99-405 M. Christ, A. Kiselev, Y. Last

Approximate Eigenvectors and Spectral Theory (368K, Postscript) 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.

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