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Hangzhou Lectures on Eigenfunctions of the Laplacian

Hangzhou Lectures on Eigenfunctions of the Laplacian

Christopher D. Sogge
Copyright Date: 2014
Pages: 256
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    Hangzhou Lectures on Eigenfunctions of the Laplacian
    Book Description:

    Based on lectures given at Zhejiang University in Hangzhou, China, and Johns Hopkins University, this 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.

    eISBN: 978-1-4008-5054-9
    Subjects: Mathematics

Table of Contents

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  1. Front Matter
    (pp. i-vi)
  2. Table of Contents
    (pp. vii-viii)
  3. Preface
    (pp. ix-xii)
  4. Chapter One A review: The Laplacian and the d’Alembertian
    (pp. 1-15)

    One of the main goals of this course is to understand well the solution of wave equation both in Euclidean space and on manifolds and then to use this knowledge to derive properties of eigenfunctions on Riemannian manifolds. This is a very classical idea. A key step in understanding properties of solutions of wave equations on manifolds will be to compute the types of distributions that include the fundamental solution of the wave operator in Minkowski space (d’Alembertian),

    $\square = {\partial ^2}/\partial {t^2} - \Delta $, (1.1.1)


    $\Delta = \sum\limits_{j = 1}^n {\partial _j^2} ,{\text{ }}{\partial _j} = \partial /\partial {x_j}$, (1.1.2)

    being the Euclidean Laplacian on${\mathbb{R}^n}$.

    In the next section we shall compute fundamental solutions for$\square $,...

  5. Chapter Two Geodesics and the Hadamard parametrix
    (pp. 16-38)

    A main goal of this course will be to study the spectrum of Laplace-Beltrami operators on compact manifolds. For the sake of simplicity, though, throughout this chapter we shall consider Laplace-Beltrami operators defined on an open subset Ω of${\mathbb{R}^n}$. We then will easily be able to lift results presented here to corresponding ones on compact manifolds, which among other things will provide us with the tools that we need to prove the sharp Weyl formula in the next chapter.

    Let us start our discussion by defining a metric on an open subset$\Omega \subset {\mathbb{R}^n}$. Everything that we shall review is...

  6. Chapter Three The sharp Weyl formula
    (pp. 39-70)

    From now on we shall mostly be concerned with analysis on compact boundaryless Riemannian manifolds. Let us quickly recall a few facts about this setting.

    First,Mis a${C^\infty }$manifold if it is a Hausdorff space for which there is a countable collection of open sets${\Omega _v} \subset M$together with homeomorphisms${k_v}:{\Omega _v} \to {{\tilde \Omega }_v} \subset {\mathbb{R}^n}$satisfying$ \cup {\Omega _v} = M$and

    ${k_{v'}} \circ k_v^{ - 1}:{k_v}({\Omega _v} \cap {\Omega _{v'}}) \to {k_{v'}}({\Omega _v} \cap {\Omega _{v'}})$is${C^\infty }$.

    Note that the above mapping is between open subsets of${\mathbb{R}^n}$. We call$y = {k_v}(x) \in {\tilde \Omega _v} \subset {\mathbb{R}^n}$the local coordinates ofxin the coordinates patch${\Omega _v}$.

    This${C^\infty }$structure allows us to define${C^\infty }$functions onMin a natural way. We say thatuis...

  7. Chapter Four Stationary phase and microlocal analysis
    (pp. 71-119)

    The purpose of this section is to go over basic techniques from the theory of stationary phase which will be useful for many of the calculations to follow. Specifically, we shall study the decay rates of oscillatory integrals of the form

    $I(\lambda ) = \int\limits_{{\mathbb{R}^n}} {{e^{{\text{i}}\lambda {\text{ }}\phi }}^{(y)}a(y)} dy,{\text{ }}\lambda \underline > 1$, (4.1.1)

    where throughout we shall assume that$\phi $is real and${C^\infty }$and$a \in C_0^\infty $.

    Difficulties can arise when$\phi $has stationary points, i.e., points in the support of a where$\nabla \phi (y) = 0$. If there are no such points the following “non-stationary phase” lemma says that the above oscillatory integral is repidly decreasing as$\lambda \to + \infty $.

    Lemma 4.1.1Suppose that...

  8. Chapter Five Improved spectral asymptotics and periodic geodesics
    (pp. 120-140)

    Let$(M,g)$be a compact boundaryless Riemannian manifold of dimension$n\underline > 2$The goal of this chapter is to prove an improved Weyl formula

    $N(\lambda ) = {(2\pi )^{ - n}}{\text{Vo}}{{\text{l}}_g}(B*M){\lambda ^n} + o({\lambda ^{n - 1}})$, (5.1.1)


    $\mathop {\lim {\text{sup}}}\limits_{\lambda \to + \infty } {\lambda ^{ - (n - 1)}}\left[ {N(\lambda ) - {{(2\pi )}^{ - n}}{\text{Vo}}{{\text{l}}_g}(B*M){\lambda ^n}} \right] = 0$, (5.1.2)

    under the assumption that the set of periodic geodesics for$(M,g)$has measure zero. Here$B*M \subset T*M$denotes the ball bundle associated with the metric

    $B*M = \left\{ {(x,\xi ) \in T*M:p(x,\xi )\underline < 1} \right\}$


    $p(x,\xi ) = \sqrt {\sum\limits_{j,k = 1}^n {{g^{{j^k}}}(x)} {\xi _j}{\xi _k}} $(5.1.3)

    denotos the principal symbol of$\sqrt { - {\Delta _g}} $. As in Remarks 3.3.2,

    ${\text{Vo}}{{\text{l}}_g}(B*M) = \iint_{P(x,\xi )\underline < 1} {d\xi dx}$

    with$d\xi dx$denoting the Liovuille measure.

    To make this assumption more precise, we recall that the cotangent bundle is the map${\Phi _t}:T*M\backslash 0 \to T*M\backslash 0$which is following for$t$along the Hamiltion vector field

    ${H_p} = \frac{{\partial p}} {{\partial \xi }}\frac{\partial } {{\partial x}} - \frac{{\partial p}} {{\partial x}}\frac{\partial } {{\partial \xi }}$...

  9. Chapter Six Classical and quantum ergodicity
    (pp. 141-164)

    Ergodic theory orginially arose in the work of physicists studying statistical mechanics at the end of the nineteenth century. The wordergodicwas introduced by Boltzmann and has as its roots two Greek words:ergon, meaning work or energy, andhodos, 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.

    The main purpose of this chapter is to prove results involving the quantum ergodicity of certain high frequency eigenfunctions. However, before doing this, we shall present some...

  10. Appendix
    (pp. 165-182)
  11. Notes
    (pp. 183-184)
  12. Bibliography
    (pp. 185-190)
  13. Index
    (pp. 191-192)
  14. Symbol Glossary
    (pp. 193-193)