Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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.

share|improve this question
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.