Spectral Theory

img1.gif

[Spectral Theory in Riemannian Geometry]   by Olivier Lablee

                                 Topics in Spectral Geometry       Michael Levitin   Dan Mangoubi  Iosif Polterovich

第一章 譜理論簡介

  1. 譜理論簡介(Spectral Geometry) (1)各領域的比較  (2)簡介 (3) 問題與反問題 (4)Open problems (5)伴隨算子
  2. 一個ODE的eigen functions and eigenvaluse      Morse理論與譜理論      Laplace譜的最近發展
  3. Laplacian operator的幾何意義      Laplacian on a manifold
  4. 振動弦   振動鼓膜
  5. Hyperbolic plane

第二章 譜定理

  1. Weak derivative與weak solution
  2. 特徵函數    特徵值
  3. Lax-Milgram定理
  4. Sobolev Space

第三章 緊緻黎曼流形上的Laplace算子與譜

  1. Basic Riemannian Geometry      geodesic      球體積
  2. Analysis on manifolds (1)Distribution and Green functions (2)Dirac distribution (3) What is the Sobolev space
  3. 圓周S^1上的Laplace算子 一維分布   S^2上的特徵值
  4. Physical examples (1)振動弦 (2)heat equation
  5. 波動方程 (1)矩形區域的鼓膜振動
  6. Flat tori
  7. Rectangular domains with boundary condition
  8. S^n上的特徵函數與譜   (1)S^3
  9. Harmonic oscillations
  10. Flat Kline bottle的譜      Spetrum for product manifolds      Hilbert basis

第四章 變分原理與應用

  1. The Rayleigh quotient
  2. Variational principles
  3. The Robin and Zaremba problems
  4. Domain monotonicity and Dirichlet-Neumann bracketing
  5. Symmatry tricks
  6. Counting functions
  7. Inequalities
  8. Weyl law and Polya conjecture
  9. The Berezin-Li-Yau inequatities

第五章 特徵函數的節點(Nodal)幾何

第六章 特徵值不等式

第七章 薛丁格方程 [Schrodinger equation] ...>Hydrogen Atom  [Eigenvalue for a Schrodinger operator]  [Morse and Spectral theory...不懂]

第八章  譜幾何的定理

  1. (1)Obata定理   (2)[Stone theorem]  Edward Witten (1) Jone-Witten理論及Floer同調群 (2)Witten形變及古典摩斯
  2. Weyl Formula      [Weyl formula for rectangular domain]

第九章  Specitial topics

  1. A first eigenvalue estimate for minimal hypersurfaces
  2. Eigenvalues of the Laplacian of compact Riemann surfaces and minimal submanifolds (丘成桐) 1980
  3. 在S^3中 所有嵌入的子流形的第一非零Neumann特徵值=2 (丘成桐) (CMC曲面的穩定性問題)
  4. 氫原子哈密頓算子在H^2(R^3)的作用

 書目 :  [Laplacian算子對應譜的最近發展]

  1. Spectral theory (Basic concepts and application)   David Borthwick   [Research]
  2. Spectral theory and Mathematical Physics   Pablo Miranda ...
  3. Spectral theory and its Applications   Bernard Helffer [ResearchGate]
  4. 譜理論手冊      C.Cheverry and N.Raymond
  5. Introduction to Spectral theory (Schrodinger operators)   Peter D.Hislop [ResearchGate] and Israel Michael Sigal
  6. Spectral theory and Application...   Aref Jeribi [ResearchGate]
  7. Functional Analysis and Spectral theory   Manfred Einsiedler  Thomas Ward
  8. Eigenvalues of the Laplacian of compact Riemann surfaces and minimal submanifolds......丘成桐 楊建平
  9. 薛丁格