The Mathematical Institute, University of Oxford, Eprints Archive

Spectra, pseudospectra, and localization for random bidiagonal matrices

Trefethen, Lloyd N. and Contedini, Marco and Embree, Mark (2000) Spectra, pseudospectra, and localization for random bidiagonal matrices. Technical Report. Unspecified. (Submitted)



There has been much recent interest, initiated by work of the physicists Hatano and Nelson, in the eigenvalues of certain random non-hermitian periodic tridiagonal matrices and their bidiagonal limits. These eigenvalues cluster along a "bubble with wings" in the complex plane, and the corresponding eigenvectors are localized in the wings, delocalized in the bubble. Here, in addition to eigenvalues, pseudospectra are analyzed, making it possible to treat the non-periodic analogues of these random matrix problems. Inside the bubble, the resolvent norm grows exponentially with the dimension. Outside, it grows subexponentially in a bounded region that is the spectrum of the infinite-dimensional operator. Localization and delocalization correspond to resolvent matrices whose entries exponentially decrease or increase, respectively, with distance from the diagonal. This article presents theorems that characterize the spectra, pseudospectra, and numerical range for the four cases of finite bidiagonal matrices, infinite bidiagonal matrices ("stochastic Toeplitz operators"), finite periodic matrices, and doubly infinite bidiagonal matrices ("stochastic Laurent operators").

This is a preprint of an article published in Communications in Pure and Applied Mathematics, copyright 2000, John Wiley & Sons, Inc. This work was supported by the UK Engineering and Physical Sciences Research Council Grant GR/M12414

Item Type:Technical Report (Technical Report)
Subjects:H - N > Linear and multilinear algebra; matrix theory
O - Z > Operator theory
H - N > Numerical analysis
Research Groups:Numerical Analysis Group
ID Code:1274
Deposited By: Lotti Ekert
Deposited On:01 Jun 2011 08:21
Last Modified:29 May 2015 18:58

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