This paper is about analytic properties of single transfer matrices originating from general block-tridiagonal or banded matrices. Such matrices occur in various applications in physics and numerical analysis. The eigenvalues of the transfer matrix describe localization of eigenstates and are linked to the spectrum of the block tridiagonal matrix by a determinantal identity. If the block tridiagonal matrix is invertible, it is shown that half of the singular values of the transfer matrix have a lower bound exponentially large in the length of the chain, and the other half have an upper bound that is exponentially small. This is a consequence of a theorem by Demko, Moss and Smith on the decay of matrix elements of the inverse of banded matrices.
Identities and exponential bounds for transfer matrices / L. G. Molinari. - In: JOURNAL OF PHYSICS. A, MATHEMATICAL AND THEORETICAL. - ISSN 1751-8113. - 46:25(2013), pp. 254004.1-254004.15. [10.1088/1751-8113/46/25/254004]
Identities and exponential bounds for transfer matrices
L.G. MolinariPrimo
2013
Abstract
This paper is about analytic properties of single transfer matrices originating from general block-tridiagonal or banded matrices. Such matrices occur in various applications in physics and numerical analysis. The eigenvalues of the transfer matrix describe localization of eigenstates and are linked to the spectrum of the block tridiagonal matrix by a determinantal identity. If the block tridiagonal matrix is invertible, it is shown that half of the singular values of the transfer matrix have a lower bound exponentially large in the length of the chain, and the other half have an upper bound that is exponentially small. This is a consequence of a theorem by Demko, Moss and Smith on the decay of matrix elements of the inverse of banded matrices.File | Dimensione | Formato | |
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