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.
Categories: quant-ph math-ph math.MP nlin.SI