the mean value for both quantities $W_{K}$ and $W_{\Gamma}$ as expressed in (\[mean\]). It also shows how well the classical dynamics describes a wave operator such as $W_{K}$ down to a detailed description of the complex phase space structure present in a mixed system like the kicked Harper map. Note, that the wave fluctuations due to higher iterates of the map are suppressed here due to the 10% damping ($v=0.9$) introduced.

Fluctuations of $\hat{K}$ and $\hat{\Gamma}$ about the mean {#sec:var}
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Although $\hat{K}$ and $\hat\Gamma$ have the same average, the fluctuations about the mean are much greater for $\hat{\Gamma}$ than for $\hat{K}$ as illustrated in Fig. \[fig:kr:k1:a\], an effect which is greatly amplified in the weak damping limit. In this section we quantify this effect for the simple model $\hat{T}=v\hat{U}$ introduced in Sec. \[sec:QKH\] by calculating mean values for $\Tr \hat{\Gamma}^2$ and $ \Tr \hat{K}^2$. Formally, these also provide us with variances of individual matrix elements $K_{ij}$ and $\Gamma_{ij}$ in any given basis, using, for example, $$\label{matrixelement}
\langle|\Gamma_{ij}|^2\rangle_N = 
\frac{1}{N^2}
\Tr \hat{\Gamma}^2.$$ Here $\langle \cdot \rangle_N$ denotes an average over the $N$-dimensional basis on the space on which $\hat{T}$ is defined.

Explicit expressions can be given for these quantities, as 