Spectral Theory of Random Schrödinger Operators

1. Front Matter

   Pages i-xxvi

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2. Spectral Theory of Self-Adjoint Operators

   • René Carmona, Jean Lacroix

   Pages 1-42

3. Schrödinger Operators

   • René Carmona, Jean Lacroix

   Pages 43-88

4. One-Dimensional Schrödinger Operators

   • René Carmona, Jean Lacroix

   Pages 89-174

5. Products of Random Matrices

   • René Carmona, Jean Lacroix

   Pages 175-240

6. Ergodic Families of Self-Adjoint Operators

   • René Carmona, Jean Lacroix

   Pages 241-298

7. The Integrated Density of States

   • René Carmona, Jean Lacroix

   Pages 299-357

8. Absolutely Continuous Spectrum and Inverse Theory

   • René Carmona, Jean Lacroix

   Pages 359-437

9. Localization in One Dimension

   • René Carmona, Jean Lacroix

   Pages 439-514

10. Localization in Any Dimension

    • René Carmona, Jean Lacroix

    Pages 515-555

11. Back Matter

    Pages 557-589

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Since the seminal work of P. Anderson in 1958, localization in disordered systems has been the object of intense investigations. Mathematically speaking, the phenomenon can be described as follows: the self-adjoint operators which are used as Hamiltonians for these systems have a ten­ dency to have pure point spectrum, especially in low dimension or for large disorder. A lot of effort has been devoted to the mathematical study of the random self-adjoint operators relevant to the theory of localization for disordered systems. It is fair to say that progress has been made and that the un­ derstanding of the phenomenon has improved. This does not mean that the subject is closed. Indeed, the number of important problems actually solved is not larger than the number of those remaining. Let us mention some of the latter: • A proof of localization at all energies is still missing for two dimen­ sional systems, though it should be within reachable range. In the case of the two dimensional lattice, this problem has been approached by the investigation of a finite discrete band, but the limiting pro­ cedure necessary to reach the full two-dimensional lattice has never been controlled. • The smoothness properties of the density of states seem to escape all attempts in dimension larger than one. This problem is particularly serious in the continuous case where one does not even know if it is continuous.

• Finite
• Hölder condition
• Identity
• Smooth function
• differential equation
• function
• operator theory
• proof
• spectral theorem
• subharmonic function
• theorem
• partial differential equations

• Department of Mathematics, University of California at Irvine, Irvine, USA

  René Carmona

• Département de Mathématiques, Université de Paris XIII, Villetaneuse, Fessy 74, France

  Jean Lacroix

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