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