# Time-dependent quantum dynamics as dynamical system and ergodic therorems

In full glory, a dynamical system is defined as a tuple $$(T,M,\Phi)$$ where $$T$$ is a monoid, written additively, $$M$$ is a set, and $$\Phi$$ is a function. $$\Phi : U \subset T \times M \to M$$ with

$$I(x) = \{t \in T : (t,x) \in U \}$$
$$\Phi(0,x) = x$$
$$\Phi(t_2, \Phi(t_1(x)) = \Phi(t_1 + t_2, x)$$ for $$t_1, t_2,t_1+t_2 \in I(x)$$

I am interested in quantum dynamics generated by a Hamiltonian that could be time-dependent, that is, a quantum state in a $$d$$-dimensional Hilbert space evolving via the Schrodinger equation \begin{align} i \partial_t |\psi(t)\rangle = H(t)|\psi(t)\rangle \end{align} with initial state $$|\psi(0)\rangle$$. The solution is $$|\psi(t)\rangle = U(t) |\psi(0)\rangle$$ where $$U(t)$$ is the unitary time-evolution operator which is formally given as a time-ordered exponential \begin{align} U(t) = \mathcal{T}e^{-i\int_0^t H(t') dt'}. \end{align}

My question is: does this constitute an example of a dynamical system?

My understanding is, if the Hamiltonian is time-independent $$H(t) = H$$, then yes. This is because $$U(t) = e^{-iHt}$$ so $$U(t+s) = U(t)U(s)$$. If the Hamiltonian has non-trivial time-dependence, then no, according to Is the initial value problem of an ODE considered as a dynamic system?. This is because $$U(t)$$ fails the semi-group property in general. Would my understanding be right?

The reason I am interested in asking this is because I am interested in ergodic theorems (e.g., Birkhoff's, point wise ergodic, Von Neumann ergodic theorems). But it seems to me they all deal with dynamical systems, in particular, with the flow map $$\Phi$$ satisfying the semigroup property. My next question is: are there any ergodic theorems in the case when the semigroup property does not hold?

For a time-dependent Hamiltonian you will need an evolution operator $$U(t_2,t_1)= \mathcal{T}e^{-i\int_{t_1}^{t_2} H(t') dt'}$$ that depends on initial and final times ($$0). With the composition property $$U(t_2,t_1)U(t_1,0)=U(t_2,0)$$ you can then consider this a non-stationary dynamical system.
The main difference between classical and quantum dynamical systems lies in the fact that the Hilbert space is typically infinite-dimensional. A formulation of quantum dynamical systems based on $$C^\ast$$ algebras is given by Claude Pillet (see section 4.4).