Consider the fermion condensate in gauge theory:

$$

\langle \bar{f}f\rangle = -i\int d^{4}x\text{Tr}[S(x,x)]

$$

where

$$

S(x,y) = \langle x |D^{-1}(x,y)|y\rangle

$$

is the fermion propagator and $D$ is the Dirac operator including the fermion mass $m$. Using the spectral representation of the Dirac operator,

$$

D(x,y) = \sum_{\lambda}\frac{\psi(x)\psi^{\dagger}(y)}{\lambda + im},

$$

one finds

$$\langle \bar{f}f\rangle = \sum_{\lambda}\frac{1}{\lambda + im}$$

How to obtain from this expression the Casher-Banks relation $\langle \bar{f}f\rangle = \pi \rho(\lambda = 0)$, where $\rho$ is the spectral density?