Welcome to MathOverflow

Question

-1

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

flag	asked Oct 13 2011 at 0:46 Physics beginner 1

1

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

Michael Renardy · Answer 1 · 2011-10-13 02:11:43Z UTC

1

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.

answered Oct 13 2011 at 2:11

Michael Renardy
4,771●1●10●17

2

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

Domain and exponential of self- adjoint operator

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

1 Answer

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.