Let $H$ be a Hilbert space. I just want an example of a skew adjoint operator $(A^*=-A)$ with uncountable spectrum.

I also want an example for unbounded differential operators. The only example I know is the Laplace operator $A=i \Delta$ in the Hilbert space $H=L^2(\mathbb{R}^n)$ with domain $D(A)=H^2(\mathbb{R}^n)$ which is skew adjoint (because $\Delta$ is self adjoint), but $A$ has a countable spectrum.

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    $\begingroup$ Why do you think the spectrum of A is countable? I believe it is the negative imaginary axis. $\endgroup$ – Nate Eldredge Jun 16 '14 at 15:15
  • $\begingroup$ Alternatively you could consider multiplication operators (and thanks to the spectral theorem that is all you need to consider). See also my answer here: math.stackexchange.com/a/779087/822 $\endgroup$ – Nate Eldredge Jun 16 '14 at 15:39
  • $\begingroup$ @NateEldredge I thought it is discrete, but I just realized it is discrete only in bounded domains as in the link you provided. $\endgroup$ – user165633 Jun 16 '14 at 21:34

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