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43
Diophantine tori and spectral asymptotics for nonselfadjoint operators
, 2005
"... We study spectral asymptotics for small nonselfadjoint perturbations of selfadjoint hpseudodifferential operators in dimension 2, assuming that the classical flow of the unperturbed part possesses several invariant Lagrangian tori enjoying a Diophantine property. We get complete asymptotic expansio ..."
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Cited by 13 (2 self)
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We study spectral asymptotics for small nonselfadjoint perturbations of selfadjoint hpseudodifferential operators in dimension 2, assuming that the classical flow of the unperturbed part possesses several invariant Lagrangian tori enjoying a Diophantine property. We get complete asymptotic expansions for all eigenvalues in certain rectangles in the complex plane in two different cases: in the first case, we assume that the strength ɛ of the perturbation is O(h δ) for some δ>0 and is bounded from below by a fixed positive power of h. In the second case, ɛ is assumed to be sufficiently small but independent of h, and we describe the eigenvalues completely in a fixed hindependent domain in the complex spectral plane.
The Fermi Golden Rule and its form at thresholds in odd dimensions
"... Let H be a Schrödinger operator on a Hilbert space H, such that zero is a nondegenerate threshold eigenvalue of H with eigenfunction Ψ0. Let W be a bounded selfadjoint operator satisfying 〈Ψ0, W Ψ0 〉> 0. Assume that the resolvent (H − z) −1 has an asymptotic expansion around z = 0 of the form typ ..."
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Cited by 13 (6 self)
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Let H be a Schrödinger operator on a Hilbert space H, such that zero is a nondegenerate threshold eigenvalue of H with eigenfunction Ψ0. Let W be a bounded selfadjoint operator satisfying 〈Ψ0, W Ψ0 〉> 0. Assume that the resolvent (H − z) −1 has an asymptotic expansion around z = 0 of the form typical for Schrödinger operators on odddimensional spaces. Let H(ε) = H + εW for ε> 0 and small. We show under some additional assumptions that the eigenvalue at zero becomes a resonance for H(ε), in the timedependent sense introduced by A. Orth. No analytic continuation is needed. We show that the imaginary part of the resonance has a dependence on ε of the form ε 2+(ν/2) with the integer ν ≥ −1 and odd. This shows how the Fermi Golden Rule has to be modified in the case of perturbation of a threshold eigenvalue. We give a number of explicit examples, where we compute the “location” of the resonance to leading order in ε. We also give results, in the case where the eigenvalue is embedded in the continuum, sharpening the existing ones. 1
FROM OPEN QUANTUM SYSTEMS TO OPEN QUANTUM MAPS
"... In this paper we show that for a class of open quantum systems satisfying a natural dynamical assumption (see §1.2) the study of the resolvent, and hence of scattering, and of resonances, can be reduced to the study of open quantum maps, that is of finite dimensional quantizations of canonical relat ..."
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Cited by 5 (2 self)
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In this paper we show that for a class of open quantum systems satisfying a natural dynamical assumption (see §1.2) the study of the resolvent, and hence of scattering, and of resonances, can be reduced to the study of open quantum maps, that is of finite dimensional quantizations of canonical relations obtained by truncation of symplectomorphisms.
Embedded eigenvalues and resonances of Schrödinger operators with two channels
 Ann. Fac. Sci. Toulouse Math
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PROBABILISTIC WEYL LAWS FOR QUANTIZED TORI
"... Abstract. For the Toeplitz quantization of complexvalued functions on a 2ndimensional torus we prove that the expected number of eigenvalues of small random perturbations of a quantized observable satisfies a natural Weyl law (1.3). In numerical experiments the same Weyl law also holds for “false ..."
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Abstract. For the Toeplitz quantization of complexvalued functions on a 2ndimensional torus we prove that the expected number of eigenvalues of small random perturbations of a quantized observable satisfies a natural Weyl law (1.3). In numerical experiments the same Weyl law also holds for “false ” eigenvalues created by pseudospectral effects.
Semiclassical spectral asymptotics for a magnetic Schrödinger operator with nonvanishing magnetic field
, 2014
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Eigenfunction localization for the 2D periodic Schrödinger operator, Int
 Département de Mathématique, Université Paris Sud, 91405 Orsay Cedex, FRANCE
, 2010
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