MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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 $H$?

share|cite|improve this question
to (2) : you should make this more precise : For instance, there is a divergent series of eigenfunctions of $H_N$ with eigenvalue zero. – jjcale Aug 23 '13 at 19:51
Let's say I look at the first $k$ eigenvalues of $H$ and compare them with the first $k$ eigenvalues of $H_N$ for $N\gg k$. – Slava Rychkov Aug 23 '13 at 20:55

Concerning your first question, see paragraph 47 on "Matrix representations of unbounded symmetric operators" of Akhiezer & Glazman's Theory of Linear Operators in Hilbert Space. Theorem 4 gives a sufficient condition: $\sum_{i}|H_{ij}|^{2}<\infty$ for each $j$. Remarkbly enough, this condition need not be preserved by a unitary transformation of $H$.

share|cite|improve this answer

Your Answer


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

Not the answer you're looking for? Browse other questions tagged or ask your own question.