Hangzhou lectures on eigenfunctions of the Laplacian / Christopher D. Sogge.

Includes bibliographical references and index.

1. A review : the Laplacian and the d'Alembertian --
2. Geodesics and the Hadamard paramatrix --
3. The sharp Weyl formula --
4. Stationary phase and microlocal analysis --
5. Improved spectral asymptotics and periodic geodesics --
6. Classical and quantum ergodicity --
Appendix --
Notes --
Bibliography --
Index --
Symbol glossary.

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.