On the structure of the set of Self-Adjoint operators acting on $L^{2}(\mathbb{R})$.

Question

Let us consider the $Hilbert$ Space $L^{2}(\mathbb{R})$ and let $SA(L^{2}(\mathbb{R})$ be the space of all self adjoint operators acting on $L^{2}$. I have worked with operators such as $X$, $P$, $X^{2}$, and $P^{2}$ which which are characterized by the commutation relation $$[X,P] = i$$ pleanty of times. Hamiltonians such as $H = \alpha X^{2} + \beta P^{2}$ ($\alpha$ and $\beta$ real) are typical. However, I would like to understand what the most general Self-adjoint operator in $SA(L^{2}(\mathbb{R})$ looks like. Am I right to assume that the most general element of $SA(L^{2}(\mathbb{R})$ is simply an operator valued fucntion $f(X,P)$ of $X$ and $P$? If so what would be necessary and sufficient conditions on $f()$ for $f(X,P)$ to be self-adjoint (I suspect that the image of $f()$ would have to be real valued). Thank you for your time.