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Spectral bounds for Dirac operators on open manifolds. (English) Zbl 1193.53119

The author considers spectral bounds for the eigenvalues of generalized Dirac first order self-adjoint elliptic differential operators on noncompact Riemannian manifolds. Some examples of generalized Dirac operators are given, such as the classical operator on the spinor bundle on a manifold with spin and the Dirac operator \(D=d+\delta \) on forms on a Riemannian manifold. The author extends results on some classical eigenvalue estimates for the Dirac operator from the compact case to noncompact Riemannian manifolds. Some of the results follow from lower bounds on the curvature endomorphism which, in particular, extends Friedrich’s estimate for manifolds with positive scalar curvature. While under the assumption that the curvature endomorphism is bounded below at infinity, the author gives a lower bound on the essential spectrum of the square of the Dirac operator. Further, on a connected 2-dimensional Riemannian manifold of genus 0 with finite area and spin structure of a certain type, an eigenvalue estimate for the square of the classical Dirac operator on spinors is given in terms of the area.

MSC:

53C27 Spin and Spin\({}^c\) geometry
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
57R15 Specialized structures on manifolds (spin manifolds, framed manifolds, etc.)
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