Quantum quasiballistic dynamics and thick point spectrum

Abstract

We obtain dynamical lower bounds for some self-adjoint operators with pure point spectrum in terms of the spacing properties of their eigenvalues. In particular, it is shown that for systems with thick point spectrum, typically in Baire’s sense, the dynamics of each initial condition (with respect to some orthonormal bases of the space) presents a quasiballistic behaviour. We present explicit applications to some Schrödinger operators.

Appendix

Proposition A.1

Let $-\infty< a<b <\infty$. If $\displaystyle \cup _{j} \{a_j\}$ is a dense subset of [a, b], then $(a_j)$ is weakly spaced.

Proof

Let $\alpha >0$. Firstly, we note that, for each $x > 1$,

$$\begin{aligned} \biggr (\frac{x}{x-1}\biggr )^\alpha + \biggr (\frac{x}{x+1}\biggr )^\alpha > 2. \end{aligned}$$

(8)

Namely, set

$$\begin{aligned} f(\alpha ) := \biggr (\frac{x}{x-1}\biggr )^\alpha + \biggr (\frac{x}{x+1}\biggr )^\alpha . \end{aligned}$$

So,

$$\begin{aligned} \biggr (\frac{x-1}{x}\biggr )^\alpha f'(\alpha )= & {} \ln \biggr (\frac{x}{x-1}\biggr ) - \biggr (\frac{x-1}{x+1}\biggr )^\alpha \ln \biggr (\frac{x+1}{x}\biggr )\\> & {} \ln \biggr (\frac{x}{x-1}\biggr ) \biggr (1-\biggr (\frac{x-1}{x+1}\biggr )^\alpha \biggr )>0. \end{aligned}$$

Since $f(0) = 2$, the inequality in (8) follows.

For each $l \ge 1$, set

$$\begin{aligned} b_l := a+ \frac{1}{l^\alpha }; \end{aligned}$$

by (8), for $l\ge 2$ one has $K_l:=b_{l-1} -2b_{l} + b_{l+ 1}>0$. Note that

$$\begin{aligned} \lim _{l \rightarrow \infty } l^{1+\alpha }(b_l -b_{l+1})=\alpha . \end{aligned}$$

(9)

Now, for l sufficiently large such that $b_l \in [a,b)$, pick $a_{j_l}$ satisfying

$$\begin{aligned} 0\le a_{j_l} - b_l \le \min \biggr \{\frac{K_l}{2},\frac{\alpha }{4l^{1+\alpha }}\biggr \}. \end{aligned}$$

(10)

Then, by (9) and (10), for l sufficiently large, one has

$$\begin{aligned} a_{j_l} - a_{j_{l+1}}= & {} (a_{j_l} - b_l) - (a_{j_{l+1}}-b_{l+1}) + (b_l - b_{l+1})\\\ge & {} -\frac{\alpha }{4(l+1)^{1+\alpha }} + \frac{3\alpha }{4l^{1+\alpha }} \ge \frac{\alpha }{2l^{1+\alpha }},\\ a_{j_l} - a_{j_{l+1}}= & {} (a_{j_l} - b_l) - (a_{j_{l+1}}-b_{l+1}) + (b_l - b_{l+1})\\\le & {} \frac{\alpha }{4l^{1+\alpha }} + \frac{7\alpha }{4l^{1+\alpha }} = \frac{2\alpha }{l^{1+\alpha }}. \end{aligned}$$

Hence,

$$\begin{aligned} \frac{\alpha }{2l^{1+\alpha }} \le a_{j_l} - a_{j_{l+1}} \le \frac{2\alpha }{l^{1+\alpha }}. \end{aligned}$$

Moreover,

$$\begin{aligned}&(a_{j_l} -a_{j_{l+1}})-(a_{j_{l+1}}- a_{j_{l+2}})\\&\quad = (a_{j_l} -2a_{j_{l+1}} + a_{j_{l+2}}) \\&\quad = a_{j_l} -b_l -2(a_{j_{l+1}} - b_{l+1}) + a_{j_{l+2}} -b_{l+2} + (b_{l} -2b_{l+1} + b_{l+2})\\&\quad \ge -2(a_{j_{l+1}} - b_{l+1}) + K_{l+1} \ge 0, \end{aligned}$$

which implies that $a_{j_l} - a_{j_{l+1}}$ goes to zero monotonically. Therefore, $(a_j)$ is weakly spaced. $\square$

Proposition A.2

Let T be a self-adjoint operator in $\mathcal {H}$, $p>0$ and let $B = \{e_n\}_{n \in {\mathbb {Z}}}$ be an orthonormal basis. Suppose that, for every $\xi \in \mathcal {H}$, $\sum _{n \in {\mathbb {Z}}} |n|^p |\langle \hbox {e}^{-itT}\xi ,e_n \rangle |^2 = \infty$ for $t = 0$ if, and only if, $\sum _{n \in {\mathbb {Z}}} |n|^p |\langle \hbox {e}^{-itT}\xi ,e_n \rangle |^2 = \infty$ for all $t \in {\mathbb {R}}$. Then,

$$\begin{aligned} G(B) = \{\xi \in {\mathcal {H}}\mid \langle \langle |X|^p \rangle \rangle _{t,\xi } \equiv \infty \text { for all } p>0\} \end{aligned}$$

is a dense $G_\delta$ set in ${\mathcal {H}}$.

Proof

One just has to show that

$$\begin{aligned} \left\{ \xi \in {\mathcal {H}}\mid \sum _{n \in {\mathbb {Z}}} |n|^p |\langle \xi ,e_n \rangle |^2 = \infty \text { for all } p>0\right\} \end{aligned}$$

is a dense $G_\delta$ set in ${\mathcal {H}}$.

Since for each $j\ge 1$, the mapping

$$\begin{aligned} {\mathcal {H}} \ni \xi \longmapsto \sum _{|n|\le j} |n|^p |\langle \xi ,e_n \rangle |^2 \end{aligned}$$

is continuous, it follows that, for each $p>0$,

$$\begin{aligned} \left\{ \xi \in {\mathcal {H}} \mid \sum _{n \in {\mathbb {Z}}} |n|^p |\langle \xi ,e_n \rangle |^2 = \infty \right\} = \bigcap _{k \ge 1} \bigcap _{j\ge 1} \left\{ \xi \in {\mathcal {H}} \mid \sum _{|n|\le j} |n|^p |\langle \xi ,e_n \rangle |^2 > k\right\} \end{aligned}$$

is a $G_\delta$ set in $\mathcal {H}$. Now, for each fixed $p>0$, $\xi \in \mathcal {H}$ and $j\in \mathbb {N}$, set

$$\begin{aligned} \xi _j := \displaystyle \sum _{|n|\le j} \langle \xi ,e_n \rangle e_n + \displaystyle \sum _{|n|>j}^\infty \frac{1}{\sqrt{|n|^{p+1}}}e_n. \end{aligned}$$

It is clear that $\xi _j \rightarrow \xi$ in $\mathcal {H}$. Moreover, for each $j \ge 1$, $\sum _{n \in {\mathbb {Z}}} n^p |\langle \xi _j ,e_n \rangle |^2 = \infty$. Thus, for each $p>0$, $\{\xi \in {\mathcal {H}} \mid \sum _{n \in {\mathbb {Z}}} |n|^p |\langle \xi ,e_n \rangle |^2 = \infty \}$ is dense in ${\mathcal {H}}$, and therefore, by Baire’s theorem,

$$\begin{aligned}&\left\{ \xi \in {\mathcal {H}}\mid \sum _{n \in {\mathbb {Z}}} |n|^p |\langle \xi ,e_n \rangle |^2 = \infty \text { for all } p>0\right\} \\&\quad = \bigcap _{p\in {\mathbb {Q}}^+} \left\{ \xi \in {\mathcal {H}} \mid \sum _{n \in {\mathbb {Z}}} |n|^p |\langle \xi ,e_n \rangle |^2 = \infty \right\} \end{aligned}$$

is a dense $G_\delta$ set in ${\mathcal {H}}$, where ${\mathbb {Q}}^+:=\{x\in {\mathbb {Q}}\mid x>0\}$. $\square$