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Simon N. Chandler-Wilde, Ratchanikorn Chonchaiya, Marko Lindner: On the Spectra and Pseudospectra of a Class of non-self-adjoint Random Matrices and Operators

Simon N. Chandler-Wilde, Ratchanikorn Chonchaiya, Marko Lindner: On the Spectra and Pseudospectra of a Class of non-self-adjoint Random Matrices and Operators


Author(s):
Simon N. Chandler-Wilde
Ratchanikorn Chonchaiya
Marko Lindner
Title:
On the Spectra and Pseudospectra of a Class of non-self-adjoint Random Matrices and Operators
Electronic source:
application/pdf
Preprint series:
Technische Universität Chemnitz, Fakultät für Mathematik (Germany). Preprint 11, 2011
Mathematics Subject Classification:
47B80 [ ]
47A10 [ ]
47B36 [ ]
Abstract:
In this paper we develop and apply methods for the spectral analysis of non-self-adjoint tridiagonal infinite and finite random matrices, and for the spectral analysis of analogous deterministic matrices which are pseudo-ergodic in the sense of E.B.Davies (Commun. Math. Phys. 216 (2001), 687-704). The paper focuses on application of these methods to study the ``hopping sign model'' introduced by J.Feinberg and A.Zee (Phys. Rev. E 59 (1999), 6433--6443), in which the main object of study are random tridiagonal matrices which have zeros on the main diagonal and random $\pm 1$'s as the other entries. We explore the relationship between spectral sets in the finite and infinite matrix cases, and between the semi-infinite and bi-infinite matrix cases, for example showing that the numerical range and $p$-norm $\eps$-pseudospectra ($\eps>0$, $p\in [1,\infty]$) of the random finite matrices converge almost surely to their infinite matrix counterparts, and that the finite matrix spectra are contained in the infinite matrix spectrum $\Sigma$. We also propose a sequence of inclusion sets for $\Sigma$ which we show is convergent to $\Sigma$, with the $n$th element of the sequence computable by calculating smallest singular values of (large numbers of) $n\times n$ matrices. We propose similar convergent approximations for the 2-norm $\eps$-pseudospectra of the infinite random matrices, these approximations sandwiching the infinite matrix pseudospectra from above and below.
Keywords:
random matrix, spectral theory, Jacobi matrix, operators on $\ell^p$
Language:
English
Publication time:
07/2011

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