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This is question is motivated by Non-closed range space of Laplace operators?. We aim to determine what kind of potential will make corresponding schrodinger operators possess non-closed range.

For Schrodinger operator with Kato-Relich potentials \begin{equation*} -\Delta + V(x): H^2(\mathbb{R}^3) \subseteq L^2(\mathbb{R}^3) \to L^2(\mathbb{R}^3), \end{equation*} where $ V(x) \in L^2(\mathbb{R}^3) + L^\infty(\mathbb{R}^3) $. What conditions can we add on $ V $ such that $ 0 $ locates in the continuous spectrum of above operators?

Thank you in advance!

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    $\begingroup$ Of course not, for example you could take $V=1$. A less trivial remark is that even if you require $V$ to go to zero at infinity, that won't be enough to obtain this conclusion. You can in fact still have pure point spectrum for such a $V$. $\endgroup$
    – Christian Remling
    Commented Sep 15, 2019 at 2:27
  • $\begingroup$ @Christian Remling Thank you for comments, could you point out the latter example with pure point spectrum? Besides, I have investigated that the Schrodinger operators with Coulomb potential and short-range potential could achieve the goal ($ 0 \in \sigma_c $). Can we have a more general result for the potential? $\endgroup$
    – Yidong Luo
    Commented Sep 15, 2019 at 3:30
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    $\begingroup$ The examples are essentially one-dimensional; the potential is spherically symmetric. You could try to check the corresponding (rather large) literature. If you want to run an internet search, I would try something like "spectral properties of 1D SO with decaying potentials" perhaps. $\endgroup$
    – Christian Remling
    Commented Sep 15, 2019 at 18:48

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