  We present a refinement of the Spectral Method by incorporating an
optimization method into it and generalize it to two space dimensions. We then
apply this Refined Spectral Method as an extremely accurate technique for
finding the bound states of the two dimensional time-independent Schrodinger
equation. We first illustrate the use of this method on an exactly solvable
case and then use it on a case which is not so. This method is very simple to
program, fast, extremely accurate (e.g. a relative error of 10^(-15) is easily
obtainable in two dimensions), very robust and stable. Most importantly, one
can obtain the energies and the wave functions of as many of the bound states
as desired with a single run of the algorithm.
