By Christopher D. Sogge
Based on lectures given at Zhejiang college in Hangzhou, China, and Johns Hopkins college, this e-book introduces eigenfunctions on Riemannian manifolds. Christopher Sogge supplies an explanation of the pointy Weyl formulation for the distribution of eigenvalues of Laplace-Beltrami operators, in addition to a higher model of the Weyl formulation, the Duistermaat-Guillemin theorem lower than ordinary assumptions at the geodesic movement. Sogge indicates that there's quantum ergodicity of eigenfunctions if the geodesic stream is ergodic.
Sogge starts with a remedy of the Hadamard parametrix earlier than proving the 1st major consequence, the pointy Weyl formulation. He avoids using Tauberian estimates and as a substitute depends on sup-norm estimates for eigenfunctions. the writer additionally supplies a quick creation to the desk bound part and the fundamentals of the idea of pseudodifferential operators and microlocal research. those are used to turn out the Duistermaat-Guillemin theorem. Turning to the comparable subject of quantum ergodicity, Sogge demonstrates that if the long term geodesic circulation is uniformly allotted, so much eigenfunctions show an analogous habit, within the feel that their mass turns into equidistributed as their frequencies visit infinity.
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Extra resources for Hangzhou Lectures on Eigenfunctions of the Laplacian (AM-188) (Annals of Mathematics Studies)
Hangzhou Lectures on Eigenfunctions of the Laplacian (AM-188) (Annals of Mathematics Studies) by Christopher D. Sogge