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B: Pseudospectra of differential operators
 J. Oper. Theory
, 2000
"... 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 27 (7 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. AMS subject classifications:34L05, 35P05, 47A10, 47A12
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 corresponding ..."
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Cited by 13 (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").
Khoruzhenko: Eigenvalue curves of asymmetric tridiagonal random matrices
 Electronic Journal of Probability 5
, 2000
"... E l e c t r o n ..."
Spectral properties of random nonselfadjoint matrices and operators
, 2001
"... We describe some numerical experiments which determine the degree of spectral instability of medium size randomly generated matrices which are far from selfadjoint. The conclusion is that the eigenvalues are likely to be intrinsically uncomputable for similar matrices of a larger size. We also desc ..."
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Cited by 7 (4 self)
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We describe some numerical experiments which determine the degree of spectral instability of medium size randomly generated matrices which are far from selfadjoint. The conclusion is that the eigenvalues are likely to be intrinsically uncomputable for similar matrices of a larger size. We also describe a stochastic family of bounded operators in infinite dimensions for almost all of which the eigenvectors generate a dense linear subspace, but the eigenvalues do not determine the spectrum. Our results imply that the spectrum of the nonselfadjoint Anderson model changes suddenly as one passes to the infinite volume limit.
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 5 (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.
Enumeration of simple random walks and tridiagonal matrices
 J. Phys. A Math. Gen
"... We present some old and new results in the enumeration of random walks in one dimension, mostly developed in works of enumerative combinatorics. The relation between the trace of the nth power of a tridiagonal matrix and the enumeration of weighted paths of n steps allows an easier combinatorial en ..."
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Cited by 2 (0 self)
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We present some old and new results in the enumeration of random walks in one dimension, mostly developed in works of enumerative combinatorics. The relation between the trace of the nth power of a tridiagonal matrix and the enumeration of weighted paths of n steps allows an easier combinatorial enumeration of paths. It also seems promising for the theory of tridiagonal random matrices. 1 Introduction. Already at the foundation of the theory of random matrices by E.Wigner, the relevance of the combinatorics of random walks was recognized [1]. The following decades witnessed an explosion of the theory and the applications of random matrices. A number of specific techniques were devised and the relation with
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 1 (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.
DETERMINANTS OF BLOCK TRIDIAGONAL MATRICES
, 712
"... An identity is proven that evaluates the determinant of a block tridiagonal matrix with (or without) corners as the determinant of the associated transfer matrix (or a submatrix of it). Key words: Block tridiagonal matrix, transfer matrix, determinant 1991 MSC: 15A15, 15A18, 15A90 1 ..."
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Cited by 1 (0 self)
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An identity is proven that evaluates the determinant of a block tridiagonal matrix with (or without) corners as the determinant of the associated transfer matrix (or a submatrix of it). Key words: Block tridiagonal matrix, transfer matrix, determinant 1991 MSC: 15A15, 15A18, 15A90 1
On large Toeplitz band matrices with an uncertain block
"... This report investigates the possible spectra of large, nite 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 aected by these perturbations, provided they have suc ..."
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This report investigates the possible spectra of large, nite 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 aected by these perturbations, provided they have suciently 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. 1