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Useful bounds on the extreme eigenvalues and vectors of matrices for Harper's operators | | Daniel Bump
; Persi Diaconis
; Angela Hicks
; Laurent Miclo
; Harold Widom
; | | Date: |
25 Aug 2015 | | Abstract: | In analyzing a simple random walk on the Heisenberg group we encounter the
problem of bounding the extreme eigenvalues of an $n imes n$ matrix of the
form $M=C+D$ where $C$ is a circulant and $D$ a diagonal matrix. The discrete
Schr"odinger operators are an interesting special case. The Weyl 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
$L^2(mathbb{R})$ 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. | | Source: | arXiv, 1508.5986 | | Services: | Forum | Review | PDF | Favorites | |
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