Extraction of spectral densities from lattice correlators
The correlator is defined as
This quantity is usually known for some discrete values. We therefore discretise the correlator ( and , with and ):
We introduce the veilbein matrix:
which, in principle, is a rectangular matrix.
Thus we can write:
If we multiply for its transpose (), we obtain:
We introduce:
$$G_{nm} = [\varepsilon^T \varepsilon]_{nm} = \sigma^2 \sum_{t=0}^{N_T-1} e^{-at\sigma(n+m)} $$
Therefore
and finally the density is given by:
It is useful to define:
such that