Christopher D. Sogge
- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691160757
- eISBN:
- 9781400850549
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691160757.003.0006
- Subject:
- Mathematics, Numerical Analysis
This chapter proves results involving the quantum ergodicity of certain high-frequency eigenfunctions. Ergodic theory originally arose in the work of physicists studying statistical mechanics at the ...
More
This chapter proves results involving the quantum ergodicity of certain high-frequency eigenfunctions. Ergodic theory originally arose in the work of physicists studying statistical mechanics at the end of the nineteenth century. The word ergodic has as its roots two Greek words: ergon, meaning work or energy, and hodos, meaning path or way. Even though ergodic theory's initial development was motivated by physical problems, it has become an important branch of pure mathematics that studies dynamical systems possessing an invariant measure. Thus, this chapter first presents some of the basic limit theorems that are key to the classical theory. It then turns to quantum ergodicity.Less
This chapter proves results involving the quantum ergodicity of certain high-frequency eigenfunctions. Ergodic theory originally arose in the work of physicists studying statistical mechanics at the end of the nineteenth century. The word ergodic has as its roots two Greek words: ergon, meaning work or energy, and hodos, meaning path or way. Even though ergodic theory's initial development was motivated by physical problems, it has become an important branch of pure mathematics that studies dynamical systems possessing an invariant measure. Thus, this chapter first presents some of the basic limit theorems that are key to the classical theory. It then turns to quantum ergodicity.
Christopher D. Sogge
- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691160757
- eISBN:
- 9781400850549
- Item type:
- book
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691160757.001.0001
- Subject:
- Mathematics, Numerical Analysis
Based on lectures given at Zhejiang University in Hangzhou, China, and Johns Hopkins University, this book introduces eigenfunctions on Riemannian manifolds. The book gives a proof of the sharp Weyl ...
More
Based on lectures given at Zhejiang University in Hangzhou, China, and Johns Hopkins University, this book introduces eigenfunctions on Riemannian manifolds. The book 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. The book shows that there is quantum ergodicity of eigenfunctions if the geodesic flow is ergodic. It begins with a treatment of the Hadamard parametrix before proving the first main result, the sharp Weyl formula. The book avoids the use of Tauberian estimates and instead relies on sup-norm estimates for eigenfunctions. It 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, the book 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.Less
Based on lectures given at Zhejiang University in Hangzhou, China, and Johns Hopkins University, this book introduces eigenfunctions on Riemannian manifolds. The book 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. The book shows that there is quantum ergodicity of eigenfunctions if the geodesic flow is ergodic. It begins with a treatment of the Hadamard parametrix before proving the first main result, the sharp Weyl formula. The book avoids the use of Tauberian estimates and instead relies on sup-norm estimates for eigenfunctions. It 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, the book 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.
