Spectral Theory

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[Spectral Theory in Riemannian Geometry]   by Olivier Lablee

  1. Introduction to Spectral Geometry   [一個ODE的eigen functions and eigenvaluse]  [Dyson的RMT]
  2. Detailed introduction to abstract spectral theory
    2.1 Linear operators  [Lax-Milgram theorem] 2.2 Closed operatore  2.3 Adjoint Operator  2.4 Spectrums and resolvent set  
    2.5 Spectral theory of compact operators  
    2.6 The spectral theorem multiplication operator ...
  3. The Laplacian on a compact Riemannian manifold    [Laplacian operator的幾何意義]
    3.1 Bacsic Riemannian Geometry
    3.2 Analysis on manifolds  [Distribution and Green functions ] [Dirac distribution] [What is the Sobolev space]
     [The Laplacian]
  4. Spectrum of the Laplacian on a compact manifold
    4.1 Physical examples [wave eq on a string] [The heat eq]  [The Schrodinger eq]  [Lesson 001...S^1上的一維分佈]
    4.2 A class of spectral problems
    4.3 Spectral theorem for the Laplacian
    4.4 A detailed proof  4.5 The minimal principle and applications
    4.6 The Schrodinger operator and the Hodge-de Rham Laplacian
    [Eigenfunction 1-dim]  [Eigenvalues  S^2]  [Calculation of the Spetrum] [Eigenvalue for a Schrodinger operator]
  5. Direct problems in spectral geometry
    5.2 [Rectangular domains] [Wave equation and Eigenequation]  5.4[n-Spheres and n=3]   [Lichnerowicz-Obata theorem]  [S^2] [Hyperbolic plane]
  6. Inverse problems in spectral geometry
  7.  [Schrodinger equation]   [薛丁格]   [Morse and Spectral theory...不懂]
  8. [Stone theorem]  Edward Witten (1) Jone-Witten理論及Floer同調群 (2)Witten形變及古典摩斯

課程 :

  1. [Lesson 001...S^1上的譜與一維分布(廣義函數)]
  2. [Lesson 002...弱導數與弱解]
  3. [Lesson 003  Sobolev space]
  4. [Lesson 004 為什麼說Green fnction是一個分佈]

 書目 :  [Laplacian算子對應譜的最近發展] [最近發展]

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