Exact Finite-Channel Schur Moments of the Wigner-Smith Matrix with a Tunnel Barrier
Abstract
We consider the Wigner--Smith time-delay matrix of a chaotic cavity with broken time-reversal symmetry, a finite number M of equivalent open channels, and a uniform tunnel barrier of reflection probability R. Exact random-matrix results for nonideal coupling are known at the level of the joint matrix distribution, whereas explicit finite-channel formulas for Schur moments are much less developed. We show that every Schur moment of the nonideal Wigner--Smith matrix factorizes into its ideal-coupling inverse-Laguerre value and a barrier factor given by a circular unitary ensemble (CUE) average of a Schur polynomial. A dual-Cauchy generating function reduces the latter to a Toeplitz determinant with a rational symbol, placing the barrier problem within the classical theory of exact CUE characteristic-polynomial ratios. For one-row partitions, specializing the exact CUE ratio formula gives a closed generating function for all barrier coefficients and therefore all finite delay-time moments in the allowed range n≤ M. The exact ratio formula decomposes into a sector visible to the semiclassical expansion and an additional contribution proportional to RM+1, which is invisible to every algebraic order in 1/M. For the second moment, this contribution exactly reproduces the exponentially small term in the known random-matrix expression for ⟨τW2⟩ that is absent from the semiclassical expansion. We also derive an exact finite sum for all one-column partitions and an exact complement symmetry for Schur averages inside the M× M square. The new contribution therefore lies not in the underlying CUE ratio identities, which are classical, but in combining them with the exact nonideal Wigner--Smith mapping and extracting the resulting finite-channel time-delay formulas and corrections to semiclassical Schur-moment conjectures.
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BibTeX
@misc{araujo2026exact,
title = {{Exact Finite-Channel Schur Moments of the Wigner-Smith Matrix with a Tunnel Barrier}},
author = {Israel F. Araujo},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.04170},
eprint = {2610.04170},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.04170},
}
RIS
TY - UNPB
TI - Exact Finite-Channel Schur Moments of the Wigner-Smith Matrix with a Tunnel Barrier
AU - Israel F. Araujo
PY - 2026
DA - 2026/10/03
PB - arXiv
DO - 10.48550/arxiv.2610.04170
UR - https://arxiv.org/abs/2610.04170
AB - We consider the Wigner--Smith time-delay matrix of a chaotic cavity with broken time-reversal symmetry, a finite number M of equivalent open channels, and a uniform tunnel barrier of reflection probability R. Exact random-matrix results for nonideal coupling are known at the level of the joint matrix distribution, whereas explicit finite-channel formulas for Schur moments are much less developed. We show that every Schur moment of the nonideal Wigner--Smith matrix factorizes into its ideal-coupling inverse-Laguerre value and a barrier factor given by a circular unitary ensemble (CUE) average of a Schur polynomial. A dual-Cauchy generating function reduces the latter to a Toeplitz determinant with a rational symbol, placing the barrier problem within the classical theory of exact CUE characteristic-polynomial ratios. For one-row partitions, specializing the exact CUE ratio formula gives a closed generating function for all barrier coefficients and therefore all finite delay-time moments in the allowed range n≤ M. The exact ratio formula decomposes into a sector visible to the semiclassical expansion and an additional contribution proportional to R^M+1, which is invisible to every algebraic order in 1/M. For the second moment, this contribution exactly reproduces the exponentially small term in the known random-matrix expression for ⟨τ_W^2⟩ that is absent from the semiclassical expansion. We also derive an exact finite sum for all one-column partitions and an exact complement symmetry for Schur averages inside the M× M square. The new contribution therefore lies not in the underlying CUE ratio identities, which are classical, but in combining them with the exact nonideal Wigner--Smith mapping and extracting the resulting finite-channel time-delay formulas and corrections to semiclassical Schur-moment conjectures.
ER -