标题： 超导电性的埃利亚斯伯格理论中$T_c$的界限。 II：色散声子 显示英文标题

标题： Bounds on $T_c$ in the Eliashberg theory of Superconductivity. II: Dispersive phonons

摘要： 标准的埃利亚斯伯格超导理论被研究，其中有效的电子-电子相互作用由通常具有色散性的声子介导，埃利亚斯伯格谱函数$\alpha^2 F(\omega)\geq 0$在小$\omega>0$时为$\propto\omega^2$并在大$\omega$时消失。 埃利亚斯伯格函数也定义了电子-声子耦合强度$\lambda:= 2 \int_0^\infty\frac{\alpha^2 F(\omega)}{\omega}d\omega$。 设定$\frac{2\alpha^2 F(\omega)}{\omega}d\omega =: \lambda P(d\omega)$，形式上定义一个具有紧支集的概率测度$P(d\omega)$，并通常假设正常态与超导态之间的相变与正常区域对向超导区域的扰动的线性稳定性边界$\mathscr{S}_{\!c}$一致，结果表明$\mathscr{S}_{\!c}$是由变分原理确定的函数$\Lambda(P,T)$的图像：如果$(\lambda,P,T)\in\mathscr{S}_{\!c}$，则$\lambda = 1/\mathfrak{k}(P,T)$，其中$\mathfrak{k}(P,T)>0$是在论文中构造的紧自伴算子$\mathfrak{K}(P,T)$在$\ell^2$序列上的最大特征值。 给定$P$，在$T$上给出了充分条件，使得映射$T\mapsto \lambda = \Lambda(P,T)$可逆。 对于足够大的$\lambda$，这得出：(i) 临界温度$T_c$作为$\lambda$和$P$的函数存在；(ii) 一个关于$T_c(\lambda,P)$的下界序列，该序列收敛到$T_c(\lambda,P)$。 还得到了$T_c(\lambda,P)$的一个上界。 它与适用于$\lambda\sim\infty$的渐近形式$T_c(\lambda,P) \sim C \sqrt{\langle \omega^2\rangle} \sqrt{\lambda}$一致，给定$P$，尽管常数$C$比尖锐常数大一个因子$\approx 2.034$。 这里， $\langle\omega^2\rangle := \int_0^\infty \omega^2 P(d\omega)$。 显示英文摘要

摘要： The standard Eliashberg theory of superconductivity is studied, in which the effective electron-electron interactions are mediated by generally dispersive phonons, with Eliashberg spectral function $\alpha^2 F(\omega)\geq 0$ that is $\propto\omega^2$ for small $\omega>0$ and vanishes for large $\omega$. The Eliashberg function also defines the electron-phonon coupling strength $\lambda:= 2 \int_0^\infty\frac{\alpha^2 F(\omega)}{\omega}d\omega$. Setting $\frac{2\alpha^2 F(\omega)}{\omega}d\omega =: \lambda P(d\omega)$, formally defining a probability measure $P(d\omega)$ with compact support, and assuming as usual that the phase transition between normal and superconductivity coincides with the linear stability boundary $\mathscr{S}_{\!c}$ of the normal region against perturbations toward the superconducting region, it is shown that $\mathscr{S}_{\!c}$ is a graph of a function $\Lambda(P,T)$ that is determined by a variational principle: if $(\lambda,P,T)\in\mathscr{S}_{\!c}$, then $\lambda = 1/\mathfrak{k}(P,T)$, where $\mathfrak{k}(P,T)>0$ is the largest eigenvalue of a compact self-adjoint operator $\mathfrak{K}(P,T)$ on $\ell^2$ sequences constructed in the paper. Given $P$, sufficient conditions on $T$ are stated under which the map $T\mapsto \lambda = \Lambda(P,T)$ is invertible. For sufficiently large $\lambda$ this yields: (i) the existence of a critical temperature $T_c$ as function of $\lambda$ and $P$; (ii) a sequence of lower bounds on $T_c(\lambda,P)$ that converges to $T_c(\lambda,P)$. Also obtained is an upper bound on $T_c(\lambda,P)$. It agrees with the asymptotic form $T_c(\lambda,P) \sim C \sqrt{\langle \omega^2\rangle} \sqrt{\lambda}$ valid for $\lambda\sim\infty$, given $P$, though with a constant $C$ that is a factor $\approx 2.034$ larger than the sharp constant. Here, $\langle\omega^2\rangle := \int_0^\infty \omega^2 P(d\omega)$.