Spectral Theory

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[Spectral Theory in Riemannian Geometry]   by Olivier Lablee

                                 Topics in Spectral Geometry       Michael Levitin   Dan Mangoubi  Iosif Polterovich

  1. 譜理論簡介(Spectral Geometry)  [一個ODE的eigen functions and eigenvaluse]  [Dyson的RMT]
    [Spec002-2HyperbolicPlane]  [EigenFunctions]  [Laplacian operator的幾何意義]
    [振動弦  振動鼓膜 與 Laplacian]  [Lesson 001...S^1上的一維分佈(廣義函數)]
  2. 譜定理 Detailed introduction to abstract spectral theory
    2.1 Linear operators  [Lax-Milgram theorem]
    [Weak derivative and weak solution] 2.2 Closed operators  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. 變分原理與應用
  4. The Laplacian on a compact Riemannian manifold
    3.1 Bacsic Riemannian Geometry
    3.2 Analysis on manifolds  [Distribution and Green functions ] [Dirac distribution] [What is the Sobolev space]
     [The Laplacian(1)] [The Laplacian on manifold(2)(flat Torus)]
    5. [習作]
  5. Spectrum of the Laplacian on a compact manifold
    4.1 Physical examples  
    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]
    [習作]
  6. Direct problems in spectral geometry
    5.2 [Wave equation and Eigenequation]
    5.4[The Laplace operator on sphere (1)n=2 (2)n=3 (3)Sphere] [Lichnerowicz-Obata theorem]  [S^2] [Hyperbolic plane]
    [習作]   [Weyl asymptotic formula(Gauss circle problem)]
  7. 特徵函數的節點(Nodal)幾何
  8. 特徵值不等式
  9. Spectral invariants,and isospectrality
  10. The Steklov problem and the Dirichlet-to-Neumann map
  11. Inverse problems in spectral geometry   
    Length spectrum and trace formula
  12.  [Schrodinger equation] ...>Hydrogen Atom  [Eigenvalue for a Schrodinger operator]  [Morse and Spectral theory...不懂]
  13. [Stone theorem]  Edward Witten (1) Jone-Witten理論及Floer同調群 (2)Witten形變及古典摩斯

習作:

  1. [Lesson 003  Sobolev space]
  2. [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 (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