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We consider the Gross-Petaevskii equation in 1 space dimension with a n-well trapping potential. We prove, in the semiclassical limit, that the finite dimensional eigenspace associated to the lowest n eigenvalues of the linear operator is slightly deformed by the nonlinear term into an almost invariant manifold M. Precisely, one has that solutions starting… (More)
Here we consider stationary states for nonlinear Schrödinger equations in any spatial dimension n with symmetric double well potentials. These states may bifurcate as the strength of the nonlinear term increases and we observe two different pictures depending on the value of the nonlinearity power: a supercritical pitchfork bifurcation, and a subcritical… (More)
We consider the small eld asymptotics of the lifetime of metastable states in Wannier-Stark problems. Assuming that at zero eld we have Bloch operators with only the rst gap open and using the regular perturbation theory, we prove that the behavior of the lifetime computed by means of the Fermi Golden Rule is proportional to the correct one with the factor… (More)
In this paper we consider embedded eigenvalues of a Schrödinger Hamiltonian in a waveguide induced by a symmetric perturbation. It is shown that these eigenvalues become unstable and turn into resonances after twisting of the waveguide. The perturbative expansion of the resonance width is calculated for weakly twisted waveguides and the influence of the… (More)
We consider a class of SchrÄ odinger equations with a symmetric double-well potential and an external, both repulsive and attractive, nonlinear perturbation. We show that, under certain conditions, the reduction of the time-dependent equation to a two-mode equation gives the dominant term of the solution with a precise estimate of the error.
We consider the time-dependent non linear Schrödinger equations with a double well potential in dimensions d = 1 and d = 2. We prove, in the semiclassical limit, that the finite dimensional eigenspace associated to the lowest two eigenvalues of the linear operator is almost invariant for any time.
We consider a Wannier-Stark problem for small eld f in the one-ladder case. We prove that a generical rst band state is a metastable state (Wannier-Bloch oscillator) with the lifetime determined by the imaginary part of the Wannier-Stark ladder. The innnite resonances of the ladder cause Bloch oscillations as a global beating eeect. For an adiabatic time =… (More)