Exact Finite-Channel Schur Moments of the Wigner-Smith Matrix with a Tunnel Barrier

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 $n\le M$. 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 $1\mathrm{/}M$. For the second moment, this contribution exactly reproduces the exponentially small term in the known random-matrix expression for $⟨{\tau }_W^2⟩$ 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 $M\times M$ 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.