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We study the stability of the Schr\"odinger equation generated bytime-dependent Hamiltonians with constant form domain. That is, we bound thedifference between solutions of the Schr\"odinger equation by the difference oftheir Hamiltonians. The stability theorem obtained in this article provides asharper bound than those previously obtained in the literature. This makes it apotentially useful tool for time-dependent problems in Quantum Physics, inparticular for Quantum Control. We apply this result to prove two theoremsabout global approximate controllability of infinite-dimensional quantumsystems. These results improve and generalise existing results oninfinite-dimensional quantum control.

On a sharper bound on the stability of non-autonomous Schr\"odinger equations and applications to quantum control