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Nonselfadjoint differential operators
, 2002
"... We describe methods which have been used to analyze the spectrum of nonselfadjoint differential operators, emphasizing the differences from the selfadjoint theory. We find that even in cases in which the eigenfunctions can be determined explicitly, they often do not form a basis; this is closely ..."
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Cited by 41 (6 self)
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We describe methods which have been used to analyze the spectrum of nonselfadjoint differential operators, emphasizing the differences from the selfadjoint theory. We find that even in cases in which the eigenfunctions can be determined explicitly, they often do not form a basis; this is closely related to a high degree of instability of the eigenvalues under small perturbations of the operator.
Spectra, Pseudospectra, and Localization for Random Bidiagonal Matrices
 Comm. Pure Appl. Math
"... There has been much recent interest, initiated by work of the physicists Hatano and Nelson, in the eigenvalues of certain random nonhermitian periodic tridiagonal matrices and their bidiagonal limits. These eigenvalues cluster along a \bubble with wings" in the complex plane, and the correspon ..."
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Cited by 16 (4 self)
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There has been much recent interest, initiated by work of the physicists Hatano and Nelson, in the eigenvalues of certain random nonhermitian 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 nonperiodic 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 in nitedimensional 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 nite bidiagonal matrices, in nite bidiagonal matrices (\stochastic Toeplitz operators"), nite periodic matrices, and doubly in nite bidiagonal matrices (\stochastic Laurent operators").
Spectral theory of pseudoergodic operators
 Commun. Math. Phys
, 2001
"... We define a class of pseudoergodic nonselfadjoint Schrödinger operators acting in spaces l 2 (X) and prove some general theorems about their spectral properties. We then apply these to study the spectrum of a nonselfadjoint Anderson model acting on l 2 (Z), and find the precise condition for 0 t ..."
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Cited by 7 (3 self)
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We define a class of pseudoergodic nonselfadjoint Schrödinger operators acting in spaces l 2 (X) and prove some general theorems about their spectral properties. We then apply these to study the spectrum of a nonselfadjoint Anderson model acting on l 2 (Z), and find the precise condition for 0 to lie in the spectrum of the operator. We also introduce the notion of localized spectrum for such operators.
Spectrum of a FeinbergZee Random Hopping Matrix by S N ChandlerWilde and E B DaviesSpectrum of a FeinbergZee Random Hopping Matrix
, 2011
"... This paper provides a new proof of a theorem of ChandlerWilde, Chonchaiya and Lindner that the spectra of a certain class of infinite, random, tridiagonal matrices contain the unit disc almost surely. It also obtains an analogous result for a more general class of random matrices whose spectra cont ..."
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Cited by 3 (1 self)
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This paper provides a new proof of a theorem of ChandlerWilde, Chonchaiya and Lindner that the spectra of a certain class of infinite, random, tridiagonal matrices contain the unit disc almost surely. It also obtains an analogous result for a more general class of random matrices whose spectra contain a hole around the origin. The presence of the hole forces substantial changes to the analysis.
Infinite Toeplitz and Laurent matrices with localized impurities
, 2000
"... This paper is concerned with the change of the spectra of innite Toeplitz and Laurent matrices under perturbations in a prescribed nite set of sites. The main result says that the spectrum of a Toeplitz matrix with a nonconstant rational symbol is not aected by small localized impurities, while suc ..."
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Cited by 2 (1 self)
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This paper is concerned with the change of the spectra of innite Toeplitz and Laurent matrices under perturbations in a prescribed nite set of sites. The main result says that the spectrum of a Toeplitz matrix with a nonconstant rational symbol is not aected by small localized impurities, while such impurities can nevertheless enlarge the spectrum of the corresponding Laurent matrix. We also study the spectra that may emerge when randomly perturbing Toeplitz or Laurent matrices in a randomly chosen single site. 1 Introduction and main results Given a complexvalued continuous function a on the complex unit circle T, a 2 C, we consider the matrices T (a) = (a j k ) 1 j;k=1 and L(a) = (a j k ) 1 j;k=1 ; where a k = 1 2 2 Z 0 a(e i ) e ik d; k 2 Z: The matrices T (a) and L(a) are referred to as the Toeplitz matrix and the Laurent matrix with the symbol a, respectively. These matrices induce bounded operators on l 2 (N) and l 2 (Z), respectively. 1 Email: aboet...
SPECTRAL BOUNDS USING HIGHER ORDER NUMERICAL RANGES
, 2004
"... We describe how to obtain bounds on the spectrum of a nonselfadjoint operator by means of what we call its higher order numerical ranges. We prove some of their basic properties and describe explain how to compute them. We finally use them to obtain new spectral insights into the nonselfadjoint A ..."
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Cited by 2 (0 self)
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We describe how to obtain bounds on the spectrum of a nonselfadjoint operator by means of what we call its higher order numerical ranges. We prove some of their basic properties and describe explain how to compute them. We finally use them to obtain new spectral insights into the nonselfadjoint Anderson model in one and two space dimensions.
On large Toeplitz band matrices with an uncertain block
, 2001
"... This report investigates the possible spectra of large, finite dimensional Toeplitz band matrices with perturbations (impurities, uncertainties) in the upperleft mm block. The main result shows that the asymptotic spectrum of such a matrix is not affected by these perturbations, provided they have ..."
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Cited by 1 (0 self)
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This report investigates the possible spectra of large, finite dimensional Toeplitz band matrices with perturbations (impurities, uncertainties) in the upperleft mm block. The main result shows that the asymptotic spectrum of such a matrix is not affected by these perturbations, provided they have sufficiently small norm. This follows from analysis of structured pseudospectra (structured spectral value sets). In contrast, for typical nonHermitian Toeplitz matrices there exist certain rankone perturbations of arbitrarily small norm that move an eigenvalue away from the asymptotic spectrum in the largedimensional limit.
Linear Algebra and its Applications 351–352 (2002) 99–116
, 2001
"... www.elsevier.com/locate/laa Can spectral value sets of Toeplitz band matrices jump? A. Böttcher a,∗, S.M. Grudsky b,1 ..."
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www.elsevier.com/locate/laa Can spectral value sets of Toeplitz band matrices jump? A. Böttcher a,∗, S.M. Grudsky b,1