We present a new convergent iterative solution for the two lowest quantum wave functions Ο ev and Ο 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 + Ξ΄V , we construct the Green's function for the modified potential. The true wave functions, Ο ev or Ο od , then satisfy 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 Ξ΄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. The effectiveness of this iterative process clearly depends on how good the trial function is, or equivalently, how small the potential difference Ξ΄V is. Although each iteration brings a correction smaller than the previous one by a factor proportional to the parameter that characterizes the smallness of Ξ΄V , it is not a power series expansion in the parameter. The exact tunneling information of the modified potential is, of course, contained in the Green's function; by adjusting the kernel of the integral equation via the energy shift at each iteration, we bring enough of this information into the calculation so that each approximate wave function is exponentially tuned. This is the underlying reason why the present method converges, while the usual power series expansion does not.