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