Abdallah Khochman

Université Bordeaux 1, Talence, Aquitaine, France

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Publications (4)2.39 Total impact

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    Abdallah Khochman
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    ABSTRACT: We consider the 3D Schrödinger operator H0 with constant magnetic field and subject to an electric potential v0 depending only on the variable along the magnetic field x3. The operator H0 has infinitely many eigenvalues of infinite multiplicity embedded in its con-tinuous spectrum. We perturb H0 by smooth scalar potentials V = O((x1, x2) −δ ⊥ x3 −δ), δ ⊥ > 2, δ > 1. We assume also that V and v0 have an analytic continuation, in the mag-netic field direction, in a complex sector outside a compact set. We define the resonances of H = H0 + V as the eigenvalues of the non-selfadjoint operator obtained from H by analytic distortions of Rx 3 . We study their distribution near any fixed real eigenvalue of H0, 2bq + λ for q ∈ N. In a ring centered at 2bq + λ with radiuses (r, 2r), we establish an upper bound, as r tends to 0, of the number of resonances. This upper bound depends on the decay of V at infinity only in the directions (x1, x2). Finally, we deduce a representation of the derivative of the spectral shift function (SSF) for the operator pair (H0, H) in terms of resonances. This representation justifies the Breit-Wigner approximation and implies a local trace formula.
    Journal of Mathematical Physics 01/2009; 50(4). · 1.30 Impact Factor
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    Abdallah Khochman
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    ABSTRACT: We study the Klein paradox for the semi-classical Dirac operator on $\R$ with potentials having constant limits, not necessarily the same at infinity. Using the complex WKB method, the time-independent scattering theory in terms of incoming and outgoing plane wave solutions is established. The corresponding scattering matrix is unitary. We obtain an asymptotic expansion, with respect to the semi-classical parameter $h$, of the scattering matrix in the cases of the Klein paradox, the total transmission and the total reflection. Finally, we treat the scattering problem in the zero mass case.
    12/2007;
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    Abdallah Khochman
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    ABSTRACT: We consider the selfadjoint operator H = H0 + V, where H0 is the free semi-classical Dirac operator on R3. We suppose that the smooth matrix-valued potential V = O(-delta), delta > 0, has an analytic continuation in a complex sector outside a compact. We define the resonances as the eigenvalues of the non-selfadjoint operator obtained from the Dirac operator H by complex distortions of R3. We establish an upper bound O(h-3) for the number of resonances in any compact domain. For delta > 3, a representation of the derivative of the spectral shift function xi(lambda,h) related to the semi-classical resonances of H and a local trace formula are obtained. In particular, if V is an electro-magnetic potential, we deduce a Weyl-type asymptotics of the spectral shift function. As a by-product, we obtain an upper bound O(h-2) for the number of resonances close to non-critical energy levels in domains of width h and a Breit-Wigner approximation formula for the derivative of the spectral shift function.
    Reviews in Mathematical Physics 01/2007; 19(10):1071-1115. · 1.09 Impact Factor
  • Vincent Bruneau, Abdallah Khochman, Georgi Raikov
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    ABSTRACT: We consider a 3D magnetic Schrödinger operator having infinitely many eigenvalues of infinite multiplicity, embedded in the continuous spectrum. We perturb this operator by a relatively compact potential and analyse the transition of these eigenvalues into a “cloud” of resonances. Several differential approaches are employed. First we consider resonances as eigenvalues of a non-selfadjoint operator by using analytic distortion. Then we study the dynamical aspect of the resonances and finally we study the behavior of the spectral shift function near the infinite-multiplicity eigenvalues.