2002/10/31 by Luca Guido Molinari, L. Molinari
Mathematics · Physics and Astronomy · #Quantum Mechanics and Non-Hermitian Physics #Quantum chaos and dynamical systems #Spectral Theory in Mathematical Physics #cond-mat #math-ph #math.MP #msc:15A57 #msc:15A90
paper · pdf · doi:10.1088/0305-4470/36/14/311
published as J. Phys. A: Math. Gen. 36 (2003) 4081-4090 · Revised text and new proposition added, relating counting function of exponents to winding numbers of eigenvalues. To appear on J. Phys. A: Math.Gen. 36 (2003)
arxiv created 2003/03/21 · openalex publication_date 2003/03/27 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30
I consider a general block-tridiagonal matrix and the corresponding transfer matrix. By allowing for a complex Bloch parameter in the boundary conditions, the two matrices are related by a spectral duality. As a consequence, I derive some analytic properties of the exponents of the transfer matrix in terms of the eigenvalues of the (non-Hermitian) block matrix. Some of them are the single-matrix analogues of results holding for Lyapunov exponents of an ensemble of block matrices, which occur in models of transport. The counting function of exponents is related to winding numbers of eigenvalues. I discuss some implications of duality for the distribution (real bands and complex arcs) and the dynamics of eigenvalues.