Consider an operator $H$ on the Hilbert space $\ell_2$ given as an infinite matrix with two pieces, one diagonal and one arbitrary: $H_{ij}=E_i\delta_{ij}+V_{ij}$. This has a physical meaning in quantum mechanics: I am considering a perturbed Hamiltonian $H=H_0+V$ in the basis of eigenvectors of $H_0$; $E_i$ are the eigenvalues of $H_0$ (assumed known and forming an increasing sequence tending to $+\infty$, so that $H_0$ is unbounded) and $V_{ij}$ are matrix elements of $V$ in this basis.

I am interested in two questions.

(1) What are conditions on $V_{ij}$ guaranteeing that $H$ be self-adjoint?

I am familiar with the discussion in Reed and Simon vol.2 about the self-adjointness of various Schroedinger operators. But there $H_0$ is usually the Laplacian or the harmonic oscillator Hamiltonian, and $V$ is a potential. I would be interested in criteria for more general Hamiltonians given as matrices as above.

(2) Consider a truncation of H of the form: $H_N=P_N H P_N$ where $P_N$ is the projector on the subspace spanned by the first $N$ eigenvectors of $H_0$ (i.e. restrict indices $i,j$ to run from $1$ to $N$). What are conditions guaranteeing that for $N\to\infty$ the eigenvalues and eigenfunctions of $H_N$ approach those of $H$?