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.

Counting the exponents of single transfer matrices / L.G. Molinari, G. Lacagnina. - In: JOURNAL OF MATHEMATICAL PHYSICS. - ISSN 0022-2488. - 52:6(2011), pp. 063501.063501.1-063501.063501.8.

### Counting the exponents of single transfer matrices

#### 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.
##### Scheda breve Scheda completa Scheda completa (DC) Lyapunov spectrum ; Anderson localization ; transfer matrix ; block tridiagonal matrix
Settore FIS/02 - Fisica Teorica, Modelli e Metodi Matematici
Settore FIS/03 - Fisica della Materia
Settore MAT/07 - Fisica Matematica
2011
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/2434/162042`
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