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arXiv:2604.28127

Nodal algebraic curves and entropy diagnostics in degenerate two-dimensional harmonic-oscillator shells

C A Escobar Ruiz, H Olivares-Pilon, A M Escobar-Ruiz

quant-phSubmitted Apr 30, 2026• Announced May 1, 2026• 0 citations• Revised May 8, 2026 (v2)
v1 · Apr 30, 2026v2 · May 8, 2026v3 · Sep 4, 2026
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Abstract

Degenerate quantum eigenspaces can support substantial changes in nodal geometry at fixed energy. We show that, for the two-dimensional isotropic harmonic oscillator, this restructuring is organized by the Hermite-constrained algebraic curve \(P_N(x,y)=0\) associated with each real shell state, $ψ_N(x,y)=e^{-αr^2/2}P_N(x,y)$. Finite singularities, \(P_N=\nabla P_N=0\), together with projective degeneracies of the leading homogeneous part, identify the strata where topology-changing events can occur. We combine these algebraic criteria with three information diagnostics: the nodal-domain entropy \(S_{\rm dom}\), the Cartesian mutual information \(I(x;y)\), and the entropic uncertainty sum \(S_r+S_p\). The first three shells reveal a clear hierarchy. The \(N=1\) shell only rotates a nodal line; the \(N=2\) shell exhibits a conic transition at \(b^2=2ac\), sharply detected by \(S_{\rm dom}\) but not by global entropies; and the \(N=3\) shell supports cubic close-branch regimes organized by the projective discriminant, with enhanced responses in \(S_{\rm dom}\) and \(I(x;y)\). Thus algebraic stratification, rather than spectral ordering, organizes nodal geometry inside a degenerate eigenspace, while entropy diagnostics quantify the associated probability redistribution and coordinate correlations. The same stratification defines experimentally testable signatures in real-phase Hermite--Gaussian structured light and approximately isotropic trapped motional systems, and suggests a geometry-sensitive verification primitive for fixed-shell bosonic-qudit gates.

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22 pages, 8 figures

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Degenerate quantum eigenspaces can support substantial changes in nodal geometry at fixed energy. We show that, for the two-dimensional isotropic harmonic oscillator, this restructuring is organized by the Hermite-constrained algebraic curve (P_N(x,y)=0) associated with each real shell state, ψN(x,y)=e−αr2/2PN(x,y)ψ_N(x,y)=e^{-αr^2/2}P_N(x,y)ψN​(x,y)=e−αr2/2PN​(x,y). Finite singularities, (P_N=\nabla P_N=0), together with projective degeneracies of the leading homogeneous part, identify the strata where topology-changing events can occur. We combine these algebraic criteria with three information diagnostics: the nodal-domain entropy (S_{\rm dom}), the Cartesian mutual information (I(x;y)), and the entropic uncertainty sum (S_r+S_p). The first three shells reveal a clear hierarchy. The (N=1) shell only rotates a nodal line; the (N=2) shell exhibits a conic transition at (b^2=2ac), sharply detected by (S_{\rm dom}) but not by global entropies; and the (N=3) shell supports cubic close-branch regimes organized by the projective discriminant, with enhanced responses in (S_{\rm dom}) and (I(x;y)). Thus algebraic stratification, rather than spectral ordering, organizes nodal geometry inside a degenerate eigenspace, while entropy diagnostics quantify the associated probability redistribution and coordinate correlations. The same stratification defines experimentally testable signatures in real-phase Hermite--Gaussian structured light and approximately isotropic trapped motional systems, and suggests a geometry-sensitive verification primitive for fixed-shell bosonic-qudit gates.
Abstract Degeneracy allows the nodal structure of a quantum state to vary without changing its energy. We study this effect for real superpositions in the fixed-energy shells of the two-dimensional isotropic harmonic oscillator. In Cartesian coordinates ( x , y ) , any such state in the N th shell has the parametrized form ψ N ( x , y ; c ) = ψ 0 ( x , y ) P N ( x , y ; c ) , where ψ 0 is the non-vanishing ground state and the real coefficients c determine the polynomial factor P N . The nodal set is exactly the algebraic curve P N = 0 . Thus, varying c reshapes this curve at fixed energy. We show that its topology can change only when it develops a finite singular point, P N = ∇ P N = 0 , or when two asymptotic nodal directions merge. We characterize this geometry using three diagnostics: the nodal-domain entropy S dom , the Cartesian mutual information I ( x ; y ) , and the entropic uncertainty sum S r + S p
Parwani and Tabia established in 2006 that in information-theoretic nonlinear extensions of the Schrödinger equation the energy shift of an excited state is carried predominantly by the nodes of the unperturbed wavefunction, while nodeless ground states receive only subleading shifts. This paper shows that for the rotation-invariant branch of that family, regulated by the small components of the Dirac spinor as proposed by Liew and Parwani, the node contribution closes in a single expression: each node of the density contributes exactly −4πC²/ε to the shift integral, where C is the slope of the unnormalised wavefunction at the node and ε is the ratio of the reduced Compton wavelength to the characteristic length of the system. The rule contains no free parameter and covers every bound state of every smooth potential. Two applications follow. For hydrogen it reproduces in three lines the closed coefficient obtained previously by radial integration. For the harmonic oscillator it gives a finite sum over the roots of the Hermite polynomials; the first two evaluate in closed form. The oscillator spectrum ceases to be equidistant, with two parameter-free level-spacing ratios, 1.4261 and 1.7514, which separate this model from the commutator branch, where the corresponding ratios are integers. The relative shift scales as the square root of the trap frequency. Two questions left open in the companion paper are settled: the momentum entering the deformed uncertainty relation is the one a time-of-flight measurement records, and the leading constant of that relation is 4, so that the Gaussian is its minimum-uncertainty state to first order. The same constant governs the entropy-power form, which is uniformly stronger for every non-Gaussian state; the corresponding statement for the product of entropy powers is formulated as a conjecture, with the obstruction to proving it identified. Throughout, other deformations are discussed under an explicit rule: parameters of different frameworks are never compared, only predictions for one and the same measured observable. The paper claims neither the family of equations nor the node mechanism as new, and reports no evidence for a nonzero deformation. Companion paper: https://doi.org/10.5281/zenodo.22072059A working log documenting all calculations and twenty-one self-corrections is included in the files.
The problem of calculating the information entropy in both position and momentum spaces for the nth stationary state of the one-dimensional quantum harmonic oscillator reduces to the evaluation of the logarithmic potential Vn(t)=−∫−∞∞(Hn(x))2 ln |x−t|e−x2 dx at the zeros of the Hermite polynomial Hn(x). Here, a closed analytical expression for Vn(t) is obtained, which in turn yields an exact analytical expression for the entropies when the exact location of the zeros of Hn(x) is known. An inequality for the values of Vn(t) at the zeros of Hn(x) is conjectured, which leads to a new, nonvariational, upper bound for the entropies. Finally, the exact formula for Vn(t) is written in an alternative way, which allows the entropies to be expressed in terms of the even-order spectral moments of the Hermite polynomials. The asymptotic (n≫1) limit of this alternative expression for the entropies is discussed, and the conjectured upper bound for the entropies is proved to be asymptotically valid.
Abstract The Shannon information entropy sum is examined as systems comprised of two-level harmonic oscillator states are passed through dephasing and depolarization quantum channels. We show how the entropy sum increases as the dephasing parameter is increased. The interpretation here is a loss of information in the system as the quantum coherences are removed. This result and interpretation also holds in the depolarization channel as the system transits to the maximally mixed state. The increase in the entropy sum is associated with a delocalization of both position and momentum densities. The behaviour of phase-space Wigner functions and their Shannon entropies is also addressed. Wigner negativity is seen to decrease and disappear with an increase of the dephasing parameter. Here, Wigner negativity is associated with the presence of quantum coherences. The imaginary component of the Wigner function entropy is shown to capture the changes in Wigner negativity. On the other hand, Wigner functions with diagonal input density matrices are non-negative in the depolarization channel. Their real-valued Shannon entropies increase with the depolarization parameter as the system approaches the maximally mixed state. The mutual information between the Wigner function and the product of position and momentum space densities is used to probe the statistical correlation between position and momentum variables. This measure decreases in both channels as the parameter increases. Thus, there is a loss of correlation as quantum coherences are eliminated in the dephasing case, and as the system transits to the maximally mixed case in the depolarizing channel.
We analyse the entropy of Hermite polynomials and orthogonal polynomials for the Freud weights w(x)=exp(-|x|^m) on the real line and show how these entropies are related to information entropy of the one-dimensional harmonic oscillator. The physical interest in such entropies comes from a stronger version of the Heisenberg uncertainty principle, due to Bialynicki-Birula and Mycielski, which is expressed as a lower bound for the sum of the information entropies of a quantum-mechanical system in the position space and in the momentum space.
An alternative method for the description of quantum superintegrable systems in two dimensions through the use of quantum algebraic techniques is introduced. It is suggested that such systems can be described in terms of a generalized deformed oscillator, characterized by a structure function specific to the system. The energy eigenvalues corresponding to a state with finite-dimensional degeneracy can then be determined directly from the properties of the relevant structure function. The validity of the method is demonstrated in the case of the isotropic harmonic oscillator in a space with constant curvature. The method can be used for constructing the quantum versions of several classical superintegrable systems, the Holt potential being given as an example.