Existence of the continuous spectrum of a possibly-unbounded, linear self-adjoint operator on a complex Hilbert space Let $\mathbf{A}$ be a possibly-unbounded, linear self-adjoint operator on an infinte-dimensional, complex separable Hilbert space $\mathcal{H}$, and suppose we know the matrix elements $\langle j|\mathbf{A}|k\rangle$ of $\mathbf{A}$ on a basis $\{|n\rangle\}_{n\in\mathbb{N}}$.
Are there theorems helping in the explicit calculation of the spectrum of $\mathbf{A}$ using the aforementioned matrix elements?
If not, are there theorems giving necessary and/or sufficient conditions for the existence of the continuous spectrum of $\mathbf{A}$?
Thanks in andvance.
 A: Suppose $\mathcal{H}=L^{2}[0,\pi]$. Suppose $\mathcal{M}$ is a countably dense subset of compactly supported $C^{\infty}$ functions on $(0,\pi)$. Apply Gram-Schmidt in order to obtain an orthonormal basis $\{ f_{k} \}_{k=1}^{\infty}$ of smooth compactly supported functions on $(0,\pi)$. The following operators $L_{\alpha,\beta}$ are selfadjoint
$$
               L_{\alpha,\beta} = -\frac{d^{2}}{dx^{2}},
$$
where $\mathcal{D}(L_{\alpha,\beta})$ consists of all twice absolutely continuous functions $f\in\mathcal{H}$ for which $f''\in\mathcal{H}$ and
$$
              \cos\alpha f(0)+\sin\alpha f'(0) = 0,\\
              \cos\beta f(\pi)+\sin\beta f'(\pi)= 0.
$$
(Here $\alpha,\beta$ are real angles.) All of these operators are selfadjoint, and they all contain $\{ f_{k} \}$ in their domains. And they all agree on these elements, i.e., $L_{\alpha,\beta}f_{k}=L_{\alpha',\beta'}f_{k}$. These operators have discrete spectrum, and the spectrum is very different as you allow $\alpha$, $\beta$ to vary. For example, the conditions
$$
                f(0) = 0,\;\; f(\pi)=0
$$
lead to eigenfunctions $\sin(nx)$ with spectrum $\{ 1,2^{2},3^{2},\cdots\}$. The conditions
$$
                f'(0)=0,\;\; f(\pi)=0
$$
lead to eigenfunctions $\cos((n+1/2)x)$ with spectrum $\{ (1/2)^{2},(3/2)^{2},(5/2)^{2},\cdots\}$. Every point of $[0,\infty)$ is in the spectrum of one of the $L_{\alpha,\beta}$.
I'm not aware of anything that will help much with the continuous spectrum.
