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

Abstract

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 2 i k L U L U R (with U L = U R T in the presence of time reversal symmetry), and introduce a novel symmetrisation procedure for the Wigner–Smith matrix: Q ̃ = U R Q U R † = ( 2 L / v ) 1 N − i U L † ∂ ω U L U R U R † , 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 (2001 Physica E 9 463). For finite length L, the exponential functional representation is used to calculate the first moments ⟨ t r ( Q ) ⟩ , ⟨ t r ( Q 2 ) ⟩ and ⟨ t r ( Q ) 2 ⟩ . Finally we derive a partial differential equation for the resolvent g ( z ; L ) = lim N → ∞ ( 1 / N ) t r z 1 N − N Q − 1 in the large N limit.