标题： Eliashberg超导理论中关于$T_c$的界限。 第一部分：$γ$模型 显示英文标题

标题： Bounds on $T_c$ in the Eliashberg theory of Superconductivity. I: The $γ$-model

摘要： 利用最近以经典相互作用布洛赫自旋链模型重新表述的埃利阿什贝格超导理论，针对 $T_c$ 模型——一种有效电子-电子相互作用与 $(g/|\omega_n-\omega_m|)^{\gamma}$ 成正比的埃利阿什贝格理论版本，得到了临界温度 $\gamma$ 的严格上下界，其中 $\omega_n-\omega_m$ 是传递的 Matsubara 频率， $g>0$ 是参考能量， $\gamma>0$ 是参数。 严格下界基于一个变分原理，该原理将$(T_c/g)^\gamma$识别为显式构造的紧致、自伴算子$\mathfrak{G}(\gamma)$的最大（正）特征值。这些下界构成一个递增序列，收敛于$T_c(g,\gamma)$。$T_c(g,\gamma)$的上界基于不动点理论，证明了当$T$大于$T_c(g,\gamma)$的上界时，正常态的线性稳定性。 显示英文摘要

摘要： Using the recent reformulation for the Eliashberg theory of superconductivity in terms of a classical interacting Bloch spin chain model, rigorous upper and lower bounds on the critical temperature $T_c$ are obtained for the $\gamma$ model -- a version of Eliashberg theory in which the effective electron-electron interaction is proportional to $(g/|\omega_n-\omega_m|)^{\gamma}$, where $\omega_n-\omega_m$ is the transferred Matsubara frequency, $g>0$ a reference energy, and $\gamma>0$ a parameter. The rigorous lower bounds are based on a variational principle that identifies $(T_c/g)^\gamma$ with the largest (positive) eigenvalue of an explicitly constructed compact, self-adjoint operator $\mathfrak{G}(\gamma)$. These lower bounds form an increasing sequence that converges to $T_c(g,\gamma)$. The upper bound on $T_c(g,\gamma)$ is based on fixed point theory, proving linear stability of the normal state for $T$ larger than the upper bound on $T_c(g,\gamma)$.