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\begin{document}

\title{Quantum dynamics for a triadic Cantor set spectrum}

\author{A. Soret\dag, E. Gurevich \ddag\ and E. Akkermans\ddag}


\address{$^{1}$Department of Physics, Technion Israel Institute of Technology, 32000 Haifa, Israel}

\date{\today}


\begin{abstract}

We study the dynamics of a wave packet for a triadic Cantor spectrum $K$. The scaling symmetry of the spectrum translates into the characteristic log-periodic feature of relevant physical quantities. Using Mellin transform, we derive the analytical expression of the time-averaged return probability, which is universal for any system with a Cantor spectrum. This result is supported by numerical results obtained recently for polaritons in a Fibonacci potential. We give a more thorough analysis in the case of the tight-binding model, in which case the dynamics of the system depend essentially on the Fourier transform on a Cantor set of monomes $x^k$, namely : $h_k(t) = \int_K e^{-ixt}x^kd\mu$, $k \in \mathbb{Z}$, which carry the fingerprint of the spectral fractal feature.


\end{abstract}

\maketitle


\section{Introduction}
\section{Time-averaged return probability}
\subsection{Analytical expression}

Note that $C(t)$ verifies asymptotically the functional equation of a scaling function : 
$$C(t) \sim \frac{1}{2}C(t/3)$$
 
\subsection{Numerical results for the Fibonacci cavity}
\section{Generalization to other quantities}

Time-averaging is equivalent to taking the wavelet transform of the spectrum, which we have shown to display a characteristic behaviour. Therefore, it is expected that other time-averaged dynamical quantities 
%$$\langle |\psi(x,t)|^2\rangle = \iint_{K}g(\epsilon')^{*}g(\epsilon)\phi_{\epsilon'}(x)^{*}\phi_{\epsilon}(x)sinc((\epsilon-\epsilon')t)d\mu d\mu$$
%where $\langle |\psi(x,t)|^2\rangle = \frac{1}{t}\int_0^t |\psi(x,t')|^2 dt'$.
would also display the same behaviour. To illustrate this : \\
- participation ratio (Fibonacci potential and triadic Cantor set spectrum)  \\
- RMS (triadic Cantor set spectrum).\\

However, only qualitative argument : more information about the eigenfunctions is required to study rigorously the cases of the participation ratio and of the RMS. 

\section{Tight-binding model}

Allows us to obtain more information on the eigenfunctions themselves, in order to study non-spectral quantity.

\medskip

The dynamics of the system depends essentially on the Fourier transforms of monomes : 
$$ h_k(t) = \int_K e^{-i x t}x^k d\mu(x)$$ 

- These functions are not scaling functions, but display a structure, which translates in some dynamical quantities\\
- Application to the RMS \\
- taking the time-average

\section{Conclusion}
\section{References}


\end{document}



