Quasi-harmonic spectra from branched Hamiltonians

TL;DR

This paper investigates the quantum spectra of branched Hamiltonians derived from a modified Emden equation, revealing that for small parameters, the energy levels are approximately but not exactly equally spaced, showing quasi-harmonic behavior.

Contribution

It introduces a detailed analysis of the quantization of branched Hamiltonians associated with the modified Emden equation, highlighting deviations from perfect harmonic spectra.

Findings

Energy spectrum is approximately equispaced for small k

Deviations from perfect harmonicity are characterized and quantified

Analytical perturbation theory confirms numerical results

Abstract

We revisit the canonical quantization to assess the spectrum of the modified Emden equation $\ddot{x} + kx\dot{x} + \omega^2 x + \frac{k^2}{9}x^3 = 0$, which is an isochronous case of the Li\'enard-Kukles equation. While its classical isochronicity and canonical quantization, leading to polynomial solutions with an exactly-equispaced spectrum have been discussed earlier, including in the recent paper [Int. J. Theor. Phys. 64, 212 (2025)], the present study focuses on the quantization of its branched Hamiltonians. For small $k$, we show numerically that the resulting energy spectrum is no longer perfectly harmonic but only approximately equispaced, exhibiting quasi-harmonic behavior characterized by deviations from uniform spacing. Our numerical results are precisely validated by analytical calculations based on perturbation theory.