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We know that in order to solve the time-dependent Schrodinger equation $i\partial_t \psi = H(t) \psi$, we need the evolution operator $$U(t) = T \exp{\left(-i\int_0^t H(t')dt'\right)}$$ where $T$ is the time ordering operator and the right hand side (RHS) denotes a formal summation. My question: Does the evolution operator exist in a mathematically rigorous sense? We may deal with imaginary time if necessary.

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If $H(t)$ is selfadjoint and bounded (thus everywhere defined), the theory is standard and quite easy to handle using the uniform operator topology. You can find all required proofs in the first or second volume of Reed and Simon's textbook on mathematical methods. If the operators $H(t)$ are unbounded, the theory is much more difficult also in view of evident problems with domains and it uses the strong operator topology. I am sure that there are classical results by Kato in his celebrated book about the general context ("Perturbation theory of linear operators"), but I suspect there are further more modern results.

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