Exact spectral integrability of a finite-channel random Schrödinger operator: from fermionization of the Dyson--Riccati gas to Kontsevich integral

We solve exactly, for arbitrary finite channel number $N$, the Riccati dynamics of a quasi-one-dimensional random matrix Schrödinger operator with Hermitian white-noise potential in the $\beta =2$ unitary symmetry class. The associated Dyson–Riccati eigenvalue gas admits an exact free-fermion representation at any sample length $L$, which in the limit $L\to \mathrm{\infty }$ expresses the density of states and the total Lyapunov growth rate exactly through a single Airy Hankel $\tau$-function providing multichannel generalization of $N=1$ Halperin–Thouless formulas. The $\tau -$function further admits an outgoing-contour Kontsevich matrix Airy integral representation. Its saddle-point and double-scaling analyses give, in a unified way, the finite-channel density Lifshitz tail, and the large-$N$ spectral edge. In particular, for $N\gg 1$ the sharp $N=\mathrm{\infty }$ edge is softened within an $N^{-4\mathrm{/}5}$ spectral window, with its profile governed by the tritronquée solution of Painlevé I. We conjecture universality of this quasi-one-dimensional scaling edge profile for a broad class of $\beta =2$ random band matrices.