2026/07/28 by Vladislav Kargin
Mathematics · #math.OA #msc:15A22 #msc:46L07 #msc:46L54 #msc:47B50 #msc:60B20
87 pages; companion to arXiv:2604.23089
arxiv created 2026/07/28 · arxiv updated 2026/07/30
Let S=∑i=1r Ai⊗ si be a matrix semicircular element, with Hermitian coefficients Ai∈ Mn(ℂ) and free standard semicircular generators si. Its scalar spectral density f is governed, through Speicher's equation (a matrix Dyson equation), by the completely positive covariance map ηS(X)=∑i AiXAi. We treat the singular regime, where the pencil ∑i Ai xi is full but not semisimple and f is unbounded at the origin, in contrast to the bounded real-analytic density of the regular case. We prove three results. (i) The leading singularity exponent at 0 is invariant under congruence Ai↦ bAib* of the pencil (b invertible), and more generally under symmetric scaling of the covariance map. (ii) For binary elements (r=2) we obtain a complete classification: in Lancaster-Rodman canonical form every indecomposable cell is of one of three types, and f(x)∼ c|x|-(n^*-1)/(n^*+1) as x→0 with an explicit constant c, where the exponent depends only on the size of the largest Jordan block (the effective chain length n^*) and not on the coupling. With (i) and the direct-sum behaviour, this classifies all full binary Hermitian pencils. (iii) The spectral classification is strictly coarser than the algebraic one: a Type III cell of size 2m with non-real β and the direct sum of two Type II cells of size m with parameter |β| have identical scalar densities, yet their covariance maps are not symmetrically scalable; the scalar spectrum cannot detect the phase of β. Each type calls for a different method: a reduction of Speicher's equation to an autonomous discrete Painlevé I (McMillan) map (Type I), a Lyapunov-Schmidt reduction at the branch point (Type II), and a gauge reduction by a diagonal unitary (Type III).