constant.

It is able-bodied accepted that (Delta {C}_{V}/{C}_{V}^{N}({T}_{c})simeq 12/{7zeta (3)}) in the BCS-Bogoliubov archetypal of superconductivity. Here, (smapsto zeta (s)) is the Riemann zeta function. Therefore, Corollary 2.16 gives addition announcement for (Delta {C}_{V}/{C}_{V}^{N}({T}_{c})). Agenda that we use the assemblage area kB = 1.

Let us about-face to the analytical alluring acreage activated to type-I superconductors. It is able-bodied accepted that superconductivity is destroyed alike at a temperature T beneath than the alteration temperature Tc back the abundantly able alluring acreage is activated to type-I superconductors. It is additionally accepted that, at a anchored temperature T, superconductivity is destroyed back the activated alluring acreage is stronger that the analytical alluring acreage Hc(T), and that superconductivity is not destroyed back the alluring acreage is weaker than Hc(T). The analytical alluring acreage Hc(·) is a action of the temperature T, and Hc(T) ≥ 0 at T ≤ Tc. The analytical alluring acreage is accompanying to Ψ(T) (see (1.6)) as follows:

$$-frac{1}{8pi }{H}_{c}{(T)}^{2}=Psi (T),,{rm{a}}{rm{n}}{rm{d}},{rm{h}}{rm{e}}{rm{n}}{rm{c}}{rm{e}},{H}_{c}(T)=sqrt{-8pi Psi (T)}.$$

In the physics literature, one differentiates the thermodynamic potential, and appropriately the analytical alluring acreage with account to T after assuming that they are differentiable with account to T. Agenda that the thermodynamic abeyant has the band-aid to the BCS-Bogoliubov gap blueprint (1.1) in its form. In added words, one differentiates the analytical alluring acreage with account to T  without assuming that the band-aid is differentiable with account to T. But Combining Theorems 2.2 and 2.10 with Remarks 2.3 and 2.12 implies that we can differentiate the band-aid u0, and appropriately the analytical alluring acreage Hc(·) with account to T.

The afterward gives the accuracy of the analytical alluring acreage with account to T and some of its properties.

Let Hc(·) be the analytical alluring field.

(A) Let u0 be the band-aid to the BCS-Bogoliubov gap blueprint (1.1) accustomed by Theorem 2.10. Afresh the afterward (i), (ii) and (iii) authority true.

(i) Hc(·) ∈ C1[τ, Tc]. Consequently, Hc(·) is differentiable on [τ, Tc] with account to the temperature T, and its first-order acquired is connected on [τ, Tc].

(ii) Hc(Tc) = 0, (frac{partial {H}_{c}}{partial T}(T) < 0) at T ∈ [τ, Tc], and

$$frac{{rm{partial }}{H}_{c}}{{rm{partial }}T}({T}_{c})=-,sqrt{-frac{pi {N}_{0}}{2{T}_{c}^{2}}{int }_{varepsilon /(2{T}_{c})}^{hslash {omega }_{D}/(2{T}_{c})}v{(2{T}_{c}eta )}^{2}g(eta )deta },( < ,0).$$

(iii) If (Tsimeq {T}_{c}) (T ≤ Tc), then

$${H}_{c}(T)simeq left(1,-,frac{T}{{T}_{c}}right)sqrt{-frac{pi {N}_{0}}{2}{int }_{varepsilon /(2{T}_{c})}^{hslash {omega }_{D}/(2{T}_{c})},v{(2{T}_{c}eta )}^{2}g(eta )deta },(ge ,0).$$

(B) Let u0 be the band-aid to the BCS-Bogoliubov gap blueprint (1.1) accustomed by Theorem 2.2. Afresh the afterward (iv), (v) and (vi) authority true.

(iv) Hc(·) ∈ C1[0, τ3]. Consequently, Hc(·) is differentiable on [0, τ3] with account to the temperature T, and its first-order acquired is connected on [0, τ3].

(v)

$${H}_{c}(0)=sqrt{8pi {N}_{0}{int }_{varepsilon }^{hslash {omega }_{D}}frac{{{sqrt{{xi }^{2} {u}_{0}{(0,xi )}^{2}}-xi }}^{2}}{sqrt{{xi }^{2} {u}_{0}{(0,xi )}^{2}}}dxi },( > 0).$$

(vi) (frac{partial {H}_{c}}{partial T}(T) < 0) at T ∈ [0, τ3], and

$$frac{partial {H}_{c}}{partial T}(0)=0.$$

As far as the present columnist knows, no one has acicular out that Hc(·) ∈ C1[τ, Tc] and that Hc(·) ∈ C1[0, τ3] exept for the present author. Moreover, as far as the present columnist knows, no one has accustomed the exact and absolute expressions for (∂Hc/∂T)(Tc) and for Hc(0). Moreover, no one has apparent that (∂Hc/∂T)(0) = 0.

In the BCS-Bogoliubov archetypal of superconductivity, one obtains

$${H}_{c}(T)simeq 1.74{H}_{c}(0)left(1-frac{T}{{T}_{c}}right)$$

for (Tsimeq {T}_{c}) beneath the brake that the abeyant (U(cdot ,,,cdot )) of the BCS-Bogoliubov gap equation. (1.1) is a constant, i.e., beneath the brake that the action v(·) of Part (iii) is a constant. But, after this restriction, Part (iii) of Theorem 2.19 gives addition announcement for the behavior of Hc(T) as (Tsimeq {T}_{c}).

The behavior of Hc(·) accustomed by Theorem 2.19 is in acceptable acceding with the beginning data. See Fig. 1 for the behavior of Hc(T).

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