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\section*{Notes du 1 juin 2015 - quantum dynamics for a Cantor spectrum, tight-binding model}

\subsection*{Introduction}

This is a summary of the discussion of June 1, on quantum dynamics for the tight-binding model, with some complementary results obtained since. After the discussion, we decided that the following questions should be examined :\\
- the dynamics of the wave packet depends on the functions : $ h_k(t) = \int_K e^{-i\epsilon t}\epsilon^k d\mu(\epsilon)$, which are not scaling functions, but do have a certain structure ; can we approximate the $h_k(t)$ by scaling functions, thus implying that the RMS is, in first approximation, a scaling function ?\\
- if not, how does the characteristics of the $h_k(t)$ affect the dynamics ?\\

After further numerical study, I found that the $h_k(t)$ can not easily be approximated by scaling functions. However, a numerical study shows that there is clearly an almost-proportional relation between $h_k(t)$ and $h_{k-1}(t/3)$, which can be justified theoretically by the relations between the zeroes and maxima of the $h_k(t)$ ; I have no better theoretical explanation so far. \\
If the $h_k(t)$ were scaling functions, so would be the RMS. That is, if we assume that :$h_k(t)\propto h_{k-1}(t/3)$, we deduce that the RMS is of the form : $\Delta x(t) = t^{-\ln 2 /\ln 3}G(\frac{\ln t}{\ln 3})$. However, since the $h_k(t)$ are not scaling functions, we do not expect $\Delta x (t)$ to have this exact behaviour, and in fact a numerical study of $\Delta x(t)$ shows that this quantity is not of that form. Nevertheless, the structure of the $h_k(t)$ (scaling relations between zeroes and maxima) does translate to the RMS. In fact, I found numerically that the RMS is a power-law modulated by oscillations, which appear to have a log-periodic envelope. 

\medskip

I give first a reminder of the framework and of previous results, then I discuss and give a partial answer to the questions above. I also did a complementary study of the Fibonacci potential, to compare the behaviour of the $h_k(t)$ in the case of a triadic Cantor spectrum and in the case of a Fibonacci potential.

\subsection*{Reminder of previous results}

We consider an infinite 1D lattice, with sites $k$ labelled from $0$ to $\infty$. We define a potential $V$ on this system such that the discrete Schrodinger operator : $H = -\Delta+V$, which acts on $l_2(\mathbb{Z})$, has a triadic Cantor set spectrum. The eigenfunctions of $H$ verify : 
$$H\phi_{\epsilon} = \epsilon\phi_{\epsilon}$$
Using transfer matrices, the above equation becomes :
$$\begin{pmatrix} \phi_{\epsilon}(k) \\
   \phi_E(k-1)  \\
  \end{pmatrix} = \begin{pmatrix}
  
   2+V_k - \epsilon & -1  \\
    1  & 0  \\
   \end{pmatrix} \begin{pmatrix}
   
    \phi_{\epsilon}(k) \\
     \phi_{\epsilon}(k-1)  \\
    \end{pmatrix}$$ 
 Imposing $\phi_{\epsilon}(0) = 0$, we find that the eigenfunctions $\{\phi_{\epsilon}\}_{\epsilon \in K}$, evaluated on the sites $k$, can be seen as polynomials in $\epsilon$ : 
\begin{equation}
\phi_{\epsilon}(k) = \phi_{\epsilon}(1)p_k(\epsilon)
\end{equation}
with 
 $$\begin{pmatrix}
      
       p_k(E) & q_k(E) \\
        p_{k-1}(E) & q_{k-1}(E)  \\
       \end{pmatrix} = \prod_{j=k}^{1} \begin{pmatrix}
        
      2+V_j - E & -1  \\
         1  & 0  \\
      
         \end{pmatrix}$$
We are interested of the evolution of a wave packet, described by the normalized wave function $\psi(t)$, evolving according to Schrodinger's equation :
$$i\frac{d\psi(t)}{dt} = H\psi(t)$$
The wave function is initially localized at the site 1 : $\langle k |\psi(t=0)\rangle = \delta(k-1)$. At time $t$, the system is in the state : $|\psi (t)\rangle = e^{-iHt}|\psi(0)\rangle$, and the value of the wave function at site $k$ is given by : 
$$|\psi(k,t)| = \int_K \phi_{\epsilon}(1)^{*}\phi_{\epsilon}(k)e^{-i\epsilon t}d\mu$$  
Using (1), we obtain :
\begin{eqnarray*}
\lefteqn{\psi(k,t) = \int_{K}\phi_{\epsilon}(1)^{*}\phi_{\epsilon}(k)e^{-i\epsilon t}d\mu}\\
& = & \int_K p_1(\epsilon)p_{k}(\epsilon)e^{-i\epsilon t}d\mu \\
& = & \sum_{j_k\leq k}c_{j_k}\int_{K}\epsilon^{j_k}e^{-i\epsilon t}d\mu
\end{eqnarray*}
with $p_k(x) = \sum_{0\leq j_k \leq k} c_{j_k}x^{j_k}$
Thus, the dynamical behaviour of the system depends essentially on the functions :
$$ h_k(t) = \int_K e^{-i\epsilon t}\epsilon^k d\mu(\epsilon)$$
aka the Fourier transforms on $K$ of the moments $x^k$.

\medskip


Expending the exponential in $h_k(t)$, we obtain :
$$ h_k(t) = \sum_{n\leq 0} \frac{(-it)^n}{n!}\mu_{n+k}$$
with $\mu_n = \int_K x^n d\mu$.

The fingerprint of the scaling symmetry of the spectrum is carried by the $\mu_n = \int_K x^n d\mu$. It can be shown that, for $n\to \infty$ (see [1])
$$\mu_n = n^{-\ln 2 / \ln 3} F(\ln n/\ln 3)(1+ O(\frac{1}{n}))$$
where $F$ is a log-periodic function of period $1$.


The $h_k(t)$ do not verify a scaling property (of the form $f(x) = bf(ax)$) ; however, there is a structure and there are some scaling invariance. \\
First, there is a recurrence relation verified by the $h_k(t)$ :
\begin{eqnarray*}
\lefteqn{h_k(t) = \int_K e^{-i\epsilon t}\epsilon^k d\mu}\\
& = & \frac{3^{-k}}{2}\int_K e^{-i\epsilon t/3}\epsilon^k d\mu+\frac{3^{-k}e^{-i 2t/3}}{2}\int_K e^{-i\epsilon t/3}(\epsilon+2)^k d\mu\\
& = & \frac{3^{-k}}{2} h_k(t/3) + \frac{3^{-k}}{2}e^{-i2 t/3}\sum_{j = 0}^{k} \binom {k} {j} h_j(t/3) 2^{k-j}
\end{eqnarray*}

From this relation, we can show by recurrence that the zeroes of $h_k(t)$ is the set : 
$$Z_k = \{t_{m,k} = \frac{\pi}{2}(2m+1)3^{k+1}, m\in \mathbb{Z}\}$$
Thus, the zeroes of $h_k(t)$ are those of $h_{k-1}(t/3)$, as illustrated in fig. 1.\\

Moreover, the $h_k(t)$ all have local maxima at points of the form $m_{n,l} = n\pi 3^l$. I did not manage to prove this rigorously for $k>0$ (the case $k=0$ is straightforward, the detail is given in annexe), but the numerics support this assumption and one can intuitively understand why it is so from the definition of $h_k(t)$ : 
$$h_k(t) = \int_K x^k e^{-ixt}d\mu = \lim_{n \to \infty}\frac{3^n}{2^n}\sum_{a_{j_n}\in K_n} \int_{a_{j_n}}^{a_{j_n}+3^{-n}} x^k e^{-ixt}dx$$

$|h_k(t)|$ is maximal at points $t$ such that the phase $e^{-ixt}$ varies as little as possible as $x$ goes through $K$. $K$ is composed of real numbers which have only $0$ and $2$ in there decomposition in base $3$, e.g. $x = \sum_{k\leq 1} \frac{\beta_k}{3^k}$ with $\beta_{k}\in\{0,2\}$. Now, for $t= n\pi 3^l$, we see that for any $x\in K$ whose decomposition in base $3$ has only $0$ after the $l$ first terms, we have $e^{-ixt}=1$ : the phase does not change. 

\medskip

At these local maxima, the amplitude of the functions $h_k(t)$ decreases with $k$  (as a consequence of Riemann-Lebesgue theorem).

\begin{figure} %1
\centering
\includegraphics[scale=0.7]{9emeh0tiersh1h2.pdf}
\caption{Numerical curves of $|h_0(t/9)|$ (blue), $|h_1(t/3)|$ (red) and $|h_{2}(t)|$ (yellow).}
\end{figure} 

We wish now to examine how this structure translates in terms of physical dynamical quantities.

\subsection*{Numerical study}

The relation between the zeroes of the $h_k(t)$ is very well verified numerically (fig. 1). \\

We observe that the amplitude of $h_k$ decreases with $k$ (see fig. 2), thus we wish to make the assumption : $h_k(t) \propto h_{k-1}(t/3)$. 

\medskip

%We also find that, regardless of the decreasing of the period with $k$, there seems to be a proportional relation between local maxima of $h_k(t)$ : if we look "from afar", it would seem that the $h_k$ are proportional to one another (see fig.10)

\begin{figure} %2
\centering
\includegraphics[scale=0.7]{h1-h2-h10-h20-cantor.pdf}
\caption{$h_1(t), h_2(t), h_{10}(t), h_{20}(t)$ for a triadic Cantor set spectrum.}
\end{figure} 


\subsection*{Approximation of $h_k(t)$ by a scaling function}

Using 
\begin{eqnarray*}
\lefteqn{h_k(t) = \int_K e^{-i\epsilon t}\epsilon^k d\mu}\\
& = & \frac{3^{-k}}{2}\int_K e^{-i\epsilon t/3}\epsilon^k d\mu+\frac{3^{-k}e^{-i 2t/3}}{2}\int_K e^{-i\epsilon t/3}(\epsilon+2)^k d\mu\\
& = & \frac{3^{-k}}{2} h_k(t/3) + \frac{3^{-k}}{2}e^{-i2 t/3}\sum_{j = 0}^{k} \binom {k} {j} h_j(t/3) 2^{k-j}
\end{eqnarray*}

we wish to keep only the dominant terms of the sum on the right side to approximate $h_k(t)$.

\medskip

Let us test the validity of the approximation : $h_k(t) = \frac{3^{-k}e^{-2it/3}}{2}2^k h_0(t/3)$. We find (see fig. 3) that this is quite accurate for small $k$, but falls for $k\leq 4$.

\begin{figure} %3
\centering
\includegraphics[scale=0.7]{h1-h0d3-cantor.pdf}
\caption{$h_1(t)$ and $\frac{1}{3}h_0(t/3)$ for a Cantor spectrum.}
\end{figure} 

%We observe that local maxima for certain $k$ are not local maxima for all $k$ ; fig. 10 shows that, for example, the time $t=281$ correspond to a local maximum for $h_1$, $h_2$, $h_3$, but is not a local maximum for $h_4$ and for higher $k$.

\medskip

I have tried a few other approximation by scaling functions, which did not work well. So far, it seems that the relations between the zeroes of $h_k$ and the behaviour at the local maxima alone can justify the numerical observation that "from afar" $h_k(t) \propto h_{k-1}(t/3)$.

\medskip

This almost re-scaling relation translates to the RMS. The details of how the wave packet is constructed is given in the annexe. The wave packet is initially localized on the first site $k=1$. Figure 4 shows the graphs of $\Delta x (t)$ and $\Delta x(t/3)$. We find that $\Delta x(t)$ displays a log-periodic structure. $\Delta x(t)$ is a power-law modulated by oscillations, which have a log-periodic envelope of period $\log 3$ ; inside the envelope, the period of the oscillations increases with time, which is consistent with the behaviour of $h_k(t)$ under the transformation $t \rightarrow t/3$. I also find that : $\Delta x (t) \approx \frac{3}{2}\Delta x (t/3)$ (see fig. 5), but I have no explanation for the value of this coefficient so far.

\begin{figure} %4
\centering
\includegraphics[scale=0.7]{rms-rmsd3-cantor.pdf}
\caption{RMS for a triadic Cantor spectrum ; blue : $\Delta x(t)$, green : $\Delta x(t/3)$. }
\end{figure}



\begin{figure} %5
\centering
\includegraphics[scale=0.7]{rms-rmsd3x32-cantor.pdf}
\caption{RMS for a triadic Cantor spectrum as a function of time, in loglog scale ; blue : $\Delta x(t)$, green : $\frac{3}{2}\Delta x(t/3)$.}
\end{figure}




\subsection*{Application to the Fibonacci potential}

We study numerically the case of a Fibonacci potential. The potential is defined with the Fibonacci algorithm. For  the numerics we choose $V(k) = \pm0.6$ (note : for $|V(k)| > 2$, the spectrum is continuous and not singular continuous). The fig. 6 is the spectrum associated to this potential.

\begin{figure} %2
\centering
\includegraphics[scale=0.7]{spectre-fibo.pdf}
\caption{Spectrum for a Fibonacci potential, $V(k) = \pm 0.6$ .}
\end{figure} 

The spectrum in this case is not a pure Cantor set, but a superposition of Cantor sets ; however, we can use the following approximate scaling symmetry : $N(\epsilon) = 3N(\epsilon/5)$ where $N(\epsilon)$ is the IDOS (see fig. 7).

\begin{figure} %3
\centering
\includegraphics[scale=0.7]{idos-fibo.pdf}
\caption{}
\end{figure}  

We find, in this case, that, as for the triadic Cantor set, the functions $h_k(t)$ seem almost proportional to one another for $k\geq1$. The function $h_0(t)$ has a very different behaviour. We find that $h_1(t) \approx h_0(t/5)$ (see fig. 8).

\medskip

Assuming the pure scaling symmetry for this spectrum ($N(\epsilon) = 3N(\epsilon/5)$), we find the following theoretical expression for $h_0(t)$ :
$$h_0^{th}(t) = \prod_{k=1}^{\infty}\frac{1+e^{-2it/5^k}+e^{-4it/5^k}}{3}$$
However, this expression does not fit well with the numerical graph ; this is due to the fact that the spectrum does not have such a simple scaling relation (see fig. 7). Nevertheless, we observe, as for the Cantor set case, that for $t<100$, the zeroes of $h_0$ do seem to be stable under the transform $t \rightarrow t/5$ (see fig. 9). We thus expect the RMS to be stable under the transform $t \rightarrow t/5$. We find numerically that $\Delta x(t)$ is a power-law modulated by log-periodic oscillations of very small amplitude for $0<t<300$. For larger $t$, the function starts to decay and has a very wild behaviour, probably due to numerical approximations.

\begin{figure} %7
\centering
\includegraphics[scale=0.7]{h0-h0th-fibo.pdf}
\caption{}
\end{figure}

\begin{figure} %8
\centering
\includegraphics[scale=0.7]{h0-h0d5-fibo.pdf}
\caption{}
\end{figure}

\section*{Annexe}

\subsection*{Maxima of $h_0(t)$ for the triadic Cantor set}

Using the sclaing symmetry of the spectrum : 
\begin{equation}
\int_k f(x) d\mu(x) = \frac{1}{2}\int_k f\left(\frac{x}{3}\right) d\mu(x)+\frac{1}{2}\int_k f\left(\frac{x+2}{3}\right) d\mu(x)
\end{equation}
we find that :
\begin{eqnarray*}
\lefteqn{h_0(t) = \int_K e^{-i\epsilon t}d\mu}\\
& = & \frac{1}{2}\int_K e^{-i\epsilon t/3}d\mu+\frac{e^{-i 2t/3}}{2}\int_K e^{-i\epsilon t/3}d\mu\\
& = & \frac{1+e^{-i2t/3}}{2} h_0(t/3) 
\end{eqnarray*}

By repeating this process, and using $h_0(0) = \int_K 1d\mu = 1$, we find the exact expression of $h_0(t)$ :

\begin{equation}
h_0(t) = \prod_{k\geq 1} \frac{1+e^{-i2t/3^k}}{2}
\end{equation}

thus :

\begin{equation}
|h_0(t)| = \prod_{k\geq 1} \cos{t/3^k}
\end{equation}

The derivative :

\begin{equation}
|h_0(t)|' = -\sum_{k\geq 1}3^{-k}\sin{t/3^k}\prod_{j\neq k} \cos{t/3^j}
\end{equation}

is null for $t$ of the form : $t = m_{n,l} = \pi n 3^l$, with $n\in \mathbb{Z}$ and $l \in \mathbb{N}^{*}$. 

\subsection*{Construction of the wave packet for the triadic Cantor set spectrum}

I first tried to build a potential $V$ whose associated spectrum would be a triadic Cantor set, but it was difficult to obtain something close enough so that the properties of the $h_k(t)$ would still be verified. Therefore, I considered the case of $V=0$, with an associated continuous spectrum, and I built a wave packet using only eigenfunctions associated to energies belonging to the triadic Cantor set embedded in the spectrum.

\medskip
 
I used a finite lattice of size $2^8 = 256$, and an associated finite Cantor set of size $256$. The Hamiltonian is a $256x256$ matrix $H$ of diagonal elements equal to $2$, off-diagonal elements $H_{i,i+1} = H_{i+1,i} = -1$ and the other elements equal to zero. The theoretical eigenfunctions are constructed using :
$$\phi_{\epsilon}(k) = \phi_{\epsilon}(1)p_k(\epsilon) $$
with $p_k(x)$ constructed as explained earlier. The boundary values $\phi_{\epsilon}(1)$ are obtained numerically from the eigenvectors of the matrix $H$.\\
The wave packet is constructed so as to be initially located on the site $k=1$ (see fig. 10 for the evolution in time).

\begin{figure} %10
\centering
\includegraphics[scale=0.7]{etalement-pack-onde-fibo-t0100t500t800.pdf}
\caption{}
\end{figure}

\section*{References}

[1] P.J. Grabner , H. Prodinger, Statistics \& Probability Letters 26 (1996) 243-248



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