000 02250cam a2200253 i 4500
001 136618
003 ISI Library, Kolkata
005 20160310123335.0
008 130823s2014 nju b 001 0 eng
020 _a9780691160788 (pbk. : alk. paper)
040 _aISI Library
_beng
082 0 4 _a515.3533
_223
_bSo682
100 1 _aSogge, Christopher D.
245 1 0 _aHangzhou lectures on eigenfunctions of the Laplacian /
_cChristopher D. Sogge.
260 _aPrinceton :
_bPrinceton University Press,
_c2014.
300 _ax, 193 p. ;
_c26 cm.
490 0 _aAnnals of mathematics studies ;
_vno 188.
504 _aIncludes bibliographical references and index.
505 0 _a1. 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.
520 _aThis 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.
650 0 _aLaplacian operator.
650 0 _aEigenfunctions.
942 _2ddc
_cBK
999 _c420197
_d420197