Counting the exponents of single transfer matrices
arXiv:1102.1641 · doi:10.1063/1.3594654
Abstract
The eigenvalue equation of a band or a block tridiagonal matrix, the tight binding model for a crystal, a molecule, or a particle in a lattice with random potential or hopping amplitudes: these and other problems lead to three-term recursive relations for (multicomponent) amplitudes. Amplitudes n steps apart are linearly related by a transfer matrix, which is the product of n matrices. Its exponents describe the decay lengths of the amplitudes. A formula is obtained for the counting function of the exponents, based on a duality relation and the Argument Principle for the zeros of analytic functions. It involves the corner blocks of the inverse of the associated Hamiltonian matrix. As an illustration, numerical evaluations of the counting function of quasi 1D Anderson model are shown.
9 pages, 2 figures
References in corpus (7)
- Random-Matrix Theory of Quantum Transport
- Determinants of Block Tridiagonal Matrices
- Non-Hermitian spectra and Anderson localization
- Fluctuations of the Lyapunov exponent in 2D disordered systems
- The Lyapunov Spectrum of a Continuous Product of Random Matrices
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