What is the most general Self-Adjoint operator acting on $L^{2}\mathbb{R}$?
My hypothesis is that the aswer to my question will be that the most general Self-Adjoint operator $A$ acting on $L^{2}(\mathbb{R})$ is a non linear combination of terms involving the so called position and momentum operators $X$ and $P$ defined as follows for some $\psi(x)\in L^{2}(\mathbb{R})$
$$ X\psi(x) = x\psi(x) \:\: (Multiplication\:\: operator)$$ $$ P\psi(x) = -i\frac{\partial}{\partial x} \psi(x)$$
these operators satisfy the commutation relation $[X,P]= i\mathbb{I}$.
Would the most general $A$ be just some general operator valued fucntion $f(X,P)$ of $X$ and $P$?