Analytic approximate eigenvalues by a new technique. Application to sextic anharmonic potentials

A new technique to obtain analytic approximant for eigenvalues is presented here by a simultaneous use of power series and asymptotic expansions is presented. The analytic approximation here obtained is like a bridge to both expansions: rational functions, as Padé, are used, combined with elementary functions are used. Improvement to previous methods as multipoint quasirational approximation, MPQA, are also developed. The application of the method is done in detail for the 1-D Schrödinger equation with anharmonic sextic potential of the form V(x)=x2+λx6 and both ground state and the first excited state of the anharmonic oscillator.