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edited Aug 22, 2017 at 1:24

Christian Remling

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This does not follow. We can find a counterexample in dimension $n=1$ (and let's also set $m=1$). If we write $x=ye^{-kt/2}$, then $y$ solves $$ -y'' - W(t) y = -\frac{k^2}{4} y , $$ and I want to interpret this as a 1D Schrödinger equation at energy $E=-k^2/4$. For a periodic potential, the spectrum has band structure and the Lyapunov exponent is positive in the gaps; in other words, we have (modulo a periodic factor) exponentially increasing solutions in those gaps. The restriction that $-\alpha\le -W \le -\beta$ is not an issue; typically, there are gaps at arbitrarily large energies.

We can now just choose such a $W$ (with a suitable constant added, so that $E=-k^2/4$ will lie in a gap), and we obtain a solution $|y|\gtrsim e^{\gamma t}$ (where the periodic factor is not close to zero). Here, the Lyapunov exponent $\gamma$ is not affected by $k$, so if $k$ was small enough compared to $\gamma$, then $x$ will also have exponential increase on average.

In fact, I could do this more carefully (using facts about the location of gaps), and it then follows that such a counterexample is possible for any given bounds $-\alpha<-\beta$ as long as I'm allowed to make $k>0$ small in comparison.

This does not follow. We can find a counterexample in dimension $n=1$ (and let's also set $m=1$). If we write $x=ye^{-kt/2}$, then $y$ solves $$ -y'' - W(t) y = -\frac{k^2}{4} y , $$ and I want to interpret this as a 1D Schrödinger equation at energy $E=-k^2/4$. For a periodic potential, the spectrum has band structure and the Lyapunov exponent is positive in the gaps; in other words, we have (modulo a periodic factor) exponentially increasing solutions in those gaps. The restriction that $-\alpha\le -W \le -\beta$ is not an issue; typically, there are gaps at arbitrarily large energies.

We can now just choose such a $W$ (with a suitable constant added, so that $E=-k^2/4$ will lie in a gap), and we obtain a solution $|y|\gtrsim e^{\gamma t}$ (where the periodic factor is not close to zero). Here, the Lyapunov exponent $\gamma$ is not affected by $k$, so if $k$ was small enough compared to $\gamma$, then $x$ will also have exponential increase on average.

This does not follow. We can find a counterexample in dimension $n=1$ (and let's also set $m=1$). If we write $x=ye^{-kt/2}$, then $y$ solves $$ -y'' - W(t) y = -\frac{k^2}{4} y , $$ and I want to interpret this as a 1D Schrödinger equation at energy $E=-k^2/4$. For a periodic potential, the spectrum has band structure and the Lyapunov exponent is positive in the gaps; in other words, we have (modulo a periodic factor) exponentially increasing solutions in those gaps. The restriction that $-\alpha\le -W \le -\beta$ is not an issue; typically, there are gaps at arbitrarily large energies.

We can now just choose such a $W$ (with a suitable constant added, so that $E=-k^2/4$ will lie in a gap), and we obtain a solution $|y|\gtrsim e^{\gamma t}$ (where the periodic factor is not close to zero). Here, the Lyapunov exponent $\gamma$ is not affected by $k$, so if $k$ was small enough compared to $\gamma$, then $x$ will also have exponential increase on average.

In fact, I could do this more carefully (using facts about the location of gaps), and it then follows that such a counterexample is possible for any given bounds $-\alpha<-\beta$ as long as I'm allowed to make $k>0$ small in comparison.

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created Aug 22, 2017 at 1:05

Christian Remling

24.2k
2
48
83

This does not follow. We can find a counterexample in dimension $n=1$ (and let's also set $m=1$). If we write $x=ye^{-kt/2}$, then $y$ solves $$ -y'' - W(t) y = -\frac{k^2}{4} y , $$ and I want to interpret this as a 1D Schrödinger equation at energy $E=-k^2/4$. For a periodic potential, the spectrum has band structure and the Lyapunov exponent is positive in the gaps; in other words, we have (modulo a periodic factor) exponentially increasing solutions in those gaps. The restriction that $-\alpha\le -W \le -\beta$ is not an issue; typically, there are gaps at arbitrarily large energies.

We can now just choose such a $W$ (with a suitable constant added, so that $E=-k^2/4$ will lie in a gap), and we obtain a solution $|y|\gtrsim e^{\gamma t}$ (where the periodic factor is not close to zero). Here, the Lyapunov exponent $\gamma$ is not affected by $k$, so if $k$ was small enough compared to $\gamma$, then $x$ will also have exponential increase on average.