  This report discusses two new ideas for using perturbation methods to solve
the time-independent Schr\"odinger equation. The first concept begins with
rewriting the perturbation equations in a form that is closely related to
matrix diagonalization methods. That allows a simple, compact derivation of the
standard (Rayleigh-Schr\"odinger) equations. But it also leads to a new
iterative solution method that is not based on the usual power series
expansion. The iterative method can also be used when the unperturbed system is
two-fold degenerate. The second concept is quite different from the first but
compatible with it. It is based on the fact that one can replace the true
Hamiltonian with a synthetic Hamiltonian having the same eigenvalues. This
approach allows one to cancel part of the perturbation and to reduce the size
of the off-diagonal matrix elements, giving better convergence of the series.
These methods are illustrated by application to three perturbed harmonic
oscillator problems--a 1-D oscillator with a linear perturbation, a 1-D
oscillator with a quartic perturbation, and a perturbed 2-D oscillator, which
involves degenerate states. The iterative method and the synthetic Hamiltonian
method are shown to give significantly better results than the standard method.
