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Von Neumann spectra near the spectral gap. (English) Zbl 0911.58034

Author’s abstract: “In this paper we study some new von Neumann spectral invariants associated to the Laplacian acting on \(L^2\) differential forms on the universal cover of a closed manifold. These invariants coincide with the Novikov-Shubin invariants whenever there is no spectral gap in the spectrum of the Laplacian, and are homotopy invariants in this case. In the presence of a spectral gap, they differ in character and value from the Novikov-Shubin invariants. Under a positivity assumption on these invariants, we prove that certain \(L^2\) theta and \(L^2\) zeta functions defined by metric dependent combinatorial Laplacian acting on \(L^2\) cochains associated with a triangulation of the manifold, converge uniformly to their analytic counterparts, as the mesh of the triangulation goes to zero”.

MSC:

58J35 Heat and other parabolic equation methods for PDEs on manifolds
58J37 Perturbations of PDEs on manifolds; asymptotics
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
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