Perturbative, asymptotic and Pade-approximant solutions for harmonic and inverted oscillators in a box

The energy levels of boxed-in harmonic and inverted oscillators are constructed from the perturbative and asymptotic solutions that are valid in the limits of small and large sizes, respectively. In order to obtain expressions for the energy levels which are valid for boxes of any size, Pade approximants constructed as interpolations between the perturbative and asymptotic solutions are used. Special attention is paid to the lowest levels. The accuracy and range of validity of each type of solution are illustrated by comparing them with the exact solution which is obtained by constructing and diagonalising the matrix of the Hamiltonian of the system in the basis of eigenfunctions of the free particle in a box. (author)