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

Solutions of the Schroedinger equation for piecewise harmonic potentials: remarks on the asymptotic behavior of the wave functions