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Dirac operators with \(W^{1,\infty }\)-potential on collapsing sequences losing one dimension in the limit

manuscripta mathematica volume 157, pages 387–410 (2018)Cite this article

Abstract

We study the behavior of the spectrum of the Dirac operator together with a symmetric \(W^{1, \infty }\)-potential on a collapsing sequence of spin manifolds with bounded sectional curvature and diameter losing one dimension in the limit. If there is an induced spin or \(\text {pin}^-\) structure on the limit space N, then there are eigenvalues that converge to the spectrum of a first order differential operator D on N together with a symmetric \(W^{1,\infty }\)-potential. In the case of an orientable limit space N, D is the spin Dirac operator \(D^N\) on N if the dimension of the limit space is even and if the dimension of the limit space is odd, then \(D = D^N \oplus -D^N\).

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  1. Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111, Bonn, Germany

    Saskia Roos

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  1. Saskia Roos
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Roos, S. Dirac operators with \(W^{1,\infty }\)-potential on collapsing sequences losing one dimension in the limit. manuscripta math. 157, 387–410 (2018). https://doi.org/10.1007/s00229-018-1003-6

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Mathematics Subject Classification

  • 53C27
  • 58J50
  • 58J60
  • 31C12
  • 53C20