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Let $T:D(T)\subset L^2(\mathbb{R})\to L^2(\mathbb{R})$ be a linear operator defined as integer power of position operator $$T:=X^n, \quad n\in\mathbb{Z}$$ Has it got any self-adjoint extensions? I'm pretty sure $X^n$ is essentially self-adjoint for $n\in\mathbb N$. But what about $n\in-\mathbb{N}$?

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It is selfadjoint on the natural domain of the vectors $\psi \in L^2(R)$ such that $R \ni x \mapsto x^n \psi(x)$ defines a vector of $L^2(R)$. This is a special case of a general analogous result regarding $f(T)$ for $T$ selfadjoint and $f$ a real valued Borel measurable function. (See, e.g., my book on spectral theory and quantum mechanics, Springer 2018 2nd edition or any other book on general spectral theory.)

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