Spectral duality and distribution of exponents for transfer matrices of block tridiagonal Hamiltonians

Abstract

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 analogue 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 on the distribution (real bands and complex arcs) and the dynamics of eigenvalues.

arXiv commentRevised 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)

References in corpus (13)

• Random-Matrix Theory of Quantum Transport
• Random Matrix Theories in Quantum Physics: Common Concepts
• Localization transitions in non-Hermitian quantum mechanics
• Distribution of Eigenvalues in Non-Hermitian Anderson Model
• Winding Numbers, Complex Currents, and Non-Hermitian Localization
• Oscillating density of states near zero energy for matrices made of blocks with possible application to the random flux problem
• Spectral Curves of Non-Hermitean Hamiltonians
• Density of states in the non-hermitian Lloyd model
• Spectral estimates for periodic Jacobi matrices
• Transfer matrices, non-Hermitian Hamiltonians and Resolvents: some spectral identities
• Regular Spacings of Complex Eigenvalues in the One-dimensional non-Hermitian Anderson Model
• Parametric Spectral Correlations of Disordered Systems in the Fourier Domain
• Lyapounov exponent and density of states of a one-dimensional non-Hermitian Schroedinger equation

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• A Unified Perspective of Complex Band Structure: Interpretations, Formulations, and Applications
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• Energy structure, density of states and transmission properties of the periodic 1D Tight-Binding lattice with a generic unit cell of $u$ sites
• Topological effect on the Anderson transition in chiral symmetry classes
• Identities and exponential bounds for transfer matrices
• On the Kolmogorov-Sinai entropy of many-body Hamiltonian systems
• Real symmetric random matrices and paths counting
• Counting the exponents of single transfer matrices
• Theory of the Anderson transition in three-dimensional chiral symmetry classes: Connection to type-II superconductors
• Critical magnetic flux for Weyl points in the three-dimensional Hofstadter model