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12
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").
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.
The Spectra of Large Toeplitz Band Matrices with a Randomly Perturbed Entry
"... This report is concerned with the union sp Tn (a) of all possible spectra that may emerge when perturbing a large n n Toeplitz band matrix Tn (a) in the (j; k) site by a number randomly chosen from some set The main results give descriptive bounds and, in several interesting situations, even pr ..."
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Cited by 3 (2 self)
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This report is concerned with the union sp Tn (a) of all possible spectra that may emerge when perturbing a large n n Toeplitz band matrix Tn (a) in the (j; k) site by a number randomly chosen from some set The main results give descriptive bounds and, in several interesting situations, even provide complete identifications of the limit of sp Tn (a) as n ! 1.
Regular spacings of complex eigenvalues in the onedimensional nonHermitian Anderson model
 Commun. Math. Phys
"... We prove that in dimension one the nonreal eigenvalues of the nonHermitian Anderson (NHA) model with a selfaveraging potential are regularly spaced. The class of selfaveraging potentials which we introduce in this paper is very wide and in particular includes stationary potentials (with probabilit ..."
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Cited by 3 (1 self)
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We prove that in dimension one the nonreal eigenvalues of the nonHermitian Anderson (NHA) model with a selfaveraging potential are regularly spaced. The class of selfaveraging potentials which we introduce in this paper is very wide and in particular includes stationary potentials (with probability one) as well as all quasiperiodic potentials. It should be emphasized that our approach here is much simpler than the one we used before. It allows us a) to investigate the above mentioned spacings, b) to establish certain properties of the integrated density of states of the Hermitian Anderson models with selfaveraging potentials, and c) to obtain (as a byproduct) much simpler proofs of our previous results concerned with nonreal eigenvalues of the NHA model. 1
On the Spectra and Pseudospectra of a Class of NonSelfAdjoint Random Matrices and Operators
, 2011
"... In this paper we develop and apply methods for the spectral analysis of nonselfadjoint tridiagonal infinite and finite random matrices, and for the spectral analysis of analogous deterministic matrices which are pseudoergodic in the sense of E. B. Davies (Commun. Math. Phys. 216 (2001), 687–704). ..."
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Cited by 2 (2 self)
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In this paper we develop and apply methods for the spectral analysis of nonselfadjoint tridiagonal infinite and finite random matrices, and for the spectral analysis of analogous deterministic matrices which are pseudoergodic in the sense of E. B. Davies (Commun. Math. Phys. 216 (2001), 687–704). As a major application to illustrate our methods we focus on the “hopping sign model” introduced by J. Feinberg and A. Zee (Phys. Rev. E 59 (1999), 6433–6443), in which the main objects of study are random tridiagonal matrices which have zeros on the main diagonal and random ±1’s as the other entries. We explore the relationship between spectral sets in the finite and infinite matrix cases, and between the semiinfinite and biinfinite matrix cases, for example showing that the numerical range and pnorm εpseudospectra (ε> 0, p ∈ [1, ∞]) 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 Σ. We also propose a sequence of inclusion sets for Σ which we show is convergent to Σ, with the nth element of the sequence computable by calculating smallest singular values of (large numbers of) n × n matrices. We propose similar convergent approximations for the 2norm εpseudospectra of the infinite random matrices, these approximations sandwiching the infinite matrix pseudospectra from above and below.
Spectral Approximation of Banded Laurent Matrices with Localized Random Perturbations
"... This paper explores the relationship between the spectra of perturbed innite banded Laurent matrices L(a)+K and their approximations by perturbed circulant matrices Cn (a) +PnKPn for large n. The entries K jk of the perturbation matrices assume values in prescribed sets ..."
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Cited by 1 (1 self)
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This paper explores the relationship between the spectra of perturbed innite banded Laurent matrices L(a)+K and their approximations by perturbed circulant matrices Cn (a) +PnKPn for large n. The entries K jk of the perturbation matrices assume values in prescribed sets
Analytic solutions of a nonlinear convective equation in population dynamics
, 2003
"... Analytic solutions are presented for a simple nonlinear convective equation of use in population dynamics. In spite of its simplicity the equation predicts rich behavior including a velocity inversion transition. Stability considerations are also presented. ..."
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Cited by 1 (0 self)
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Analytic solutions are presented for a simple nonlinear convective equation of use in population dynamics. In spite of its simplicity the equation predicts rich behavior including a velocity inversion transition. Stability considerations are also presented.
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