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Notes for Spectral Geometry by Yaiza Canzani. 2016 CRM Summer School in Quebec City. Laval University. 3 Abstract I wrote these lectures using the material in set of notes \Analysis on manifolds via the Laplacian" available at http://www.math.harvard.edu/ canzani/docs/Laplacian.pdf. The material there corresponds to a year long course, so it contains more topics and many more details. The following references were important sources for these notes: Eigenvalues in Riemannian geometry. By I. Chavel. • Old and new aspects in Spectral Geometry. By M. Craiveanu, M. Puta and T. Ras- • sias. The Laplacian on a Riemannian manifold. By S. Rosenberg. • Local and global analysis of eigenfunctions on Riemannian manifolds. By S. Zelditch. • Enjoy!! Syllabus Abstract . 3 1 What makes the Laplacian special? 7 1.1 Daily life problems . 7 1.2 You want to solve the Helmholtz equation . 9 1.3 Inverse problem: Can you hear the shape of a drum? . 10 1.3.1 Hearing the length of a guitar string . 12 1.4 Direct problems . 14 1.4.1 The first eigenvalue: Rayleigh Conjecture . 14 1.4.2 Counting eigenvalues: Weyl's Law . 15 1.5 A hard problem: understanding the eigenfunctions . 18 2 The Laplacian on a Riemannian manifold 21 2.1 Definition of the Laplacian on a manifold . 21 2.1.1 Definition of a manifold . 21 2.1.2 Definition of a Riemannian metric . 24 2.1.3 Definition of the Laplacian . 27 2.2 Spectral Theory for the Laplacian . 28 2.2.1 Spectral Theorem . 30 2.2.2 Eigenvalue characterization (Exercise 4) . 30 2.3 Examples of eigenfunctions and eigenvalues . 31 2.3.1 Circle . 31 2.3.2 Torus . 31 2.3.3 Eigenfunctions on the sphere (Exercise 3) . 32 3 Hearing the geometry of a manifold 35 3.1 Heat Equation . 35 3.1.1 Solutions to Heat equation (Exercise 4) . 35 3.1.2 Definition of the fundamental solution . 35 3.2 Weyl's Law and other high energy asymptotics . 38 3.3 Isospectral manifolds . 40 4 Exercises 43 4.1 Exercise 0: Recovering the eigenvalues of a guitar string . 43 4.2 Exercise 1: Weyl's Law for a rectangle . 43 4.3 Exercise 2: Weyl's law on the Torus . 44 4.4 Exericise 3: Eigenfunctions on the Sphere . 44 4.5 Exercise 4: Characterization of eigenvalues . 45 4.6 Exercise 5: Temperature decreases with time and solutions are unique . 46 4.7 Exercise 5: Nodal domains and first eigenfunctions . 46 CHAPTER 1 What makes the Laplacian special? In this chapter we motivate the study of the Laplace operator. To simplify exposition, we do this by concentrating on planar domains. 1.1 Daily life problems n 1 Let Ω R be a connected domain and consider the operator ∆ acting on C (Ω) that ⊂ simply differentiates a function ' C1(Ω) two times with respect to each position 2 variable: n X @2' ∆' = : @x2 i=1 i Figure: Pierre-Simon de Laplace 8 What makes the Laplacian special? Here are some examples where the Laplacian plays a key role: Heat diffusion. If you are interested in understanding how would heat propagate n along Ω R then you should solve the Heat Equation ⊂ 1 @ ∆u(x; t) = u(x; t) c @t where c is the conductivity of the material of which Ω is made of, and u(x; t) is the temperature at the point x Ω at time t. 2 You could also think you have an insulated region Ω (it could be a wire, a ball, etc.) and apply certain given temperatures on the edge @Ω. If you want to know what the temperature will be after a long enough period of time (that is, the steady state temperature distribution), then you need to find a solution of the heat equation that be independent of time. The steady state temperature solution will be a function u(x1; : : : ; xn; t) such that ∆u = 0: Wave propagation. Now, instead of applying heat to the surface suppose you cover it with a thin layer of some fluid and you wish to describe the motion of the surface of the fluid. Then you will need to solve the Wave equation 1 @2 ∆u(x; t) = u(x; t) c @t2 where pc is the speed of sound in your fluid, and u(x; t) denotes the height of the wave above the point x at time t. You could also think of your domain Ω as the membrane of a drum, in which case its boundary @Ω would be attached to the rim of the drum. Suppose you want to study what will happen with the vibration you would generate if you hit it. Then, you @2 should also solve the wave equation ∆u(x; t) = @2t u(x; t) for your drum, but this time you want to make sure that you take into account that the border of the membrane is fixed. Thus, you should also ask your solution to satisfy u(x; t) = 0 for all points x @Ω. 2 Quantum particles. If you are a bit more eccentric and wish to see how a quantum particle moves inside Ω (under the assumption that there are no external forces) then you need to solve the Schr¨odingerEquation 2 ~ @ ∆u(x; t) = i~ u(x; t) −2m @t where ~ is Planck's constant and m is the mass of the free particle. Normalizing u so that u( ; t) 2 = 1 one interprets u(x; t) as a probability density. That is, if k · kL (Ω) A Ω then the probability that your quantum particle be inside A at time t is given ⊂ by R u(x; t) 2dx: A j j n Why not another operator? The Laplacian on R commutes with translations and rotations. That is, if T is a translation or rotation then ∆(' T ) = (∆') T . Something ◦ ◦ 1.2 You want to solve the Helmholtz equation 9 more striking occurs, if S is any operator that commutes with translations and rotations Pm j then there exist coefficients a1; : : : ; am making S = j=1 aj ∆ . Therefore, it is not surprising that the Laplacian will be a main star in any process whose underlying physics are independent of position and direction such as heat diffusion and wave propagation n in R . 1.2 You want to solve the Helmholtz equation There are of course many more problems involving the Laplacian, but we will focus on these ones to stress the importance of solving the eigenvalue problem (also known as Helmholtz equation) ∆' = λϕ. − It is clear that if one wants to study harmonic functions then one needs to solve the equation ∆' = λϕ with λ = 0: − So the need for understanding solutions of the Helmholtz equation for problems such as the static electric field or the steady-state fluid flow is straightforward. In order to attack the heat diffusion, wave propagation and Schr¨odingerproblems described above, a standard method (inspired by Stone-Weierstrass Theorem) is to look for solutions u(x; t) of the form u(x; t) = α(t)'(x). For instance if you do this and look at the Heat equation then you must have ∆'(x) α0(t) = x Ω; t > 0: '(x) α(t) 2 This shows that there must exist a λ R such that 2 α0 = λα and ∆' = λϕ. − − Therefore ' must be an eigenfunction of the Laplacian with eigenvalue λ and α(t) = −λt −λ t e . Once you have these particular solutions uk = e k 'k you use the superposition principle to write a general solution X −λkt u(x; t) = ake 'k(x) k where the coefficients ak are chosen depending on the initial conditions. You could do the same with the wave equation (we do it in detail for a guitar string in Section 1.3.1) or with the Schr¨odingerequation and you will also find particular solutions of the form uk(x; t) = αk(t)'k(x) with 8 e−λkt Heat eqn; <> p i λ t ∆'k = λk'k and αk(t) = e k Wave eqn; − > :eiλkt Schr¨odingereqn: 10 What makes the Laplacian special? 1.3 Inverse problem: Can you hear the shape of a drum? Inverse problem: If I know (more or less) the Laplace eigenvalues of a domain, what can I deduce of its geometry? Suppose you have perfect pitch. Could you derive the shape of a drum from the music you hear from it? More generally, can you determine the structural soundness of an object by listening to its vibrations? This question was first posed by Schuster in 1882. As Berger says in his book A panoramic view of Riemannian Geometry, \Already in the middle ages bell makers knew how to detect invisible cracks by sounding a bell on the ground before lifting it up to the belfry. How can one test the resistance to vibrations of large modern structures by non- destructive essays?... A small crack will not only change the boundary shape of our domain, one side of the crack will strike the other during vibrations invalidating our use of the simple linear wave equation. On the other hand, heat will presumably not leak out of a thin crack very quickly, so perhaps the heat equation will still provide a reasonable approximation for a short time..." An infinite sequence of numbers determines via Fourier analysis an integrable function. It wouldn't be that crazy if an infinite sequences of eigenvalues would determine the shape of the domain. Unfortunately, the answer to the question can you hear the shape of a drum? is no.
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