TITLE: Convergence of spectra under strong convergence of operators
QUESTION [11 upvotes]: Say $\left\{A_n\right\}$ is a sequence of bounded self-adjoint operators on a separable Hilbert space, converging in strong operator topology to a (bounded, self-adjoint) operator $A$. Denote the spectrum of $A_n$ by $\sigma_n$, and the spectrum of $A$ by $\sigma$. Under what conditions does it follow that $\sigma_n\rightarrow\sigma$ in Hausdorff metric? Any references will be appreciated.

REPLY [2 votes]: Even if it might be to late for answering. The following work provides a characterization of the convergence of the spectrum with respect to the Hausdorff metric. The continuity is satisfied whenever the norms of all polynomials (up to degree 2) of the operator behave continuous. One could do much better: In particular quantitative estimates are provided. Even there it can be read as a characterization.
"Continuity of the spectrum of a field of self-adjoint operators"
by Siegfried Beckus and Jean Bellissard
http://arxiv.org/abs/1507.04641
A tool to prove this continuity is the theory about continuous fields of C*-algebras. In the paper the result can be read also as follows:
The spectrum of a field of bounded self-adjoint operators behaves continuous if and only if the related field of C*-algebras is continuous.