带隧穿势垒的Wigner-Smith矩阵的精确有限通道Schur矩
Exact Finite-Channel Schur Moments of the Wigner-Smith Matrix with a Tunnel Barrier
- data cybernetics ssc GmbH
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文针对带隧穿势垒的混沌腔，将Wigner-Smith矩阵的Schur矩分解为理想耦合值与CUE平均势垒因子，导出精确有限通道时延矩公式及对半经典猜想的指数小修正。
AI中文摘要：
我们考虑具有时间反演对称性破缺的混沌腔的Wigner--Smith时延矩阵，该腔具有有限数量$M$的等效开放通道，以及反射概率为$R$的均匀隧穿势垒。非理想耦合的精确随机矩阵结果在联合矩阵分布层面已知，而Schur矩的显式有限通道公式则发展得少得多。我们证明非理想Wigner--Smith矩阵的每个Schur矩都能分解为其理想耦合逆Laguerre值与一个由圆酉系综（CUE）中Schur多项式的平均值给出的势垒因子的乘积。一个对偶Cauchy生成函数将后者化简为具有有理符号的Toeplitz行列式，从而将势垒问题置于经典精确CUE特征多项式比值理论之中。对于单行分拆，专门化精确CUE比值公式给出了所有势垒系数的封闭生成函数，从而给出了允许范围$n\le M$内的所有有限时延矩。精确比值公式分解为一个半经典展开可见的扇区和一个与$R^{M+1}$成比例的额外贡献，该贡献对$1/M$的每个代数阶都不可见。对于第二矩，该贡献精确重现了已知随机矩阵表达式$\langle\tau_W^2\rangle$中半经典展开所缺失的指数小项。我们还推导了所有单列分拆的精确有限和，以及$M\times M$正方形内Schur平均值的精确互补对称性。因此，新贡献不在于基础的CUE比值恒等式（这些是经典的），而在于将它们与精确的非理想Wigner--Smith映射相结合，并提取由此产生的有限通道时延公式以及对半经典Schur矩猜想的修正。
英文摘要：
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 $1/M$. For the second moment, this contribution exactly reproduces the exponentially small term in the known random-matrix expression for $\langleτ_W^2\rangle$ 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.
