Let $A$ be a self  adjoint operator on a Hilbert space $\mathcal{H}$ and let $D(A)$ be its domain. If $\psi \in D(A)$ then $exp(itA) \psi \in D(A)$ iff $A$ is bounded ?
Thank you guys, sorry if the question is too trivial ! ;)
Physics beginner
Let $A$ be a self  adjoint operator on a Hilbert space $\mathcal{H}$ and let $D(A)$ be its domain. If $\psi \in D(A)$ then $exp(itA) \psi \in D(A)$ iff $A$ is bounded ? Thank you guys, sorry if the question is too trivial ! ;) Physics beginner 


No. If A is selfadjoint, then exp(itA) maps D(A) to D(A) regardless whether A is bounded. You should read a text on semigroup theory for linear operators, for instance Pazy's book. 

