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21
Nonselfadjoint differential operators
, 2002
"... 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 41 (6 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.
Nonelliptic quadratic forms and semiclassical estimates for nonselfadjoint operators
 SPECTRAL PROJECTIONS AND RESOLVENTS FOR QUADRATIC OPERATORS
"... We consider a class of pseudodifferential operators with a doubly characteristic point, where the quadratic part of the symbol fails to be elliptic but obeys an averaging assumption. Under suitable additional assumptions, semiclassical resolvent estimates are established, where the modulus of the sp ..."
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Cited by 5 (2 self)
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We consider a class of pseudodifferential operators with a doubly characteristic point, where the quadratic part of the symbol fails to be elliptic but obeys an averaging assumption. Under suitable additional assumptions, semiclassical resolvent estimates are established, where the modulus of the spectral parameter is allowed to grow slightly more rapidly than the semiclassical parameter. 1 Introduction and Statement of Results 1.1 Quadratic forms and singular spaces Recently, there has been a renewed interest in the analysis of spectra and resolvents of nonselfadjoint operators with double characteristics. The study of pseudodifferential operators with double characteristics has a long and distinguished tradition in the
Wave Packet Pseudomodes Of Variable Coefficient Differential Operators
 PROC. R. SOC. LOND. A
"... The pseudospectra of nonselfadjoint linear ordinary differential operators with variable coe# cients are considered. It is shown that when a certain winding number or twist condition is satisfied, closely related to Hormander's commutator condition for partial differential equations, #pseudoe ..."
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Cited by 3 (0 self)
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The pseudospectra of nonselfadjoint linear ordinary differential operators with variable coe# cients are considered. It is shown that when a certain winding number or twist condition is satisfied, closely related to Hormander's commutator condition for partial differential equations, #pseudoeigenfunctions of such operators for exponentially small values of # exist in the form of localized wave packets. In contrast to related results of Davies and of Dencker, Sjöstrand, and Zworski, the symbol need not be smooth.
S: Spectral Behaviour of a simple nonselfadjoint operator
, 2001
"... Abstract. We investigate the spectrum of a typical nonselfadjoint differential operator AD = −d 2 /dx 2 ⊗ A acting on L 2 (0, 1) ⊗ C 2, where A is a 2 × 2 constant matrix. We impose Dirichlet and Neumann boundary conditions in the first and second coordinate respectively at both ends of [0, 1] ⊂ ..."
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Cited by 2 (0 self)
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Abstract. We investigate the spectrum of a typical nonselfadjoint differential operator AD = −d 2 /dx 2 ⊗ A acting on L 2 (0, 1) ⊗ C 2, where A is a 2 × 2 constant matrix. We impose Dirichlet and Neumann boundary conditions in the first and second coordinate respectively at both ends of [0, 1] ⊂ R. For A ∈ R 2×2 we explore in detail the connection between the entries of A and the spectrum of AD, we find necessary conditions to ensure similarity to a selfadjoint operator and give numerical evidence that suggests a nontrivial spectral evolution. 1.
On weak and strong solution operators for evolution equations coming from quadratic operators. arXiv.org; submitted
, 2014
"... Abstract. We identify, through a change of variables, solution operators for evolution equations with generators given by certain simple firstorder differential operators acting on Fock spaces. This class of operators properly includes, through unitary equivalence, the wellstudied set of elliptic ..."
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Cited by 2 (1 self)
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Abstract. We identify, through a change of variables, solution operators for evolution equations with generators given by certain simple firstorder differential operators acting on Fock spaces. This class of operators properly includes, through unitary equivalence, the wellstudied set of elliptic and weakly elliptic quadratic differential operators acting on L2(Rn). We demonstrate a variety of sharp results on boundedness, decay, and return to equilibrium for these solution operators, connecting the shorttime behavior with the range of the symbol and the longtime behavior with the eigenvalues of these operators. This is particularly striking when it allows for the definition of solution operators which are compact and regularizing for large times for certain operators whose spectrum is the entire complex plane. Contents
On nonround points of the boundary of the numerical range and an application to nonselfadjoint Schrödinger operators
 Journal of Spectral theory
, 2014
"... Abstract. We show that nonround boundary points of the numerical range of an unbounded operator (i.e. points where the boundary has infinite curvature) are contained in the spectrum of the operator. Moreover, we show that nonround boundary points, which are not corner points, lie in the essential ..."
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Cited by 1 (1 self)
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Abstract. We show that nonround boundary points of the numerical range of an unbounded operator (i.e. points where the boundary has infinite curvature) are contained in the spectrum of the operator. Moreover, we show that nonround boundary points, which are not corner points, lie in the essential spectrum. This generalizes results of Hübner, Farid, Spitkovsky and Salinas and Velasco for the case of bounded operators. We apply our results to nonselfadjoint Schrödinger operators, showing that in this case the boundary of the numerical range can be nonround only at points where it hits the essential spectrum. 1.
LARGETIME ASYMPTOTICS OF SOLUTIONS TO THE KRAMERSFOKKERPLANCK EQUATION WITH A SHORTRANGE POTENTIAL
"... Abstract. In this work, we use scattering method to study the KramersFokkerPlanck equation with a potential whose gradient tends to zero at the infinity. For shortrange potentials in dimension three, we show that complex eigenvalues do not accumulate at lowenergies and establish the lowenergy r ..."
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Abstract. In this work, we use scattering method to study the KramersFokkerPlanck equation with a potential whose gradient tends to zero at the infinity. For shortrange potentials in dimension three, we show that complex eigenvalues do not accumulate at lowenergies and establish the lowenergy resolvent asymptotics. This combined with high energy pseudospectral estimates valid in more general situations gives the largetime asymptotics of the solution in weighted L2 spaces. 1.