Results 1  10
of
12
Pseudospectra of linear operators
 SIAM Rev
, 1997
"... Abstract. If a matrix or linear operator A is far from normal, its eigenvalues or, more generally, its spectrum may have little to do with its behavior as measured by quantities such as ‖An ‖ or ‖exp(tA)‖. More may be learned by examining the sets in the complex plane known as the pseudospectra of A ..."
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Cited by 113 (8 self)
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Abstract. If a matrix or linear operator A is far from normal, its eigenvalues or, more generally, its spectrum may have little to do with its behavior as measured by quantities such as ‖An ‖ or ‖exp(tA)‖. More may be learned by examining the sets in the complex plane known as the pseudospectra of A, defined by level curves of the norm of the resolvent, ‖(zI − A) −1‖. Five years ago, the author published a paper that presented computed pseudospectra of thirteen highly nonnormal matrices arising in various applications. Since that time, analogous computations have been carried out for differential and integral operators. This paper, a companion to the earlier one, presents ten examples, each chosen to illustrate one or more mathematical or physical principles.
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
Pseudospectra of the convectiondiffusion operator
 SIAM J. Appl. Math
, 1994
"... Abstract. The spectrum of the simplest 1D convectiondiffusion operator is a discrete subset of the negative real axis, but the pseudospectra are regions in the complex plane bounded approximately by parabolas. Put another way, the norm of the resolvent is exponentially large as a function of the Pd ..."
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Cited by 25 (7 self)
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Abstract. The spectrum of the simplest 1D convectiondiffusion operator is a discrete subset of the negative real axis, but the pseudospectra are regions in the complex plane bounded approximately by parabolas. Put another way, the norm of the resolvent is exponentially large as a function of the Pdclet number throughout a certain approximately parabolic region. These observations have a simple physical basis, and suggest that conventional spectral analysis for convectiondiffusion operators may be of limited value in some applications. Key words, convectiondiffusion operator, Pdclet number, pseudospectra AMS subject classifications.
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").
Dynamics and Predictability of Stommel's box model: A phase space perspective with implications for decadal climate variability
 Tellus
, 1998
"... The dynamics and predictability of Stommel's (1961) box model of the thermohaline circulation is studied. This nonlinear model with idealized geometry of the North Atlantic is solved exactly. A phase space analysis of the model reveals that the optimal perturbation affecting longterm climate variab ..."
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Cited by 13 (1 self)
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The dynamics and predictability of Stommel's (1961) box model of the thermohaline circulation is studied. This nonlinear model with idealized geometry of the North Atlantic is solved exactly. A phase space analysis of the model reveals that the optimal perturbation affecting longterm climate variability is provided by high latitude haline forcing in the Atlantic ocean, although this perturbation has little resemblance with the most unstable mode of the system and the leading EOF. Furthermore, the predictability problem is investigated by means of singular vector analysis and the evolution of the probability distribution function. Uncertainties in the oceanic initial conditions do increase in the phase space of the model. In the stochastically forced box model with identical oceanic initial conditions, the climate predictability is examined for the damped persistence forecast. We find that the loss of the predictability is related to the different stages of the variance evolution which is also measured by the relative entropy. Our analysis shows that the nonnormal system matrix of Stommel's model does affect the dynamics and predictability of the system which is useful for the interpretation of longterm climate variability and predictability. 1
Wave Packet Pseudomodes of Twisted Toeplitz Matrices
, 2004
"... The pseudospectra of banded, nonsymmetric Toeplitz or circulant matrices with varying coefficients are considered. Such matrices are characterized by a symbol that depends on both position (x) and wave number (k). It is shown that when a certain winding number or twist condition is satisfied, relate ..."
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Cited by 12 (1 self)
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The pseudospectra of banded, nonsymmetric Toeplitz or circulant matrices with varying coefficients are considered. Such matrices are characterized by a symbol that depends on both position (x) and wave number (k). It is shown that when a certain winding number or twist condition is satisfied, related to Hörmander’s commutator condition for partial differential equations, εpseudoeigenvectors of such matrices for exponentially small values of ε exist in the form of localized wave packets. The symbol need not be smooth with respect to x, just differentiable at a point (or less).
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.
Norms of Inverses, Spectra, and Pseudospectra of Large Truncated WienerHopf Operators and Toeplitz Matrices
, 1997
"... . This paper is concerned with WienerHopf integral operators on L p and with Toeplitz operators (or matrices) on l p . The symbols of the operators are assumed to be continuous matrix functions. It is well known that the invertibility of the operator itself and of its associated operator imply ..."
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Cited by 6 (2 self)
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. This paper is concerned with WienerHopf integral operators on L p and with Toeplitz operators (or matrices) on l p . The symbols of the operators are assumed to be continuous matrix functions. It is well known that the invertibility of the operator itself and of its associated operator imply the invertibility of all sufficiently large truncations and the uniform boundedness of the norms of their inverses. Quantitative statements, such as results on the limit of the norms of the inverses, can be proved in the case p = 2 by means of C algebra techniques. In this paper we replace C algebra methods by more direct arguments to determine the limit of the norms of the inverses and thus also of the pseudospectra of large truncations in the case of general p. Contents 1. Introduction 2 2. Structure of Inverses 4 3. Norms of Inverses 7 4. Spectra 15 5. Pseudospectra 17 6. Matrix Case 22 7. Block Toeplitz Matrices 23 8. Pseudospectra of Infinite Toeplitz Matrices 27 References 30 ...
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.