3 research outputs found
Explicit bounds on eigenfunctions and spectral functions on manifolds hyperbolic near a point
We derive explicit bounds for the remainder term in the local Weyl law for
locally hyperbolic manifolds, we also give the estimates of the derivative of
this remainder. We use these to obtain explicit bounds for the C^k-norms of the
L^2-normalised eigenfunctions in the case spectrum of the Laplacian is
discrete, e.g. for closed Riemannian manifolds. We also derive bounds for the
local heat trace. Our estimates are purely local and therefore also hold for
any manifold at points near which the metric is locally hyperbolic.Comment: 23 pages, 2 figure
Computation of eigenvalues, spectral zeta functions and zeta-determinants on hyperbolic surfaces
These are lecture notes from a series of three lectures given at the summer
school "Geometric and Computational Spectral Theory" in Montreal in June 2015.
The aim of the lecture was to explain the mathematical theory behind
computations of eigenvalues and spectral determinants in geometrically
non-trivial contexts.Comment: 33 pages, LaTeX with sample Mathematica code included, some typos
correcte