Unrestricted Linear Stability of the Massless Skyrmion
Year: 2026
Type: preprint
Abstract: Unrestricted Linear Stability of the Massless Skyrmion Description / Abstract This record contains the manuscript, exact rational profile data, and computer-assisted verification scripts proving unrestricted three-dimensional linear orbital stability for the degree-one massless $SU(2)$ Skyrmion. We establish nonnegativity and nondegeneracy, modulo three spatial translations and three internal isorotations, of the full three-dimensional static Hessian at the degree-one massless hedgehog solution. The static quadratic form is shown to be coercive in the homogeneous energy space $\dot{H}^1(\mathbb{R}^3; \mathfrak{su}(2))$ on the orthogonal complement of the six-dimensional symmetry kernel. Consequently, the relativistic linearized wave evolution is orbitally stable in $\dot{H}^1 \times L^2$ for arbitrary finite-energy three-dimensional perturbations, removing the corotational symmetry restriction of prior stability results. The proof decomposes perturbations across all grand-spin angular channels $K \ge 0$ into decoupled polar and axial parity sectors. The axial quadratic form is manifestly nonnegative for all $K \ge 1$, with its $K = 1$ kernel spanned by the three isorotations. The radial $K = 0$ sector uses the spectral nonnegativity theorem of Creek, Donninger, Schlag, and Snelson, extended to exclude homogeneous $\dot{H}^1$ zero resonances. In the coupled $K = 1$ polar sector, we construct an exact analytical matrix ground-state factorization into a nonnegative integral of squares using the translation Jacobi field and a vacuum-changing infinitesimal chiral Jacobi field. Because the chiral mode approaches a nonzero constant at infinity, it does not belong to $\dot{H}^1$, leaving translations as the unique physical kernel in $K = 1$. For all higher polar channels ($K \ge 2$), a scalar Liouville transformation reduces the quadratic form to a $2 \times 2$ comparison matrix $M_\Lambda(r)$ that increases monotonically with $\Lambda = K(K + 1)$ for $\Lambda \ge 6$. Using the published rigorous Chebyshev rational polynomial profile enclosure of Creek, Donninger, Schlag, and Snelson, a standard-library Python verifier executing 128-bit outward-rounded integer arithmetic across 8,192 closed boxes on the compactified radial domain certifies $M_\Lambda(r) \ge \frac{3}{64r^2} I$ over the entire half-line. Because the massless essential spectrum begins at zero, a compact weighted embedding $\dot{H}^1 \hookrightarrow L^2(W\,dx)$ using the $O(r^{-6})$ decay of the background weight upgrades channelwise nonnegativity and kernel identification to full homogeneous $\dot{H}^1$ coercivity.
Author: Alexander, Cohen
Institution: Stanford University
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225579020