Log-anharmonic oscillator and its large-N solution

TL;DR

This paper tests the large-N expansion technique on a log-anharmonic oscillator with a soft-core potential, demonstrating its effectiveness and sensitivity to interaction asymptotics, and approximating the anharmonicity with a logarithmic function.

Contribution

It introduces a novel application of the large-N method to a log-anharmonic oscillator, showing its accuracy and practicality for solving Schrödinger equations with complex potentials.

Findings

Large-N expansion is effective for the log-anharmonic oscillator.

The method's accuracy depends on the asymptotic behavior of the potential.

The anharmonicity can be approximated by a logarithmic function for small alpha.

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.