What can be said about the commutator of an operator with itself at different times?

In general for some smooth and bounded $$\hat{V}$$

$$\left[\hat{V}(t_1), \hat{V}(t_2) \right] \neq 0 \text{ if } t_1 \neq t_2$$

But what more can be said about commutators of this type? I am particularly interested in the case where the two times are "not far apart", say $$|t_2 - t_1| \leq \epsilon$$

for some sufficiently small but not infinitesimal $$\epsilon$$. I know from other answers that if we define $$A_t(\epsilon) = \left[\hat{V}(t+\epsilon), \hat{V}(t) \right]$$

then as $$\epsilon \rightarrow 0, A_t(\epsilon)\rightarrow 0$$. But I am interested in what bounds we can place on $$\left[\hat{V}(t_1), \hat{V}(t_2) \right]$$ provided $$\hat{V}$$ is smooth. My (very naive) approach was to note (letting $$V_1=\hat{V}(t_1)$$): \begin{align} \left[V_1, V_2 \right]\cdot|t_2-t_1|^{-1} = (V_1-V_2)(V_1+V_2)|t_2-t_1|^{-1}-(V_1^2 -V_2^2)|t_2-t_1|^{-1} \end{align}

and if we say $$||V|| for $$t\in[t_1, t_2]$$, the above is roughly equal to (for small $$|t_2-t_1|$$

\begin{align} \left[V_1, V_2 \right]\cdot|t_2-t_1|^{-1} = \frac{dV}{dt}(2C)-(V_1^2 -V_2^2)|t_2-t_1|^{-1} \end{align}

and I could probably go on simplifying. But clearly this is not a good way to place bounds, and I am interested in what clean results there might be.

EDIT:

In order to limit the possibilities, I am interested in 'common' Hamiltonians. Let use assume that:

1. $$H(t) = K + V(t)$$, i.e. the only time dependence is in the potential
2. $$V(t) = g(t)\cdot V(x)$$, the time dependence can be written as a time-dependent coupling strength for the smooth and bounded potential $$V$$. We can assume $$g(t)$$ is a polynomial of time.
3. If it helps to have concrete examples, I am interested in the Newton potential where $$\nabla^2V(x) = c_1 \cdot \psi^* \psi$$, and also (but less so) the harmonic oscillator with a time varying frequency.

I am interested in results that assume more stringent conditions as long as it's clear what those are.

• Sorry for the naive question, is then $V$ just an operator that depends on a parameter? Is there some characteristic of this dependence due to the interpretation of $t$ as time? Or are you just considering a smooth operator valued function $t\mapsto V(t)$? May 15, 2020 at 18:22
• Do you have any conditions in mind to place on the Hamiltonian? If it can be arbitrary it seems difficult to say anything. May 15, 2020 at 18:27
• @knzhou I will edit the question with some conditions. In general I am interested in common Hamiltonians
– gabe
May 15, 2020 at 18:58
• @user2723984 I don't see any characteristic that would depend on the interpretation as time, so you can consider it as a smooth parameter. Maybe someone else will chime in to say why that shouldn't be the case.
– gabe
May 15, 2020 at 18:59
• @gabe then I think this is essentially a math question, maybe the people at math SE can answer it better May 15, 2020 at 19:22