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

## TL;DR

This paper analyzes solutions to the Schrödinger equation with piecewise harmonic potentials, clarifying asymptotic behaviors of eigenfunctions and providing explicit energy level results relevant for nanostructure electron confinement.

## Contribution

It offers elementary methods to solve the Schrödinger equation for piecewise potentials and corrects common misconceptions about eigenfunction asymptotics.

## Key findings

- Clarifies asymptotic behavior of wave functions in piecewise harmonic potentials
- Provides explicit energy level formulas for nanostructure models
- Highlights common errors in quantum harmonic oscillator analysis

## Abstract

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.