an:06464958
Zbl 1322.81050
Novaes, Marcel
Statistics of time delay and scattering correlation functions in chaotic systems. I: Random matrix theory
EN
J. Math. Phys. 56, No. 6, 062110, 6 p. (2015).
00346670
2015
j
81Q50 81Q20 81U35 15B52 60B20
random matrix theory; scattering theory
Given a quantum mechanical, chaotic cavity with M open channels the scattering matrix \(S(E)\) is an energy dependent, complex \(M\times M\) matrix and the Wigner time delay matrix is defined as \( Q:= -i\hbar S^\dagger\frac{dS}{dE}\). In this article the author calculates, for a system without time reversibility, the random matrix expectation values \(\langle s_\lambda(Q)\rangle\) where \(s_\lambda(Q)\) is a general Schur function of \(Q\). From these expressions random matrix expectation values for general polynomials
\[
\mathcal M_{n_1,n_2,\ldots} = \frac{1}{M}\text{Tr}[Q^{n_1}]\frac{1}{M}\text{Tr}[Q^{n_2}]\ldots
\]
can be derived. In particular, for the time delay moments \(\mathcal M_n\), the formula
\[
\langle\mathcal M_n\rangle =\tau_D\frac{M^{n-1}} {n!}\sum_{k=0}^{n-1} (-1)^k\binom{n-1}{k}\frac{[M-k]^n}{[M+k]_n}
\]
is obtained. Here, \(\tau_D\) is the classical dwell time, and
\[
[x]^n = x(x+1)\dots(x+n-1),~~[x]_n = x(x-1)\dots(x-n+1)
\]
are raising and falling factorials. In a subsequent article of the author [J. Math. Phys. 56, No. 6, Article ID 062109, 14 p. (2015; Zbl 1322.81049)], these expressions for \(\langle\mathcal M_n\rangle\) are compared to semiclassical approximations for energy averages of the time delay moments.
Tobias Weich (Paderborn)
Zbl 1322.81049