I have some questions about the position operator used in physics, to avoid physical context I formulate the questions like this:
Definition 1$^{[1]}$: Let $(\mathcal E, \langle\cdot, \cdot\rangle)$ a Hilbert space over the field of complex numbers. A position operator in $\mathcal E$ is a self-adjoint operator $\hat X:\mathcal E_X\subseteq\mathcal E\rightarrow\mathcal E$ such that for every $x\in\mathbb R$ there are a vector $\varphi_x\in \mathcal E_X$ such that $\hat X(\varphi_x) = x ~ \varphi_x$.
Question 1: What Hilbert spaces admit a position operator?
Can you provide or reference examples of how looks $\hat X$ and $\varphi_x$ in such spaces?
Little context
A physicist have, by axioms, a Hilbert space that admit a position operator and say that such space is [isomorphic to] $L^2(\mathbb R; \mathbb C)$. The position operator is therefore defined as $(\hat X(\Psi))(x) = x~\Psi(x)$, but I read (see) that there are no function $\varphi_y\in L^2(\mathbb R; \mathbb C)$ such that $x~\varphi_y(x) = y~\varphi_y(x)$, then I ask:
Question 2: The space $L^2(\mathbb R; \mathbb C)$ admit some position operator like the above defined?
Notes
[1] – From axioms of quantum mechanics definition 1 is, a priori, the only that one can say about the position operator but a physicist also use some other relations, like the commutator with linear momentum. Therefore I imagine that not all operators satisfying definition 1 serves as position operator for a physicist.
Best Answer
With your own definition of a position operator (Definition 1), we must have: $$\forall x,y\in\Bbb R\quad x\langle\varphi_x,\varphi_y\rangle=\langle\hat X(\varphi_x),\varphi_y\rangle=\langle\varphi_x,\hat X(\varphi_y)\rangle=y\langle\varphi_x,\varphi_y\rangle$$ Hence $(\varphi_x)_{x\in\Bbb R}$ must be a family of pairwise orthogonal vectors. If you implicitely wanted these vectors to be non-zero, such a family (hence such an operator) exists iff the Hilbert dimension of $\cal E$ is at least the cardinality $\mathfrak c$ of the continuum.
The Hilbert dimension of $L^2(\Bbb R;\Bbb C)$ is $\aleph_0$$<\mathfrak c.$
The simplest example of a Hilbert space of Hilbert dimension $\mathfrak c$ is the space $\mathcal E:=\ell^2(\Bbb R;\Bbb C)$ of $\Bbb R$-indexed families $z=(z_x)_{x\in\Bbb R}\in\Bbb C^{\Bbb R}$ of complex numbers such that $\sum_{x\in\Bbb R}|z_x|^2<\infty$ (such families necessarily have a countable or finite support), and a position operator on this space is given by $(\hat X(z))_x=xz_x,$ for all $z\in\mathcal E_X:=\{z\in\mathcal E\mid\sum_{x\in\Bbb R}x^2|z_x|^2<\infty\}.$