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Quasi-harmonic spectra from branched Hamiltonians

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

TL;DR

This work analyzes the quantum spectra of isochronous Liénard systems, focusing on the modified Emden equation and its branched Hamiltonians. Using von Roos operator ordering and two branches corresponding to $\ell=-\tfrac{1}{3},-\tfrac{2}{3}$, the authors derive and study the resulting Schrödinger problems in momentum space, revealing that branched quantization generally yields quasi-harmonic spectra with spacings near $2\omega$ rather than exact equidistance. For $\epsilon=\tfrac{1}{4}$, polynomial/truncated solutions can exist under a specific Hermite condition, while for $\epsilon\neq \tfrac{1}{4}$ the spectrum requires numerical or perturbative treatment; in the small-$k$ regime, perturbation theory provides analytic corrections to the isotonic baseline and matches numerics for low-lying levels. The results show that classical isochrony does not necessarily imply a perfectly harmonic quantum spectrum in branched Hamiltonian systems, highlighting nuanced spectral features of momentum-dependent mass Hamiltonians with displaced centers or singular potentials. The findings advance understanding of quasi-harmonic behavior in nonstandard quantizations of nonlinear oscillators and inform broader studies of branched or position-momentum-structured quantum systems.

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.

Quasi-harmonic spectra from branched Hamiltonians

TL;DR

This work analyzes the quantum spectra of isochronous Liénard systems, focusing on the modified Emden equation and its branched Hamiltonians. Using von Roos operator ordering and two branches corresponding to , the authors derive and study the resulting Schrödinger problems in momentum space, revealing that branched quantization generally yields quasi-harmonic spectra with spacings near rather than exact equidistance. For , polynomial/truncated solutions can exist under a specific Hermite condition, while for the spectrum requires numerical or perturbative treatment; in the small- regime, perturbation theory provides analytic corrections to the isotonic baseline and matches numerics for low-lying levels. The results show that classical isochrony does not necessarily imply a perfectly harmonic quantum spectrum in branched Hamiltonian systems, highlighting nuanced spectral features of momentum-dependent mass Hamiltonians with displaced centers or singular potentials. The findings advance understanding of quasi-harmonic behavior in nonstandard quantizations of nonlinear oscillators and inform broader studies of branched or position-momentum-structured quantum systems.

Abstract

We revisit the canonical quantization to assess the spectrum of the modified Emden equation , 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 , 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.
Paper Structure (10 sections, 1 theorem, 37 equations, 4 figures, 3 tables)

This paper contains 10 sections, 1 theorem, 37 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Hamiltonian aspects
  3. Canonical quantization of the branched Hamiltonians
  4. Possibility of a polynomial solution
  5. Non-polynomial solutions and energy quantization
  6. $\epsilon = \frac{1}{4}$
  7. $\epsilon\neq\frac{1}{4}$
  8. Perturbation theory for small anharmonic corrections
  9. Discussion
  10. Uniqueness of the system satisfying the conditions (\ref{['Cond']}) and (\ref{['Chiellini_condition']}) when $f(x)$ and $g(x)$ are polynomials

Key Result

Proposition A.1

Let $f(x)$ be a real polynomial and $I(x)=\int_0^x x' f(x') dx'$. Then the isochronicity condition (Cond) which is equivalent to with fixed $\omega > 0$, is compatible with the Chiellini condition (Chiellini_condition) if and only if $f(x)=kx$. In particular, no polynomial of degree $\ge 2$ and no affine linear $kx+b$ with $b\ne 0$ works.

Figures (4)

  • Figure 1: Effective potentials $V^\pm_{\rm eff}(\xi)$ for $\epsilon = \frac{1}{4}$.
  • Figure 2: Effective potentials $V^\pm_{\rm eff}(\xi)$ for $\epsilon = \frac{1}{2}$.
  • Figure 3: Eigenfunctions $(\phi_+)_n(\xi)$ for $n=0,1,2$, taking $\omega = 10$, $k=1$, and $\epsilon = \frac{1}{2}$. The corresponding eigenvalues read $E^+_0 \simeq 16.85$, $E^+_1 \simeq 36.75$, and $E^+_2 \simeq 56.68$.
  • Figure 4: Eigenfunctions $(\phi_-)_n(\xi)$ for $n=0,1,2$, taking $\omega = 10$, $k=1$, and $\epsilon = \frac{1}{2}$. The corresponding eigenvalues read $E^-_0 \simeq 17.29$, $E^-_1 \simeq 37.39$, and $E^-_2 \simeq 57.46$.

Theorems & Definitions (1)

  • Proposition A.1