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