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Erschienen in:
2017 | OriginalPaper | Buchkapitel
Useful Bounds on the Extreme Eigenvalues and Vectors of Matrices for Harper’s Operators
verfasst von : Daniel Bump, Persi Diaconis, Angela Hicks, Laurent Miclo, Harold Widom
Erschienen in: Large Truncated Toeplitz Matrices, Toeplitz Operators, and Related Topics
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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 L2(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.