  The widely used large-scale diagonalization method using harmonic oscillator
basis functions (an instance of the Rayleigh-Ritz method, also called a
spectral method, configuration-interaction method, or ``exact diagonalization''
method) is systematically analyzed using results for the convergence of Hermite
function series. We apply this theory to a Hamiltonian for a one-dimensional
model of a quantum dot. The method is shown to converge slowly, and the
non-smooth character of the interaction potential is identified as the main
problem with the chosen basis, while on the other hand its important advantages
are pointed out. An effective interaction obtained by a similarity
transformation is proposed for improving the convergence of the diagonalization
scheme, and numerical experiments are performed to demonstrate the improvement.
Generalizations to more particles and dimensions are discussed.
