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Spectra and Pseudospectra: the behavior of nonnormal matrices and operators
, 2005
"... Eigenvalues, latent roots, proper values, characteristic values—four synonyms for a set of numbers that provide much useful information about a matrix or operator. A huge amount of research has been directed at the theory of eigenvalues (localization, perturbation, canonical forms,...), at applicati ..."
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Cited by 112 (13 self)
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Eigenvalues, latent roots, proper values, characteristic values—four synonyms for a set of numbers that provide much useful information about a matrix or operator. A huge amount of research has been directed at the theory of eigenvalues (localization, perturbation, canonical forms,...), at applications (ubiquitous), and at numerical computation. I would like to begin with a very selective description of some historical aspects of these topics, before moving on to pseudoeigenvalues, the subject of the book under review. Back in the 1930s, Frazer, Duncan, and Collar of the Aerodynamics Department of the National Physical Laboratory (NPL), England, were developing matrix methods for analyzing flutter (unwanted vibrations) in aircraft. This was the beginning of what became known as matrix structural analysis [9], and led to the authors ’ book Elementary Matrices and Some Applications to Dynamics and Differential Equations, published in 1938 [10], which was “the first to employ matrices as an engineering tool ” [2]. Olga Taussky worked in Frazer’s group at NPL during the Second World War, analyzing 6 × 6 quadratic eigenvalue problems (QEPs)
Fractal upper bounds on the density of semiclassical resonances
 Duke Math. Journal
"... Let P = −h 2 ∆g +V (x) be a selfadjoint Schrödinger operator on a compact Riemannian nmanifold, (X, g), V ∈ C ∞ (X; R). The spectral asymptotics as h → 0 are given by the celebrated Weyl law – see [10] and [16] for recent advances and numerous references. If we assume that the zero energy surface ..."
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Cited by 35 (16 self)
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Let P = −h 2 ∆g +V (x) be a selfadjoint Schrödinger operator on a compact Riemannian nmanifold, (X, g), V ∈ C ∞ (X; R). The spectral asymptotics as h → 0 are given by the celebrated Weyl law – see [10] and [16] for recent advances and numerous references. If we assume that the zero energy surface is nondegenerate,
Geometric control in the presence of a black box
 J. Amer. Math. Soc
"... ABSTRACT. We apply the “black box ” scattering theory to problems in control theory for the Schrödinger equation, and in high energy eigenvalue scarring. 1. ..."
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Cited by 26 (4 self)
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ABSTRACT. We apply the “black box ” scattering theory to problems in control theory for the Schrödinger equation, and in high energy eigenvalue scarring. 1.
Semiclassical analysis for the KramersFokkerPlanck equation
 Comm. PDE
"... On étudie des estimations semiclassiques sur la résolvente d’opérateurs qui ne sont ni elliptiques ni autoadjoints, que l’on utilise pour étudier le problème de Cauchy. En particulier on obtient une description précise du spectre pres de l’axe imaginaire, et des estimations de résolvente à l’intérie ..."
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Cited by 24 (14 self)
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On étudie des estimations semiclassiques sur la résolvente d’opérateurs qui ne sont ni elliptiques ni autoadjoints, que l’on utilise pour étudier le problème de Cauchy. En particulier on obtient une description précise du spectre pres de l’axe imaginaire, et des estimations de résolvente à l’intérieur du pseudospectre. On applique ensuite les résultats à l’opérateur de KramersFokkerPlanck. We study some accurate semiclassical resolvent estimates for operators that are neither selfadjoint nor elliptic, and applications to the Cauchy problem. In particular we get a precise description of the spectrum near the imaginary axis and precise resolvent estimates inside the pseudospectrum. We apply our results to the KramersFokkerPlanck operator.
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).
Contraction semigroups of elliptic quadratic differential operators
 Math. Z
"... Abstract. We study the contraction semigroups of elliptic quadratic differential operators. Elliptic quadratic differential operators are the nonselfadjoint operators defined in the Weyl quantization by complexvalued elliptic quadratic symbols. We establish in this paper that under the assumption ..."
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Cited by 7 (7 self)
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Abstract. We study the contraction semigroups of elliptic quadratic differential operators. Elliptic quadratic differential operators are the nonselfadjoint operators defined in the Weyl quantization by complexvalued elliptic quadratic symbols. We establish in this paper that under the assumption of ellipticity, as soon as the real part of their Weyl symbols is a nonzero nonpositive quadratic form, the norm of contraction semigroups generated by these operators decays exponentially in time.