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

Abstract

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. 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 (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.

Spectral asymptotics for Kac–Murdock–Szegő matrices

“…KMS and related matrices have received a lot of attention lately with applications to such various fields as statistical mechanics [24,15], differential equations [5,42,43,36,28,41,17], quantum integrable systems [1,10,8] and the Heisenberg group [13,14], among others. There has been a renewed interest in matrices of this type which followed a renewed interest in Toeplitz forms and their connections to orthogonal polynomials.…”

Sums of twisted circulants

“…is the average of the images of X ±1 and Y ±1 in one of the n-dimensional irreducible representations of H(n). The papers [1] and [2] present several different proofs that the norm of M (r) is bounded above by 1 − O( 1 n ). Once this is done, a straightforward analysis of the one-dimensional representations of H(n) reveals that those eigenvalues can be as large as 1 − O( 1n 2 ), so the bound on M (r) shows that the behavior of random walk is governed by the one-dimensional representations and a mixing time of O(n 2 ) is established.…”

“…In this paper we generalize the first approach utilized in [2] in two different ways. Our main contribution is the analysis of simple random walks on H(n) with different minimal, symmetric generating sets.…”

“…Our approach, following [2], is to generalize a version of the Heisenberg Uncertainty Principle due to Donoho and Stark [4]. We discuss this further in Section 1.3.…”

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The rate of convergence of simple random walk on the Heisenberg group over Z/nZ with a standard generating set was determined by Bump et al [1,2] to be O(n 2 ). We extend this result to random walks on the same groups with an arbitrary minimal symmetric generating set.

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