In general I am trying to determine when the self-adjoint operator $M$ of multiplication by the independent variable in $L^2 (\mathbb{R})$ has a symmetric restriction to a dense linear manifold (non-closed subspace) of a given subspace.

For example - consider the subspace $B(A)$ of A-bandlimited functions - this is the subspace which is the image of $L^2 [-A , A]$ under Fourier transform. If one considers the linear manifold $\mathrm{Dom} (M_A)$ which is the image of $C_0 ^\infty (-A ,A)$ under Fourier transform then the restriction $M_A := M | _{\mathrm{Dom} (M_A)}$ is a densely defined symmetric operator in $B(A)$ whose closure has deficiency indices (1,1). There is a whole class of related examples which arise from model subspaces of the classical Hardy Hilbert space of the upper half plane.

Suppose that $\mathfrak{D} \subset L^2 (\mathbb{R})$ is a linear manifold on which any polynomial of the mulitplication operator $M$ is defined and that the position uncertainty $\Delta M [ \phi ] $ is bounded below by $\epsilon >0$ for all $\phi \in \mathfrak{D}$. For example if $H := - \frac{d^2}{dx^2} + V(x)$ is a self-adjoint Schrodinger Hamiltonian for $V \geq 0$, and one considers the spectral subspace $B_V (A) := \chi _{[0,A]} (H) L^2 (\mathbb{R})$ (here $\chi _I$ is the characteristic function of the set I and we use the functional calculus), then one can construct a dense domain of vectors $\mathfrak{D} \subset B_V (A)$ such that $\mathfrak{D}$ is contained in the domain of $p(M)$ for any polynomial $p$ and one can apply the Heisenberg uncertainty relation to show that $\Delta M [\phi] \geq \frac{1}{2\sqrt{A}} $ for all $\phi \in \mathfrak{D}$.

One can then consider the linear manifold $\mathrm{Dom} (M_S) := \bigvee ( p(M) \phi | \ \ \phi \in \mathfrak{D} )$ here p is any polynomial and $S$ is the subspace which is the closure of $\mathrm{Dom} (M_S)$. Then $M_S := M | _{\mathrm{Dom} (M_S)}$ is a densely defined symmetric operator in $S$. I am interesting in determining, especially in the examples related to Schrodinger operators described above, when (i) $S$ is not all of $L^2 (\mathbb{R})$ and (ii) when $M_S$ is not self-adjoint.