  We review a new iterative procedure to solve the low-lying states of the
Schroedinger equation, done in collaboration with Richard Friedberg. For the
groundstate energy, the $n^{th}$ order iterative energy is bounded by a finite
limit, independent of $n$; thereby it avoids some of the inherent difficulties
faced by the usual perturbative series expansions. For a fairly large class of
problems, this new procedure can be proved to give convergent iterative
solutions. These convergent solutions include the long standing difficult
problem of a quartic potential with either symmetric or asymmetric minima.
