  We discuss the solutions of the Schroedinger equation for piecewise
potentials, given by the harmonic oscillator potential for $\vert x\vert >a$
and an arbitrary function for $\vert x\vert <a$, using elementary methods. The
study of this problem sheds light on usual errors when discussing the
asymptotic behavior of the eigenfunctions of the quantum harmonic oscillator
and can also be used for the analysis of the eigenfunctions of the hydrogen
atom. We present explicit results for the energy levels of a potential of this
class, used to model the confinement of electrons in nanostructures.
