Is $\hat X^n$ a good observable?

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 $$n\in-\mathbb{N}$$?

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.)