Accurate analytic approximations to eigenvalues anharmonic potentials x2+λx8

Abstract

Precise analytic approximations have been found for the eigenvalues of the ground state in the anharmonic potentials x2+λx8. Rational functions combined with fractional powers are used to determine the new approximations. First, power series in the parameter has been found using an extension of the quantum mechanics perturbation technique recently published. Modifications of these new technologies has been also developed to obtain asymptotic expansions in λ. The analytic functions here determined are just a bridge between both expansions. The maximum relative error of the best analytic approximation here determined is 0.006 (less than 1%). However, most of the relative errors for other values of λ, are smaller than 0.3%. Positive values of λ are only considered.