  Recently was introduced in the literature a procedure to obtain ans\"atze,
free of parameters, for the eigenfunctions of the time-independent
Schr\"odinger equation with symmetric convex potential. In the present work, we
test this technique in regard to $x^{2\kappa}$-type potentials. We study the
behavior of the ans\"atze regarding the degree of the potential and to the
intervening coupling constant. Finally, we discuss how the results could be
used to establish the upper bounds of the relative errors in situations where
intervening polynomial potentials.
