Wigner-Smith matrix, exponential functional of the matrix Brownian motion and matrix Dufresne identity

We consider a multichannel wire with a disordered region of length $L$ and a reflecting boundary. The reflection of a wave of frequency $omega$ is described by the scattering matrix $mathcal{S}(omega)$, encoding the probability amplitudes to be scattered from one channel to another. The Wigner-Smith time delay matrix $mathcal{Q}=-mathrm{i}, mathcal{S}^daggerpartial_omegamathcal{S}$ 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, $mathcal{S}=mathrm{e}^{2mathrm{i}kL}mathcal{U}_Lmathcal{U}_R$ (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: $widetilde{mathcal{Q}} =mathcal{U}_R,mathcal{Q},mathcal{U}_R^dagger = (2L/v),mathbf{1}_N -mathrm{i},mathcal{U}_L^daggerpartial_omegabig(mathcal{U}_Lmathcal{U}_Rbig),mathcal{U}_R^dagger$, 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, $Ltoinfty$, 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 $langlemathrm{tr}(mathcal{Q})rangle$, $langlemathrm{tr}(mathcal{Q}^2)rangle$ and $langlebig[mathrm{tr}(mathcal{Q})big]^2rangle$. Finally we derive a partial differential equation for the resolvent $g(z;L)=lim_{Ntoinfty}(1/N),mathrm{tr}big{big( z,mathbf{1}_N - N,mathcal{Q}big)^{-1}big}$ in the large $N$ limit.