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Domain Perturbations, Shift of Eigenvalues and Capacity

Authors
Journal
Journal of Functional Analysis
0022-1236
Publisher
Elsevier
Publication Date
Volume
170
Issue
1
Identifiers
DOI: 10.1006/jfan.1999.3496

Abstract

Abstract The notion of capacity of a subspace which was introduced in [16] is used to prove new estimates on the shift of the eigenvalues which arises if the form domain of a self-adjoint and semibounded operator is restricted to a smaller subspace. The upper bound on the shift of the spectral bound given in [16] is improved and another lower bound is proved which leads to a generalization of Thirring's inequality if the underlying Hilbert space is an L 2-space. Moreover we prove a similar capacitary upper bound for the second eigenvalue. The results are applied to elliptic constant coefficient differential operators of arbitrary order. Finally it is given a capacitary characterization for the shift of the spectral bound being positive which works for operators with spectral bound of arbitrary type.

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