{"product_id":"9781400850549","title":"Hangzhou Lectures on Eigenfunctions of the Laplacian (AM-188)","description":"\u003cp\u003eBased 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.\u003c\/p\u003e\u003cp\u003e\u003c\/p\u003e\u003cp\u003eSogge 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.\u003c\/p\u003e\u003cp\u003e\u003c\/p\u003e","brand":"Princeton University Press","offers":[{"title":"Default Title","offer_id":47121524883696,"sku":"9781400850549","price":49.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0737\/7593\/9824\/files\/9781400850549_p0.jpg?v=1763712855","url":"https:\/\/shop-qa.barnesandnoble.com\/products\/9781400850549","provider":"Barnes \u0026 Noble (DEV)","version":"1.0","type":"link"}