J. -F. Bony

Université Paris-Sud 11, Paris, Ile-de-France, France

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Publications (6)2 Total impact

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    J. -F. Bony, V. Bruneau, G. Raikov
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    ABSTRACT: In this survey article we consider the operator pair $(H,H_0)$ where $H_0$ is the shifted 3D Schr\"odinger operator with constant magnetic field, $H : = H_0 + V$, and $V$ is a short-range electric potential of a fixed sign. We describe the asymptotic behavior of the Krein spectral shift function (SSF) $\xi(E; H,H_0)$ as the energy $E$ approaches the Landau levels $2bq$, $q \in {\mathbb Z}_+$, which play the role of thresholds in the spectrum of $H_0$. The main asymptotic term of $\xi(E; H,H_0)$ as $E \to 2bq$ with a fixed $q \in {\mathbb Z}_+$ is written in the terms of appropriate compact Berezin-Toeplitz operators. Further, we investigate the relation between the threshold singularities of the SSF and the accumulation of resonances at the Landau levels. We establish the existence of resonance free sectors adjoining any given Landau level and prove that the number of the resonances in the complementary sectors is infinite. Finally, we obtain the main asymptotic term of the local resonance counting function near an arbitrary fixed Landau level; this main asymptotic term is again expressed via the Berezin-Toeplitz operators which govern the asymptotics of the SSF at the Landau levels.
    Spectral and Scattering Theory and Related Topics, RIMS Kôkyûroku Bessatsu B45 edited by F. Hiroshima, 04/2014: chapter Resonances and Spectral Shift Function Singularities for Magnetic Quantum Hamiltonians: pages 77-100; RIMS, 2014., ISBN: 1881-6193
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    J. -F. Bony, F. Hérau, L. Michel
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    ABSTRACT: We study a semiclassical random walk with respect to a probability measure with a finite number n_0 of wells. We show that the associated operator has exactly n_0 exponentially close to 1 eigenvalues (in the semiclassical sense), and that the other are O(h) away from 1. We also give an asymptotic of these small eigenvalues. The key ingredient in our approach is a general factorization result of pseudodifferential operators, which allows us to use recent results on the Witten Laplacian.
  • J -F Bony, S Fujiie, T Ramond, M Zerzeri
    MRIT 6th Workshop---Math-for-Industry Tutorial: Spectral theories of non-Hermitian operators and their application, 01/2009: pages 13; Kyushu Univ. Fac. Math., Fukuoka.
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    ABSTRACT: グローバルCOEプログラム「マス・フォア・インダストリ教育研究拠点」 MI: Global COE Program Education-and-Research Hub for Mathematics-for-Industry MRIT 6th Workshop Math-for-Industry Tutorial : Spectral theories of non-Hermitian operators and their application We give a semiclassical expansion of the Schrödinger group in terms of the resonances created by a non-degenerate potential maximum. This formula implies that the imaginary part of the resonances gives the decay rate of states for large time of order of the logarithm of the semiclassical parameter.
  • Source
    J. F. Bony, V. Bruneau, G. Raikov
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    ABSTRACT: We consider the 3D Schr\"odinger operator $H = H_0 + V$ where $H_0 = (-i\nabla - A)^2$, $A$ is a magnetic potential generating a constant magnetic field of strength $b>0$, and $V$ is a short-range electric potential which decays superexponentially with respect to the variable along the magnetic field. We show that the resolvent of $H$ admits a meromorphic extension from the upper half-plane to an appropriate complex manifold ${\mathcal M}$, and define the resonances of $H$ as the poles of this meromorphic extension. We study their distribution near any fixed Landau level $2bq$, $q \in {\mathbb N}$. First, we obtain a sharp upper bound of the number of resonances in a vicinity of $2bq$. Moreover, under appropriate hypotheses, we establish corresponding lower bounds which imply the existence of an infinite number of resonances, or the absence of resonances in certain sectors adjoining $2bq$. Finally, we deduce a representation of the derivative of the spectral shift function (SSF) for the operator pair $(H,H_0)$ as a sum of a harmonic measure related to the resonances, and the imaginary part of a holomorphic function. This representation justifies the Breit-Wigner approximation, implies a trace formula, and provides information on the singularities of the SSF at the Landau levels.
    Annales- Institut Fourier 05/2006; 57 (2007):629-671. · 0.64 Impact Factor
  • J.-F. Bony
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    ABSTRACT: Rsum. On travaille dans le cadre de l‘analyse semi-classique. Considrons p(x, hDx) p(x, hD_{x}) , une perturbation de -h2D -h^{2}\Delta qui est analytique l‘infini. On suppose que dans la surface d‘nergie E0 > 0, les points critiques du symbole p(x, x) p(x, \xi) forment une sous-varit C \mathcal C et que p est non dgnr dans l‘espace normal C \mathcal C .En utilisant les rsultats de [6] et [18], on obtient une majoration du nombre de rsonances dans des disques de rayon d \delta centrs en E proche de E0, o d \delta satisfait Ch < d \delta < 1/C pour une constante C > 0. En gnralisant la formule de trace de Sjstrand qui exprime la trace d‘une diffrence d‘oprateurs en fonction des rsonances, on trouve une minoration du nombre de rsonances proches de E0.
    Annales Henri Poincare 07/2002; 3(4):693-710. · 1.37 Impact Factor