  An algebraic non-perturbative approach is proposed for the analytical
treatment of Schr\"{o}dinger equations with a potential that can be expressed
in terms of an exactly solvable piece with an additional potential. Avoiding
disadvantages of standard approaches, new handy recursion formulae with the
same simple form both for ground and excited states have been obtained. As an
illustration the procedure, well adapted to the use of computer algebra, is
successfully applied to quartic anharmonic oscillators by means of very simple
algebraic manipulations. The trend of the exact values of the energies is
rather well reproduced for a large range of values of the coupling constant
(g=0.001-10000).
