Continuity of the Spectrum of a Field of Self-Adjoint Operators |
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Authors: | Siegfried Beckus Jean Bellissard |
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Affiliation: | 1.Mathematisches Institut,Friedrich-Schiller-Universit?t Jena,Jena,Germany;2.Georgia Institute of Technology, School of Mathematics,Atlanta,USA |
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Abstract: | Given a family of self-adjoint operators ({(A_t)_{t in T}}) indexed by a parameter t in some topological space T, necessary and sufficient conditions are given for the spectrum ({sigma(A_t)}) to be Vietoris continuous with respect to t. Equivalently the boundaries and the gap edges are continuous in t. If (T, d) is a complete metric space with metric d, these conditions are extended to guarantee Hölder continuity of the spectral boundaries and of the spectral gap edges. As a corollary, an upper bound is provided for the size of closing gaps. |
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