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Spectral asymptotics for the Laplacian and lattice point counting

John Toth
McGill University

Thursday, January 24, 2002
102 Bradley Hall, 4 pm
Tea 3:30 pm, Math Lounge

Abstract: Let (M,g) be a compact Riemann manifold and -\Delta its Laplace-Beltrami operator. The spectral counting function, N(\lambda), counts (with multiplicities) the number of Laplace eigenvalues less than \lambda. The Weyl formula states that to leading order:
N(\lambda) \sim_{\lambda \rightarrow \infty} c \lambda^{n/2}.
We will review some known results concerning the error term R(\lambda) := N(\lambda) - c \lambda^{n/2} and then discuss recent work (joint with Y. Petridis) on bounds for R(\lambda) in the case of a flat torus or a Heisenberg manifold. In these cases, asymptotics for N(\lambda) can be reduced to certain lattice point counting problems.

This talk will be accessible to undergraduates.