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

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, $\psi_N(x,y)=e^{-\alpha r^2/2}P_N(x,y)$. 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.