USEFUL BOUNDS ON THE EXTREME EIGENVALUES AND VECTORS OF MATRICES FOR HARPER'S OPERATORS

Daniel Bump , Persi Diaconis (1) , Angela Hicks (2) , Laurent Miclo (3) , Harold Widom

1 Department of Statistics [Stanford]
2 MATH - UC San Diego - Department of Mathematics [Univ California San Diego]
3 IMT - Institut de Mathématiques de Toulouse UMR5219

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.

Domaines: Probabilités [math.PR]

Daniel Bump, Persi Diaconis, Angela Hicks, Laurent Miclo, Harold Widom. USEFUL BOUNDS ON THE EXTREME EIGENVALUES AND VECTORS OF MATRICES FOR HARPER'S OPERATORS. Large truncated Toeplitz matrices, Toeplitz operators, and related topics, 2017. ⟨hal-02022492⟩