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Useful Bounds on the Extreme Eigenvalues and Vectors of Matrices for Harper’s Operators

  • Daniel BumpEmail author
  • Persi Diaconis
  • Angela Hicks
  • Laurent Miclo
  • Harold Widom
Chapter
Part of the Operator Theory: Advances and Applications book series (OT, volume 259)

Abstract

In analyzing a simple random walk on the Heisenberg group we encounter the problem of bounding the extreme eigenvalues of an n×n matrix of the form M = C+D where C is a circulant and D a diagonal matrix. The discrete Schrödinger operators are an interesting special case. TheWeyl and Horn bounds are not useful here. This paper develops three different approaches to getting good bounds. The first uses the geometry of the eigenspaces of C and D, applying a discrete version of the uncertainty principle. The second shows that, in a useful limit, the matrix M tends to the harmonic oscillator on L 2(ℝ) and the known eigenstructure can be transferred back. The third approach is purely probabilistic, extending M to an absorbing Markov chain and using hitting time arguments to bound the Dirichlet eigenvalues. The approaches allow generalization to other walks on other groups.

Keywords

Heisenberg group almost Mathieu operator Fourier analysis random walk 

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Daniel Bump
    • 1
    Email author
  • Persi Diaconis
    • 1
  • Angela Hicks
    • 2
  • Laurent Miclo
    • 3
  • Harold Widom
    • 4
  1. 1.Department of MathematicsStanford UniversityStanfordUSA
  2. 2.Mathematics DepartmentLehigh UniversityBethlehemUSA
  3. 3.Institut de Mathématiques de ToulouseUniversité Paul SabatierToulouse Cedex 9France
  4. 4.Department of MathematicsUC Santa CruzSanta CruzUSA

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