Approximate analytical functions for the eigenvalues of the Schrödinger equation with cubic anharmonic potential

The energy spectrum of the cubic anharmonic potential V(x)=x2+λ|x|3 is studied. Precise approximations of the eigenvalues of the first three energy states has been determined. For the evaluation, the multipoint quasirational approximation (MPQA) technique is used, obtaining approximations with polynomial functions and elementary functions, by using a bridge function as a combination of the power series and asymptotic expansions. The approximations of the eigenvalues found are compared with the exact values. The calculated values of the approximations for different values of the coupling parameter are discussed. The present work shows that the approximations of the first three energy states with cubic anharmonic potential, has an accuracy of around 1%. By using the optimized parameters,the efficiency of the approximations is increased for the eigenvalues.