Eigenvalues and eigenfunctions for the ground state of polynomial potentials

Analytic approximations for the ground state eigenvalues and eigenfunctions of polynomial potentials are found using an extended two-point quasi-rational approximation technique. In this procedure, the approximants are obtained through the power series and asymptotic expansion of the logarithmic derivative of the ground state eigenfunction, leaving the energy eigenvalue as a free parameter. A first approximation to the energy is obtained by imposing the condition that the rational approximating function must not have defects. Later, an iteration procedure leads to very precise energy eigenvalues. The method is described in detail using several explicit potentials as examples.