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Large deviations asymptotics and the spectral theory of multiplicatively regular Markov processes
 Electron. J. Probab
"... In this paper we continue the investigation of the spectral theory and exponential asymptotics of primarily discretetime Markov processes, following Kontoyiannis and Meyn [32]. We introduce a new family of nonlinear Lyapunov drift criteria, which characterize distinct subclasses of geometrically er ..."
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Cited by 51 (10 self)
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In this paper we continue the investigation of the spectral theory and exponential asymptotics of primarily discretetime Markov processes, following Kontoyiannis and Meyn [32]. We introduce a new family of nonlinear Lyapunov drift criteria, which characterize distinct subclasses of geometrically ergodic Markov processes in terms of simple inequalities for the nonlinear generator. We concentrate primarily on the class of multiplicatively regular Markov processes, which are characterized via simple conditions similar to (but weaker than) those of DonskerVaradhan. For any such process Φ = {Φ(t)} with transition kernel P on a general state space X, the following are obtained. Spectral Theory: For a large class of (possibly unbounded) functionals F: X → C, the kernel ̂ P (x, dy) = e F (x) P (x, dy) has a discrete spectrum in an appropriately defined Banach space. It follows that there exists a “maximal ” solution (λ, ˇ f) to the multiplicative Poisson equation, defined as the eigenvalue problem ̂ P ˇ f = λ ˇ f. The functional Λ(F) = log(λ) is convex, smooth, and its convex dual Λ ∗ is convex, with compact sublevel sets.
Nonselfadjoint harmonic oscillator, compact semigroups and pseudospectra
 J. Operator Theory
"... We provide new information concerning the pseudospectra of the complex harmonic oscillator. Our analysis illustrates two different techniques for getting resolvent norm estimates. The first uses the JWKB method and extends for this particular potential some results obtained recently by E.B. Davies. ..."
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Cited by 21 (2 self)
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We provide new information concerning the pseudospectra of the complex harmonic oscillator. Our analysis illustrates two different techniques for getting resolvent norm estimates. The first uses the JWKB method and extends for this particular potential some results obtained recently by E.B. Davies. The second relies on the fact that the bounded holomorphic semigroup generated by the complex harmonic oscillator is of HilbertSchmidt type in a maximal angular region. In order to show this last property, we deduce a nonselfadjoint version of the classical Mehler’s formula.
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").
InputOutput Analysis and Control Design Applied to a Linear Model of Spatially Developing Flows
 Appl. Mech. Rev
"... A framework for the inputoutput analysis, model reduction and control design of spatially developing shear flows is presented using the Blasius boundarylayer flow as an example. An inputoutput formulation of the governing equations yields a flexible formulation for treating stability problems a ..."
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Cited by 15 (5 self)
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A framework for the inputoutput analysis, model reduction and control design of spatially developing shear flows is presented using the Blasius boundarylayer flow as an example. An inputoutput formulation of the governing equations yields a flexible formulation for treating stability problems and for developing control strategies that optimize given objectives. Model reduction plays an important role in this process since the dynamical systems that describe most flows are discretized partial differential equations with a very large number of degrees of freedom. Moreover, as system theoretical tools, such as controllability, observability and balancing has become computationally tractable for largescale systems, a systematic approach to model reduction is presented. I.
Schrödinger operators with complexvalued potentials and no resonances
 Duke Math Jour
"... Abstract. In dimension d ≥ 3, we give examples of nontrivial, compactly supported, complexvalued potentials such that the associated Schrödinger operators have no resonances. If d = 2, we show that there are potentials with no resonances away from the origin. These Schrödinger operators are isophas ..."
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Cited by 15 (9 self)
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Abstract. In dimension d ≥ 3, we give examples of nontrivial, compactly supported, complexvalued potentials such that the associated Schrödinger operators have no resonances. If d = 2, we show that there are potentials with no resonances away from the origin. These Schrödinger operators are isophasal and have the same scattering phase as the Laplacian on R d. In odd dimensions d ≥ 3 we study the fundamental solution of the wave equation perturbed by such a potential. If the space variables are held fixed, it is superexponentially decaying in time. 1.
Spectral asymptotics for large skewsymmetric perturbations of the harmonic oscillator
, 2008
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LiebThirring estimates for nonselfadjoint Schrödinger operators
 J. Math. Phys
, 2008
"... Abstract. For general nonsymmetric operators A, we prove that the moment of order γ ≥ 1 of negative realparts of its eigenvalues is bounded by the moment of order γ of negative eigenvalues of its symmetric part H = 1 2 [A + A ∗]. As an application, we obtain LiebThirring estimates for non selfad ..."
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Cited by 13 (0 self)
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Abstract. For general nonsymmetric operators A, we prove that the moment of order γ ≥ 1 of negative realparts of its eigenvalues is bounded by the moment of order γ of negative eigenvalues of its symmetric part H = 1 2 [A + A ∗]. As an application, we obtain LiebThirring estimates for non selfadjoint Schrödinger operators. In particular, we recover recent results by Frank, Laptev, Lieb and Seiringer [11]. We also discuss moment of resonances of Schrödinger selfadjoint operators. 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).
Eigenvalue bounds for Schrödinger operators with complex potentials
 Bull. London Math. Soc
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Infinitesimal form boundedness and Trudinger’s subordination for the Schrödinger operator
"... Abstract. We give explicit analytic criteria for two problems associated with the Schrödinger operator H = − ∆ + Q on L 2 (R n) where Q ∈ D ′ (R n) is an arbitrary real or complexvalued potential. First, we obtain necessary and sufficient conditions on Q so that the quadratic form 〈Q·, · 〉 has zer ..."
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Cited by 10 (3 self)
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Abstract. We give explicit analytic criteria for two problems associated with the Schrödinger operator H = − ∆ + Q on L 2 (R n) where Q ∈ D ′ (R n) is an arbitrary real or complexvalued potential. First, we obtain necessary and sufficient conditions on Q so that the quadratic form 〈Q·, · 〉 has zero relative bound with respect to the Laplacian. For Q ∈ L 1 loc (Rn), this property can be expressed in the form of the integral inequality: ∣ u(x)  2 Q(x)dx∣ ∣ ≤ ǫ ∇u2 L 2 (R n) + C(ǫ) u2 L 2 (R n) , ∀u ∈ C ∞ 0 (R n), Rn for an arbitrarily small ǫ> 0 and some C(ǫ)> 0. One of the major steps here is the reduction to a similar inequality with nonnegative function ∇(1−∆) −1 Q  2 +(1−∆) −1 Q  in place of Q. This provides a complete solution to the infinitesimal form boundedness problem for the Schrödinger operator, and leads to new broad classes of admissible distributional potentials Q, which extend the usual Lp and Kato classes, as well as those based on the wellknown conditions of Fefferman–Phong and Chang–Wilson–Wolff. Secondly, we characterize Trudinger’s subordination property where C(ǫ) in the above inequality is subject to the condition C(ǫ) ≤ c ǫ−β (β> 0) as ǫ → +0. Such quadratic form inequalities can be understood entirely in the framework of Morrey–Campanato spaces, using mean oscillations of ∇(1−∆) −1 Q and (1−∆) −1 Q on balls or cubes. A version of this condition where ǫ ∈ (0, +∞) is equivalent to the multiplicative inequality: