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I am reading literature on self-adjoint extensions of Hamiltonians (particle interaction) and I came across the following statement (in context of separating total momentum $P$):

Operator $H$ defined by $(Hf)(P)=\frac{P^2}{m+2}f(P)$ is a self-adjoint operator in $L_2(\mathbb{R}^{3})$, i.e. $f \in L_2(\mathbb R ^3)$.

So from this I figure that $H$ is defined on $L_2(\mathbb R^3)$, but we can obviously find some $f \in L_2(\mathbb R ^3)$ s.t. $\frac{P^2}{m+2}f(P)\notin L_2(\mathbb R ^3)$, in other words $H(L_2(\mathbb R ^3))$ is larger than $L_2(\mathbb R ^3)$. So the operator is not defined on the whole domain (since it has to be closed w.r.t. Hilbert space it is acting on in order to be well-defined), and its domain $D(H)$ has to be restricted. However, the paper just states that $H$ is self-adjoint on $L_2(\mathbb R ^3)$ and goes ahead.

So my question is, am I missing something here? Should the domain really be restricted, or is this operator indeed self-adjoint as it is?

Thank you in advance!

nakajuice
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    (with $p^2 = |p|^2$) it is a (densely defined) unbounded self-adjoint operator $L^2(\mathbb{R}^3) \to L^2(\mathbb{R}^3)$. It is indeed very similar to the derivative operators ($H$ is, up to a constant $\frac{-1}{4 \pi^2(m+2)}$, the Laplacian written in the Fourier domain, the Fourier transform being unitary $L^2(\mathbb{R}^3 \to L^2(\mathbb{R}^3$) – reuns Jul 04 '16 at 19:00
  • So from your comment I deduce that the domain is not $L_2 (\mathbb R ^3)$, but something like $ D(H) = {f : \frac{P^2}{m+2}f(P) \in L_2 (\mathbb R ^3) }$? – nakajuice Jul 04 '16 at 20:23
  • Yes. You know that a linear operator $T : X \to X$ is fully determined by its values $T(x)$ for $|x|= 1$ ? Then you can think to the densely defined unbounded operators as those having some sort of singularities on $|x|=1$ – reuns Jul 04 '16 at 20:35
  • Thank you for clarification, the statement and lack of elaboration in the paper was somewhat misleading for a novice like me who is more used to work with bounded ops. – nakajuice Jul 04 '16 at 21:06

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