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
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Sign up to join this communityLet $A$ 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 self-adjoint, 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.