Consider the integral \begin{align} I(\beta)=\int_{-\infty}^{\infty}\frac{\text{d}\epsilon}{\pi}\frac{\rho(\epsilon,\beta)}{1+e^{\beta \epsilon}} \end{align} where $\rho(\epsilon,\beta)$ is a generic function of both $\epsilon$ and $\beta$.

When $\rho$ is sufficiently smooth and not a function of $\beta$, we can use the Sommerfeld expansion to study the $\beta\to\infty$ (low temperature) behaviour \begin{align} \label{eq2} I(\beta)\underset{\beta\to\infty}{=}\int_{-\infty}^0\frac{\text{d}\epsilon}{\pi} \rho(\epsilon)+\frac{\pi^2}{12\beta^2}\rho'(0)+O(\beta^{-4}).\tag{1} \end{align} This is sometimes explained as the mnemonic expansion of the Fermi-Dirac distribution \begin{align} \frac{1}{1+e^{\beta\epsilon}} \underset{\beta\to\infty}{=} \theta(-\epsilon)-\frac{\pi^2}{6\beta^2}\delta'(\epsilon)+O(\beta^{-4})\tag{2} \end{align}

Two questions:

  1. Can the conditions for Eq. (1) to hold be expressed in terms of exchange between integration and the series expansion in Eq. (2)? E.g. dominated convergence for the fermi-dirac distribution.

  2. Can we follow a similar expansion of the Fermi-Dirac distribution when $\rho$ is also a function of $\beta$? In this case, what are the analytic properties $\rho$ must satisfy for this to hold? E.g. if $\rho$ is bounded (like for a spectral function).


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