Nelson's analytic vector theorem - Imported Knowlpedia Content

Statement

Let $A$ be a symmetric operator on a complex Hilbert space. Suppose the domain of $A$ contains a dense set $D$ such that every $v\in D$ is analytic for $A$ :

v\in\bigcap_{n\geq 0}\mathcal D(A^n) \quad\text{and}\quad \sum_{n=0}^{\infty}\frac{t_v^n}{n!}\lVert A^n v\rVert<\infty

for some $t_v>0$ . Nelson's analytic vector theorem states that $A$ is essentially self-adjoint. In particular, its closure is the unique self-adjoint operator extending $A$ from its stated domain.

Lie-algebra form

For a representation of a finite-dimensional real Lie algebra by skew-symmetric operators $d\pi(X_j)$ on a common dense invariant domain, form the Nelson Laplacian $\Delta=-\sum_j d\pi(X_j)^2$ . Essential self-adjointness of $\Delta$ , obtainable from a dense set of analytic vectors for $\Delta$ , supplies analytic vectors for the Lie-algebra action and integrates it to a unitary representation of the simply connected Lie group. It also yields the essential skew-adjointness of the infinitesimal generators Nelson, §§8–10.

Why analyticity matters

Symmetry alone does not imply essential self-adjointness: symmetric differential operators on incomplete domains may have several self-adjoint extensions. Analyticity supplies convergent power-series control strong enough to prove uniqueness of the unitary evolution. The theorem is therefore a domain criterion, not merely a regularity statement about vectors already known to lie in a unitary representation.

Conventions and scope

References

Edward Nelson, Analytic vectors, Annals of Mathematics 70 (1959), 572–615. DOI record. Relevant: the analytic-vector criterion and §§8–10 on Lie-algebra exponentiation.
Michael Reed and Barry Simon, Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness, Academic Press, 1975. Publisher record. Relevant: Theorem X.39 and §X.6.