On page 200 of "A Course in Mathematical Physics, v3: Quantum Mechanics of Atoms and Molecules", Walter Thirring poses the problem of showing that (the components of) the quantum Runge-Lenz (QRL) vector is essentially self-adjoint on a certain domain. He provides a solution of this problem, which consists entirely of the statement that the indicated domain contains the span of the Hermite functions, on which the QRL vector is already self-adjoint, but he does not provide a proof that the QRL vector is essentially self-adjoint on the span of the Hermite functions, and I cannot come up with one. Can someone provide such a proof or a reference?
To be specific, the QRL vector is formally given by $$ \mathbf{F} = (1/2) [ \mathbf{P \times L - L \times p} ] + m\alpha \mathbf{x}/\vert\mathbf{x}\vert ,$$ where $\mathbf{L}$ is the angular momentum operator $$\mathbf{L} = \mathbf{x \times p}$$ and $\mathbf{x}$ and $\mathbf{p}$ are the position and momentum operators. (In the Schrodinger representation, $\mathbf{x}$ is multiplication by (vector) $\mathbf{x}$ and $\mathbf{p}$ is $(-i/\hbar)$ times the gradient operator.)
Every proof strategy I have attempted has foundered on the fact that the QRL vector is a third degree polynomial in $\mathbf{x}$ and $\mathbf{p}$. One observation is that one may take $\alpha = 0$, since the $\mathbf{x}/\vert\mathbf{x}\vert$ term is bounded, and so will not change any domain of essential self-adjointness. In that case, the QRL vector is homogeneous of second order in $\mathbf{p}$, so it will commute with complex conjugation and will therefore have at least one self-adjoint extension.