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

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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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