Energy eigenvalues for free and confined triple-well potentials

Abstract

Some confined and unconfined (free) one--dimensional triple--well potentials are analyzed with two different numerical approaches. Confinement is achieved by enclosing the potential between two impenetrable walls. The unconfined (free) system is recovered as the positions of the walls move to infinity. The numerical solutions of the Schrödinger equation for the symmetric and asymmetric potentials without confinement, are comparable in precision with those obtained anaylitically. For the symmetric triple--well potentials, $V(x)=\alpha x^2-\beta x^4+x^6$, it is found that there are sets of two or three quasi-degenerate eigenvalues depending on the parameters $\alpha$ and $\beta$. A heuristic analysis shows that if the conditions $\alpha=(\beta^2/4) \pm 1$ (with $\alpha >0$ and $\beta>0$) are satisfied, then there are sets of three eigenvalues with similar energy. An interesting behavior is found when one impenetrable wall is fixed and the other is moved to different positions. In summary, the number of local minima that the potential has in the confined region determines a two- or three-fold degeneracy.