Publications (4)2.39 Total impact
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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 continuous 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 magnetic field direction, in a complex sector outside a compact set. We define the resonances of H = H0 + V as the eigenvalues of the nonselfadjoint 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 BreitWigner approximation and implies a local trace formula.Journal of Mathematical Physics 01/2009; 50(4). · 1.30 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: We study the Klein paradox for the semiclassical Dirac operator on $\R$ with potentials having constant limits, not necessarily the same at infinity. Using the complex WKB method, the timeindependent 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 semiclassical 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;  [Show abstract] [Hide abstract]
ABSTRACT: We consider the selfadjoint operator H = H0 + V, where H0 is the free semiclassical Dirac operator on R3. We suppose that the smooth matrixvalued 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 nonselfadjoint operator obtained from the Dirac operator H by complex distortions of R3. We establish an upper bound O(h3) 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 semiclassical resonances of H and a local trace formula are obtained. In particular, if V is an electromagnetic potential, we deduce a Weyltype asymptotics of the spectral shift function. As a byproduct, we obtain an upper bound O(h2) for the number of resonances close to noncritical energy levels in domains of width h and a BreitWigner approximation formula for the derivative of the spectral shift function.Reviews in Mathematical Physics 01/2007; 19(10):10711115. · 1.09 Impact Factor 
Article: Perturbation of a magnetic Schrödinger operator near an embedded infinitemultiplicity eigenvalue
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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 nonselfadjoint 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 infinitemultiplicity eigenvalues.
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7  Citations  
2.39  Total Impact Points  
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Institutions

2009

Université Bordeaux 1
Talence, Aquitaine, France
