Log-anharmonic oscillator and its large-N solution

Abstract

The large-N expansion technique is tested via an anomalous, soft-core potential which admits the tunneling through its central barrier. The precision of the approximation is found sensitive to the asymptotic component of the interaction. Once chosen in the most common harmonic-oscillator form, and once complemented by the short range part represented by the general power-law anharmonicity $\sim |x|^\alpha$, we found that the latter power-law spike may be well approximated by an elementary logarithmic function, in the limit of the smallest $\alpha \to 0$ at least. In such a model, the large-N method is found applicable and offering still an efficient and user-friendly method of the solution of the Schr\"{o}dinger equation.