Counting the exponents of single transfer matrices

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, and other problems lead to three-term recursive relations for (multicomponent) amplitudes. Amplitudes n steps ap...

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Bibliographic Details
Published in:Journal of mathematical physics 2011-06, Vol.52 (6), p.063501-063501-8
Main Authors: Molinari, Luca Guido, Lacagnina, Giuseppe
Format: Article
Language:English
Subjects:
Eigenvalues
Exact sciences and technology
Lattice theory
Mathematical methods in physics
Mathematics
Matrix
Molecules
Physics
Sciences and techniques of general use
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Summary: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, 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.
ISSN:0022-2488
1089-7658
DOI:10.1063/1.3594654