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

Abstract

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 S(ω), encoding the probability amplitudes to be scattered from one channel to another. The Wigner-Smith time delay matrix Q = −i S † ∂ ω S is another important matrix, which encodes temporal aspects of the scattering process. In order to study its statistical properties, we split the scattering matrix in terms of two unitary matrices, S = e 2ikL U L U R (with U L = U T R in the presence of time reversal symmetry), and introduce a novel symmetrisation procedure for the Wigner-Smith matrix :where k is the wave vector and v the group velocity. We demonstrate that Q can be expressed under the form of an exponential functional of a matrix Brownian motion. For semi-infinite wires, L → ∞, using a matricial extension of the Dufresne identity, we recover straightforwardly the joint distribution for Q's eigenvalues of Brouwer and Beenakker [Physica E 9, 463 (2001)]. For finite length L, the exponential functional representation is used to calculate the first moments tr(Q) , tr(Q 2 ) and tr(Q)2 . Finally we derive a partial differential equation for the resolvent g(z; L) = lim N →∞ (1/N ) tr z 1 N − N Q −1 in the large N limit.