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arXiv:2512.22510

Quasi-harmonic spectra from branched Hamiltonians

Aritra Ghosh, Bijan Bagchi, A. Ghose-Choudhury, Partha Guha, Miloslav Znojil

quant-phmath-phnlin.SISubmitted Dec 27, 2025• Announced Dec 30, 2025
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Abstract

We revisit the canonical quantization to assess the spectrum of the modified Emden equation $\ddot{x} + kx\dot{x} + ω^2 x + \frac{k^2}{9}x^3 = 0$, which is an isochronous case of the Liénard-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.

Journal reference
Phys. Lett. A 570, 131289 (2026)
DOI
10.1016/j.physleta.2025.131289

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We revisit the canonical quantization to assess the spectrum of the modified Emden equation x¨+kxx˙+ω2x+k29x3=0\ddot{x} + kx\dot{x} + ω^2 x + \frac{k^2}{9}x^3 = 0x¨+kxx˙+ω2x+9k2​x3=0, which is an isochronous case of the Liénard-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 kkk, 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.
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In this paper, we present an analysis of the equation ẍ−(1/2x)ẋ2+2ω2x−1/8x=0, where ω > 0 and x = x(t) is a real-valued variable. We first discuss the appearance of this equation from a position-dependent-mass scenario in which the mass profile goes inversely with x, admitting a singularity at x = 0. The associated potential is also singular at x = 0, splitting the real axis into two halves, i.e., x > 0 and x < 0. The dynamics is exactly solvable for both the branches and so for definiteness, we stick to the x > 0 branch. Performing a canonical quantization in the position representation and upon employing the ordering strategy of the kinetic-energy operator due to von Roos, we show that the problem is isospectral to the isotonic oscillator. Thus, the quantum spectrum consists of an infinite number of equispaced levels. The spacing between the energy levels is found to be insensitive to the specific choices of the ambiguity parameters that are employed for ordering the kinetic-energy operator à la von Roos.
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