Eigenfunctions of the Laplacian on a Riemannian Manifold (CBMS Regional Conference Series in Mathematics) by Steve Zelditch (PDF)

Description

Eigenfunctions of the Laplacian of a Riemannian manifold can be described in terms of vibrating membranes as well as quantum energy eigenstates. This book is an introduction to both the local and global analysis of eigenfunctions. The local analysis of eigenfunctions pertains to the behavior of the eigenfunctions on wavelength scale balls. After re-scaling to a unit ball, the eigenfunctions resemble almost-harmonic functions. Global analysis refers to the use of wave equation methods to relate properties of eigenfunctions to properties of the geodesic flow.The emphasis is on the global methods and the use of Fourier integral operator methods to analyze norms and nodal sets of eigenfunctions. A somewhat unusual topic is the analytic continuation of eigenfunctions to Grauert tubes in the real analytic case, and the study of nodal sets in the complex domain.The book, which grew out of lectures given by the author at a CBMS conference in 2011, provides complete proofs of some model results, but more often it gives informal and intuitive explanations of proofs of fairly recent results. It conveys inter-related themes and results and offers an up-to-date comprehensive treatment of this important active area of research.

User’s Reviews

Editorial Reviews: Review The exposition is very clear and elegant, although it reaches rather technical results. –G. V. Rozenblum, Mathematical ReviewsWe have a very serious work of scholarship, covering (if you’ll pardon the pun) quite a spectrum of analysis (largely of the hard kind, as opposed to the soft kind), and useful to many audiences, from advanced students to experienced insiders. It’s also distinctive as a springboard to any number of deeper studies flowing from the material in the book’s individual chapters. It promises to be a very valuable resource. –Michael Berg, MAA Reviews About the Author Steve Zelditch, Northwestern University, Evanston, IL.

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