# Chapter 6: Correlation Functions and Spectra

The DVR method is a pseudospectral method for calculating eigenvalues and eigenfunctions with Gaussian quadrature accuracy. The DVR procedure is:

- Construct the matrix representation of in a truncated basis of N orthogonal polynomials,
- Diagonalize by a unitary transformation, i.e. where
- Compute V(λ) Since λ is diagonal,
- Calculate the kinetic energy matrix matrix in the basis of orthogonal polynomials, which we denote . In order that the Hamiltonian H = T+V be expressed completely in the pointwise basis, , must now be transformed by the same transformation u that was used to transform the matrix to the pointwise basis:
- Construct the DVR Hamiltonian by adding the kinetic and potential energies:

- Diagonalize the DVR Hamiltonian, to obtain eigenvalues and eigenfunctions.