What are this anomalous skin effect described by Mattis-Bardeen? What was the phenomenological evidence of this anomalous skin effect [1]? What experiment have been made to show it?
[1] Mattis, D. C., and John Bardeen. "Theory of the anomalous skin effect in normal and superconducting metals." Physical Review 111.2 (1958): 412.
 A: The anomalous skin effect was first described by De Haas-van Alphen during their measure of low temperature magnetization in a sample of Bismunto and consist in anomalous magnetization oscillation that are impossible to derive via classical way. 
The first description of this phenomenon was given by Onsager that using the Landau quantization of the energy levels of free electron gas in hard magnetic field, called Landau Tubes to calculate a relation between the magnetic field and the oscillations period.
The microscopic description of this phenomenon was given Mattis-Bardeen [1] using the BCS framework and the final equation for the complex conductivity in superconducting thin films was given by [2]. Using all of this things is possible to compute the ratio of the surface impedance in the extreme anomalous limit in the superconductivity state to that in the normal state. The ratio is defined as
\begin{equation}
\frac{Z_s}{Z_N}=\left(\frac{\sigma_1-i\sigma_2}{\sigma_N}\right)^{1/3}.
\end{equation}
This expression should be useful not only for interpretation of measurements on the anomalous skin effect but also for absorption in thin superconducting films.
[1] Mattis, D. C., and John Bardeen. "Theory of the anomalous skin effect in normal and superconducting metals." Physical Review 111.2 (1958): 412.
[2] Glover III, R. E., and Ms Tinkham. "Conductivity of Superconducting Films for Photon Energies between 0.3 and 4 0 k T c." Physical Review 108.2 (1957): 243.
