Spectral estimates for Schrödinger and Dirac-type operators on Riemannian manifolds. (English) Zbl 0791.58094

From a Hilbertian theorem (which generalizes a Theorem of S. Gallot and D. Meyer), we deduce a general criterion to compare the spectra of two operators \(T\) and \(T'\) acting on Hilbert spaces \(H\) and \(H'\), under the assumption that they verify Kato’s inequality with respect to a map \(\omega: H' \to H\). We apply this principle in several contexts. We obtain estimates for the spectra of Schrödinger and Dirac-type operators by the spectrum of the Riemannian base manifold. Our results contain, as particular cases, previous estimates given by other authors (S. Gallot and D. Meyer for the Hodge-de Rham Laplacian acting on differential forms, T. Friedrich for Dirac operator acting on spinors). The criterion, applied to Riemannian submersions, or coverings, or local quasi-isometries, gives estimates of the spectrum of the total space in terms of the spectrum of the base manifold.
Reviewer: M.Bordoni (Roma)


58J50 Spectral problems; spectral geometry; scattering theory on manifolds
35P15 Estimates of eigenvalues in context of PDEs
53C20 Global Riemannian geometry, including pinching
53C27 Spin and Spin\({}^c\) geometry
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