Download Annals of Mathematics Studies: Hangzhou Lectures on Eigenfunctions of the Laplacian by Christopher D. Sogge DJV, EPUB
9780691160788 0691160783 Based on lectures given at Zhejiang University in Hangzhou, China, and Johns Hopkins University, this book introduces eigenfunctions on Riemannian manifolds. Christopher Sogge gives a proof of the sharp Weyl formula for the distribution of eigenvalues of Laplace-Beltrami operators, as well as an improved version of the Weyl formula, the Duistermaat-Guillemin theorem under natural assumptions on the geodesic flow. Sogge shows that there is quantum ergodicity of eigenfunctions if the geodesic flow is ergodic. Sogge begins with a treatment of the Hadamard parametrix before proving the first main result, the sharp Weyl formula. He avoids the use of Tauberian estimates and instead relies on sup-norm estimates for eigenfunctions. The author also gives a rapid introduction to the stationary phase and the basics of the theory of pseudodifferential operators and microlocal analysis. These are used to prove the Duistermaat-Guillemin theorem. Turning to the related topic of quantum ergodicity, Sogge demonstrates that if the long-term geodesic flow is uniformly distributed, most eigenfunctions exhibit a similar behavior, in the sense that their mass becomes equidistributed as their frequencies go to infinity., Princeton University Press is proud to have published the Annals of Mathematics Studies since 1940. One of the oldest and most respected series in science publishing, it has included many of the most important and influential mathematical works of the twentieth century. The series continues this tradition into the twenty-first century as Princeton University Press publishes the major works of the new millennium. To mark the continued success of the series, all books are again available in paperback. For a complete list of titles, please visit the Princeton University Press web site: www.press.princeton.edu. The most recently published volumes include: Chow Rings, Decomposition of the Diagonal, and the Topology of Families, Claire Voisin, Spaces of PL Manifolds and Categories of Simple Maps, Friedhelm Waldhausen, Bjørn Jahren, and John Rognes, Degenerate Diffusion Operators Arising in Population Biology, Charles L. Epstein and Rafe Mazzeo, The Gross-Zagier Formula on Shimura Curves, Xinyi Yuan, Shou-wu Zhang, and Wei Zhang, Mumford-Tate Groups and Domains: Their Geometry and Arithmetic, Mark Green, Phillip A. Griffiths, and Matt Kerr, The Decomposition of Global Conformal Invariants, Spyros Alexakis, Some Problems of Unlikely Intersections in Arithmetic and Geometry, Umberto Zannier, Convolution and Equidistribution: Sato-Tate Theorems for Finite-Field Mellin Transforms, Nicholas M. Katz, Fréchet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces, Joram Lindenstrauss, David Preiss, and Jaroslav Tiser, The Ambient Metric, Charles Fefferman and C. Robin Graham, Hypoelliptic Laplacian and Orbital Integrals, Jean-Michel Bismut, Computational Aspects of Modular Forms and Galois Representations: How One Can Compute in Polynomial Time the Value of Ramanujan's Tail at a Prime, edited by Bas Edixhoven & Jean-Marc Couveignes, Weyl Group Multiple Dirichlet Series: Type A Combinatorial Theory, Ben Brubaker, Daniel Bump, and Solomon Friedberg Book jacket.
9780691160788 0691160783 Based on lectures given at Zhejiang University in Hangzhou, China, and Johns Hopkins University, this book introduces eigenfunctions on Riemannian manifolds. Christopher Sogge gives a proof of the sharp Weyl formula for the distribution of eigenvalues of Laplace-Beltrami operators, as well as an improved version of the Weyl formula, the Duistermaat-Guillemin theorem under natural assumptions on the geodesic flow. Sogge shows that there is quantum ergodicity of eigenfunctions if the geodesic flow is ergodic. Sogge begins with a treatment of the Hadamard parametrix before proving the first main result, the sharp Weyl formula. He avoids the use of Tauberian estimates and instead relies on sup-norm estimates for eigenfunctions. The author also gives a rapid introduction to the stationary phase and the basics of the theory of pseudodifferential operators and microlocal analysis. These are used to prove the Duistermaat-Guillemin theorem. Turning to the related topic of quantum ergodicity, Sogge demonstrates that if the long-term geodesic flow is uniformly distributed, most eigenfunctions exhibit a similar behavior, in the sense that their mass becomes equidistributed as their frequencies go to infinity., Princeton University Press is proud to have published the Annals of Mathematics Studies since 1940. One of the oldest and most respected series in science publishing, it has included many of the most important and influential mathematical works of the twentieth century. The series continues this tradition into the twenty-first century as Princeton University Press publishes the major works of the new millennium. To mark the continued success of the series, all books are again available in paperback. For a complete list of titles, please visit the Princeton University Press web site: www.press.princeton.edu. The most recently published volumes include: Chow Rings, Decomposition of the Diagonal, and the Topology of Families, Claire Voisin, Spaces of PL Manifolds and Categories of Simple Maps, Friedhelm Waldhausen, Bjørn Jahren, and John Rognes, Degenerate Diffusion Operators Arising in Population Biology, Charles L. Epstein and Rafe Mazzeo, The Gross-Zagier Formula on Shimura Curves, Xinyi Yuan, Shou-wu Zhang, and Wei Zhang, Mumford-Tate Groups and Domains: Their Geometry and Arithmetic, Mark Green, Phillip A. Griffiths, and Matt Kerr, The Decomposition of Global Conformal Invariants, Spyros Alexakis, Some Problems of Unlikely Intersections in Arithmetic and Geometry, Umberto Zannier, Convolution and Equidistribution: Sato-Tate Theorems for Finite-Field Mellin Transforms, Nicholas M. Katz, Fréchet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces, Joram Lindenstrauss, David Preiss, and Jaroslav Tiser, The Ambient Metric, Charles Fefferman and C. Robin Graham, Hypoelliptic Laplacian and Orbital Integrals, Jean-Michel Bismut, Computational Aspects of Modular Forms and Galois Representations: How One Can Compute in Polynomial Time the Value of Ramanujan's Tail at a Prime, edited by Bas Edixhoven & Jean-Marc Couveignes, Weyl Group Multiple Dirichlet Series: Type A Combinatorial Theory, Ben Brubaker, Daniel Bump, and Solomon Friedberg Book jacket.