- 99-405 M. Christ, A. Kiselev, Y. Last
- Approximate Eigenvectors and Spectral Theory
Oct 24, 99
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Abstract. We develop relations between spectral and eigenfunction
properties of self-adjoint operators and properties
of approximate eigenvectors of those operators. In particular,
we establish a new general criterion for studying continuity
properties of spectral measures. It can be viewed as a
generalization of the Weyl criterion, and provides information
on pointwise continuity properties of spectral measures
(particularly, on positivity of upper $\alpha$-derivatives).
The criterion is formulated as a necessary and sufficient
condition involving certain sequences of approximate eigenvectors.
We also show that appropriately chosen sequences of approximate
eigenvectors converge to generalized eigenfunctions in an
appropriate weak sense. We apply these results to study
spectral properties of some concrete Schr\"odinger operators.