Park City Lectures on Eigenfuntions

These lectures are devoted to nodal geometry of eigenfunctions φλ of the Laplacian ∆g of a Riemannian manifold (M, g) of dimension m and to the associated problems on L norms of eigenfunctions. The manifolds are generally assumed to be compact, although the problems can also be posed on non-compact complete Riemannian manifolds. The emphasis of these lectures is on real analytic Riemannian manifolds. We use real analyticity because the study of both nodal geometry and L norms simplifies when the eigenfunctions are analytically continued to the complexification of M . Although we emphasize the Laplacian, analogous problems and results exist for Schrödinger operators −~2 2 ∆g+V for certain potentials V . We now state state the main results, some classical and some relatively new, that we concentrate on in these lectures. The study of eigenfunctions of ∆g and −~ 2 2 ∆g + V on Riemannian manifolds is a branch of harmonic analysis. In these lectures, we emphasize high frequency (or semi-classical) asymptotics of eigenfunctions and their relations to the global dynamics of the geodesic flow G : S∗M → S∗M on the unit cosphere bundle of M . Here and henceforth we identity vectors and covectors using the metric. As in Ze3 [Ze3] we give the name “Global Harmonic Analysis” to the use of global wave equation methods to obtain relations between eigenfunction behavior and geodesics. Some of the principal results in semi-classical analysis are purely local and do not exploit this connection. The relations between geodesics and eigenfunctions belongs to the general correspondence principle between classical and quantum mechanics. The correspondence principle has evolved since the origins of quantum mechanics Sch2 [Sch] into a systematic theory of Semi-Classical Analysis and Fourier integral operators, of which HoI,HoII,HoIII,HoIV [HoI, HoII, HoIII, HoIV] and Zw [Zw] give systematic presentations; see also § FORMAT 1.13 for further references. Quantum mechanics provides not only the intuition and techniques for the study of eigenfunctions, but in large part also provides the motivation. Readers who are unfamiliar with quantum mechanics are encouraged to read the physics literature. Standard texts are Landau-Lifschitz LL [LL] and Weinberg Wei [Wei]. The reader might like to see the images recently produced by physicists using quantum microscopes to directly observe nodal sets of excited hydrogen atoms St [St].