Program
 Eigenvalues and Eigenfunctions of the Laplacian Student No.： 50 Time： Mon/Wed/Fri 15:30-17:00pm, 08-07~08-21 Instructor： Steve Zelditch  [Northwestern University] Place： Conference Room 4, Floor 2, Jin Chun Yuan West Building Starting Date： 2013-8-7 Ending Date： 2013-8-21

You can download the lecture notes written by Prof.  Steve Zelditch by visit

http://ymsc.tsinghua.edu.cn/article.asp?channel=3&classid=36&parentid=17

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Description:

The course is about the eigenvalue problem \Delta \phi = - \lambda^2 \phi on a Riemannian manifold. It also discusses the Schrodinger equation. The first result is the Weyl law for counting eigenvalues. The main tools are the Hadamard parametrix construction of the fundamental solution of the wave equation, the method of stationary phase, and Tauberian theory. More extended methods give the pointwise Weyl law and the sharp bounds on sup norms of eigenfunctions. These bounds are only sharp for certain metrics like the round sphere. The next result is a theorem due to Sogge and myself which determines the geometric conditions for sharpness of the bounds. I also discuss this theorem for Schrodinger equations, where it is work in progress. I then discuss applications to nodal (zero) sets of eigenfunctions and present the currently best results in the direction of Yau's conjecture that the hypersurface measure is of the order of the square root of the eigenvalue. I will try to mention new results on lower Lp norms and the very different geometry of externals for high and low Lp norms.

Prerequisite:

Basic analysis on manifolds, e.g. vector fields, flows etc. Some acquaintance with Riemannian geometry, e.g. geodesic flow as the Hamilton flow on the cotangent bundle with respect to the length square function of the convector. Also the distance function and volume form of a metric. Most of the PDE is hyperbolic equations and it would help to know the fundamental solution of the wave equation on Euclidean space. Some familiarity with spherical harmonics would be useful. Also basic real analysis is needed, e.g. bounds for integral operators on Hilbert spaces in terms of the kernel (Schur-Young). Some familiarity with the method of stationary phase would be useful.

Reference:

C.D.Sogge, Fourier Integrals in Classical Analysis, Cambridge Tracts in Mathematics

L. H\"ormander, The analysis of linear partial differential operators. I. Distribution theory and Fourier analysis. Classics in Mathematics. Springer-Verlag, Berlin, 2003.