  We present a new convergent iterative solution for the two lowest quantum
wave functions $\psi_{ev}$ and $\psi_{od}$ of the Hamiltonian with a quartic
double well potential $V$ in one dimension. By starting from a trial function,
which is by itself the exact lowest even or odd eigenstate of a different
Hamiltonian with a modified potential $V+\delta V$, we construct the Green's
function for the modified potential. The true wave functions, $\psi_{ev}$ or
$\psi_{od}$, then satisfies a linear inhomogeneous integral equation, in which
the inhomogeneous term is the trial function, and the kernel is the product of
the Green's function times the sum of $\delta V$, the potential difference, and
the corresponding energy shift. By iterating this equation we obtain successive
approximations to the true wave function; furthermore, the approximate energy
shift is also adjusted at each iteration so that the approximate wave function
is well behaved everywhere. We are able to prove that this iterative procedure
converges for both the energy and the wave function at all $x$.
