# Chapter 6: Correlation Functions and Spectra

In the Fourier Grid Hamiltonian (FGII) method (Marston, 1989) the coordinates x, x' are discretized at N evenly spaced points:

and thus, the Hamiltonian is an N x N matrix with corresponding to position The potential energy matrix elements are given by the prescription that the off-diagonal elements vanish and that the diagonal elements are given by the local evaluation of V(x): The kinetic energy matrix elements are given (for N even) by:

Adding the kinetic and potential energies we obtain:

and thus, the Hamiltonian is an N x N matrix with corresponding to position The potential energy matrix elements are given by the prescription that the off-diagonal elements vanish and that the diagonal elements are given by the local evaluation of V(x): The kinetic energy matrix elements are given (for N even) by:

Adding the kinetic and potential energies we obtain: