Generalized coherent states in arbitrary potential

It is well known that for Hamiltonian of harmonic oscillator there exist special states which saturate the Heisenberg uncertainity principle and under the time evolution follow closely classical trajectories. Of course they are wavepackets with finite volume in phase space, but that volume is not expanding with time evolution. These states are the famous coherent states: $$| z \rangle = e^{za^{\dagger}-z^*a} |0 \rangle,$$ where $| 0 \rangle$ is the ground state of $H = a^{\dagger}a$ and $z \in \mathbb C$. My question is whether generalized coherent states with similar properties (holding possibly only approximately) exist for other Hamiltonians, either $$H= \frac{p^2}{2m}+V(x)$$ with arbitrary potential or even more generally in other quantum theories (also with infinte number of degrees of freedom). If they do, is there a general method for finding them? Do they share some nice analytic properties of coherent states (I suppose not, unless there is rich symmetry structure).

Remark: Title of my post was edited by moderation from "semiclassical" to "generalized coherent". I am not completely sure whether that is correct. I did a little bit of research and my impression is that what people usually mean by this second term are special states transforming between each other according to a represenation of some group and together forming a resolution of identity - essentially a topic of representation theory. This property is indeed the case in example I have shown, and the group under consideration is Heisenberg group. However I am interested if special states behaving approximately classicaly exist in general, when the underlying algebra of observables is not so simple.

To answer your question in its narrowest interpretation: yes but it depends on the potential (as you correctly guessed).

Perelomov generalized the notion of coherent states so it can be applied to a large number of Lie groups.

Your question actually hinges on the observation that different definitions of coherent states happen to coincide for HW.

Coherent states can be defined for HW as minimum uncertainty states, and then generalize to the notion of "intelligent" states. This definition has also been used when the Perelomov method (see below) does not work.

Coherent states can also be defined, in a spirit closer to the definition you gave, as transformed ground states; this is the generalization of Perelomov, and is applicable to a wide variety of groups. This is possibly the generalization you are looking for, and there is a lot of literature on spin coherent states for instance (as the next most common type of coherent states in applications). Perelomov coherent states also saturate the uncertainty relations suitably-defined observables.

Lastly, CS can be defined as eigenstates of the lowering operators; in this case they only exist for non-compact groups and were generalized to $SU(1,1)$ by Barut and Girardello.

Given the close relation between special functions and representations of Lie algebras, it is often possible to find an algebraic structure from which one can obtain coherent states when the potential yields analytical solutions to the Schrodinger equation in terms of special functions. Note that this is neither a necessary nor a sufficient condition.

Note also there are "coherent-like" states for various types of potentials: Morse, Poechl-Teller and the likes. There are also variations that will produce different types of distributions (this comes about by adjusting the energy levels of a 1d potential). Literature on all these can be found on arXiv.