an:06464957
Zbl 1322.81049
Novaes, Marcel
Statistics of time delay and scattering correlation functions in chaotic systems. II: Semiclassical approximation
EN
J. Math. Phys. 56, No. 6, 062109, 14 p. (2015).
00346670
2015
j
81Q50 81Q20 81U35 15B52 60B20
time delay matrix; semiclassical approximation; scattering system
The scattering properties of a quantum mechanical, chaotic cavity with \(M\) open channels can be described by its scattering matrix \(S(E)\) which is an energy dependent, complex \(M\times M\) matrix. In this article, the correlation function
\[
C_n(\epsilon,M) := \frac{1}{M}\left\langle\text{Tr}\left[S^\dagger\left(E-\frac{\epsilon\hbar}{2\tau_D}\right)S\left(E+\frac{\epsilon\hbar}{2\tau_D}\right)\right]^n\right\rangle,
\]
where \(\tau_D\) is the classical dwell time and \(\langle\bullet \rangle\) denotes the average over \(E\), is studied for a system without time reversibility and a semiclassical approximation is calculated. This calculation is based on a semiclassical approximation of the scattering matrix which takes into account a systematic pairing of classical trajectories and which has been established in physics literature [\textit{R. A. Jalabert et al.}, Phys. Rev. Lett. 65, 2442 (1990); \textit{K. Richter} and \textit{M. Sieber}, Phys. Rev. Lett. 89, 206801 (2002); \textit{S. M??ller et al.} New J. Phys. 9, 12 (2007)].
From the semiclassical expression of \(C_n(\epsilon,M)\), the author derives an expression for the energy averages of the time delay moments \(\langle\mathcal M_m\rangle\) where \(\mathcal M_m := \frac{1}{M}\text{Tr}(Q^m)\) and \(Q\) being the Wigner time delay matrix \(Q:= -i\hbar S^\dagger\frac{dS}{dE}\). The explicit expression of \(\langle \mathcal M_m\rangle\) is of complicated nature. It is nevertheless possible to compare the expression for the first 8 moments (\(m\leq 8\)) to expressions which have been obtained by the author via random matrix theory [J. Math. Phys. 56, No. 6, 062110, 6 p. (2015; Zbl 1322.81050)] and show that they coincide.
Tobias Weich (Paderborn)
Zbl 1322.81050