论文标题
Wigner-Smith矩阵，矩阵布朗运动和矩阵dufresne身份的指数功能
Wigner-Smith matrix, exponential functional of the matrix Brownian motion and matrix Dufresne identity
论文作者
论文摘要
我们考虑了一条多通道电线，其长度为$ l $的无序区域和反射边界。频率$ω$的浪潮的反射由散射矩阵$ \ Mathcal {s}（ω）$描述，编码要散布在一个通道到另一个通道的概率幅度。 Wigner-Smith时间延迟矩阵$ \ MATHCAL {q} = - \ MATHRM {i} \，\ MATHCAL {S}^\ DAGGE \ DAGGE \ DAGGE \partial_Ω\ Mathcal {S} $是另一个重要的矩阵编码散射过程的时间范围。为了研究其统计属性，我们根据两个单位矩阵分配了散射矩阵，$ \ Mathcal {s} = \ Mathrm {e}^{2 \ Mathrm {i} kl} \ Mathcal} \ Mathcal {U} $ \ MATHCAL {U} _l = \ MATHCAL {U} _r^\ MATHRM {T} $在TRS的存在下），并为Wigner-Smith矩阵介绍一个新颖的对称过程：$ \ widetilde { = \ Mathcal {U} _r \，\ Mathcal {Q} \，\ Mathcal {u} _r^\ dagger =（2l/v）\，\ Mathbf {1} _n - \ Mathrm {i} \，\ Mathcal {u} _l^\ dagger \partial_Ω我们证明$ \ widetilde {\ Mathcal {q}} $可以以矩阵布朗尼运动的指数功能的形式表示。对于半无限电线，$ l \ to \ infty $，使用Dufresne身份的临床扩展，我们直接恢复了$ \ Mathcal {q} $ of Brouwer和Brouwer和Bayakker的特征的联合分布[Physica E 9（2001）p。 463]。对于有限的长度$ l $，指数函数表示形式用于计算第一瞬间$ \ langle \ langle \ mathrm {tr}（\ Mathcal {q}）\ rangle $，$ \ langle \ langle \ sathrm {trm {tr}（ $ \ langle \ big [\ mathrm {tr}（\ mathcal {q}）\ big]^2 \ rangle $。最后，我们得出了分解$ g（z; l）= \ lim_ {n \ to \ infty}（1/n）\，\ mathrm {tr} \ big \ big \ big {z \， N \，\ Mathcal {q} \ big）^{ - 1} \ big \} $在大$ n $ limit中。
We consider a multichannel wire with a disordered region of length $L$ and a reflecting boundary. The reflection of a wave of frequency $ω$ is described by the scattering matrix $\mathcal{S}(ω)$, encoding the probability amplitudes to be scattered from one channel to another. The Wigner-Smith time delay matrix $\mathcal{Q}=-\mathrm{i}\, \mathcal{S}^\dagger\partial_ω\mathcal{S}$ is another important matrix encoding temporal aspects of the scattering process. In order to study its statistical properties, we split the scattering matrix in terms of two unitary matrices, $\mathcal{S}=\mathrm{e}^{2\mathrm{i}kL}\mathcal{U}_L\mathcal{U}_R$ (with $\mathcal{U}_L=\mathcal{U}_R^\mathrm{T}$ in the presence of TRS), and introduce a novel symmetrisation procedure for the Wigner-Smith matrix: $\widetilde{\mathcal{Q}} =\mathcal{U}_R\,\mathcal{Q}\,\mathcal{U}_R^\dagger = (2L/v)\,\mathbf{1}_N -\mathrm{i}\,\mathcal{U}_L^\dagger\partial_ω\big(\mathcal{U}_L\mathcal{U}_R\big)\,\mathcal{U}_R^\dagger$, where $k$ is the wave vector and $v$ the group velocity. We demonstrate that $\widetilde{\mathcal{Q}}$ can be expressed under the form of an exponential functional of a matrix Brownian motion. For semi-infinite wires, $L\to\infty$, using a matricial extension of the Dufresne identity, we recover straightforwardly the joint distribution for $\mathcal{Q}$'s eigenvalues of Brouwer and Beenakker [Physica E 9 (2001) p. 463]. For finite length $L$, the exponential functional representation is used to calculate the first moments $\langle\mathrm{tr}(\mathcal{Q})\rangle$, $\langle\mathrm{tr}(\mathcal{Q}^2)\rangle$ and $\langle\big[\mathrm{tr}(\mathcal{Q})\big]^2\rangle$. Finally we derive a partial differential equation for the resolvent $g(z;L)=\lim_{N\to\infty}(1/N)\,\mathrm{tr}\big\{\big( z\,\mathbf{1}_N - N\,\mathcal{Q}\big)^{-1}\big\}$ in the large $N$ limit.
