Asymptotic Analysis

In this paper we calculate the second order term of the Neumann operator for the high frequency scattering of a wave by a strictly convex obstacle Ω using the techniques introduced by G. Lebeau and used in previous works. This second term is used to calculate the scattering by Ω⊂R², and the result is compared to previous results of V. Babich and I. Molotkov. We generalize the calculation when Ω⊂R³.

We give some partial results regarding to evolutionary dispersive Ginzburg–Landau equations associated to the static case considered in Béthuel, Brezis and Hélein [1]. We treat the partially dissipative case. Moreover we perform an exact “semi-classical” limit. We obtain alternatively the Laplace, heat and wave equations as limit equation, depending on the scaling that we consider.

We consider the semiclassical Stark effect for a family of asymmetric unstable double well models and we study the crossing and anticrossing of the field dependent resonances in the complex field plane. We prove that a Bender-Wu type singularity crosses the real axis when the internal barrier is nearly twice “larger” than the external one and the beating period is close to the shorter life-time of the resonances. At this critical point we have the anticrossing-crossing transition and for larger instability we have the single well localization.

We study the sum of negative eigenvalues M(h,μ) of the Schrödinger operator H_V(h,μ) in L²(R³) with a decaying at infinity potential V and a homogeneous magnetic field of intensity μ. The parameter h denotes the Planck constant. Assuming that V has a finite number of Coulomb singularities and is smooth outside them, we establish the two-term asymptotics of M(h,μ) as h→0 and μ→∞. The second term is determined only by the singularities of V and it turns out to be the same as in the case μ=0.