How is Schrödinger evolution experimentally verified?

Nice question. Let's split this into three parts:

(1) The Schrödinger equation has features that exist more broadly as fundamental principles of quantum mechanics. Briefly, these are: all information about the state is in the wavefunction; linearity; inner product; self-adjoint observables; unitary evolution; completeness.

(2) The Schrödinger equation has specific features that are not more generally valid in quantum mechanics. For example, it's nonrelativistic.

(3) Collapse.

Taking these in order:

  1. We see no evidence that any of these are violated, but we also don't have any good ways of singling out any particular principle from this list and testing it quantitatively. To do that, we would need a test theory that differs from quantum mechanics but is viable, and right now nobody has any idea how to construct a useful test theory. The trouble is that there are no-go theorems showing that quantum mechanics is very brittle. That is, if you try to introduce any violation of these basic principles, no matter how small, the whole thing seems to break down. Some material about this is given in Aaronson, "Is Quantum Mechanics An Island In Theoryspace?," .

  2. We find, for example, that the Schrödinger equation predicts the energy levels of the hydrogen atom pretty accurately. The errors are at about the level we expect based on the fact that the Schrödinger equation is nonrelativistic, and an electron in a hydrogen atom has $v\sim0.01c$. We can verify various other effects in the Schrödinger equation, such as tunneling and the behavior of the two-state system.

  3. The part about collapse isn't actually part of the theory, it's just a feature of the Copenhagen interpretation. The Copenhagen interpretation is optional. There is absolutely no evidence for any actual physical process resembling the collapse described by the Copenhagen interpretation. All actual observations can be explained in other ways, e.g., in explanations based on decoherence.

By checking that the results of measurements correspond to what the deterministic evolution predicts: note that this is the same as the case for classical mechanics. The difference is that the predictions are inherently probabilistic.