Exact spectral integrability of a finite-channel random Schrödinger operator: from fermionization of the Dyson--Riccati gas to Kontsevich integral
Abstract
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 β=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→ ∞ expresses the density of states and the total Lyapunov growth rate exactly through a single Airy Hankel τ-function providing multichannel generalization of N=1 Halperin--Thouless formulas. The τ−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≫ 1 the sharp N=∞ edge is softened within an N−4/5 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 β=2 random band matrices.
Related
- Inverse spectral results for the magnetic Schrödinger operator in low dimensions
- Optimization Inverse Spectral Problem for the Schrödinger Operator on a Compact Graph
- A spectral-element method for computing many eigenvalues and their asymptotics of the Schrödinger operator with Robin boundary condition
- Precise spectral asymptotics, exponential localization, and spectral gap estimates for the three-boson lattice Schrödinger operator
- Sharp lower bounds for the periodic maximal Schrödinger operator in higher dimensions
BibTeX
@misc{fyodorov2026exact,
title = {{Exact spectral integrability of a finite-channel random Schrödinger operator: from fermionization of the Dyson--Riccati gas to Kontsevich integral}},
author = {Yan V. Fyodorov and Pierre Le Doussal},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.27154},
eprint = {2610.27154},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.27154},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Exact spectral integrability of a finite-channel random Schrödinger operator: from fermionization of the Dyson--Riccati gas to Kontsevich integral
AU - Yan V. Fyodorov
AU - Pierre Le Doussal
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.27154
UR - https://arxiv.org/abs/2610.27154
AB - 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 β=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→ ∞ expresses the density of states and the total Lyapunov growth rate exactly through a single Airy Hankel τ-function providing multichannel generalization of N=1 Halperin--Thouless formulas. The τ−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≫ 1 the sharp N=∞ edge is softened within an N^−4/5 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 β=2 random band matrices.
ER -