  We define Sturmian basis functions for the harmonic oscillator and
investigate whether recent insights into Sturmians for Coulomb-like potentials
can be extended to this important potential. We also treat many body problems
such as coupling to a bath of harmonic oscillators. Comments on coupled
oscillators and time-dependent potentials are also made.
  It is argued that the Sturmian method amounts to a non-perturbative
calculation of the energy levels, but the limitations of the method is also
pointed out, and the cause of this limitation is found to be related to the
divergence of the potential. Thus the divergent nature of the anharmonic
potential leads to the Sturmian method being less acurate than in the Coulomb
case. We discuss how modified anharmonic oscillator potentials, which are well
behaved at infinity, leads to a rapidly converging Sturmian approximation.
