## Domain and exponential of self- adjoint operator

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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This question would be better for math.stackexchange.com, as it is not really "research level". – Nate Eldredge Oct 13 2011 at 5:00

## 1 Answer

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.

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No semigroup theory is needed here. Just note that the spectral theorem reduces the question to the case where $\mathcal{H}$ is an $L^2$ space and $A$ is a multiplication operator. – Nate Eldredge Oct 13 2011 at 4:59
The perhaps only interesting thing here is that $\exp(-iA)\psi$ can not be defined by the exponential series for all vectors in $D(A)$. One really needs the definition via the spectral calculus, i.e. the integral over the projection valued measure of $A$. – Stefan Waldmann Oct 13 2011 at 7:37