Displaying similar documents to “Scattering amplitude for the Schrödinger equation with strong magnetic field”

Microlocalization of resonant states and estimates of the residue of the scattering amplitude

Jean-François Bony, Laurent Michel (2003)

Journées équations aux dérivées partielles

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We obtain some microlocal estimates of the resonant states associated to a resonance z 0 of an h -differential operator. More precisely, we show that the normalized resonant states are 𝒪 ( | Im z 0 | / h + h ∞ ) outside the set of trapped trajectories and are 𝒪 ( h ∞ ) in the incoming area of the phase space. As an application, we show that the residue of the scattering amplitude of a Schrödinger operator is small in some directions under an estimate of the norm of the spectral projector. Finally we prove...

Isotropic hypoellipticity and trend to the equilibrium for the Fokker-Planck equation with high degree potential

Frédéric Hérau (2002)

Journées équations aux dérivées partielles

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We consider the Fokker-Planck equation with a confining or anti-confining potential which behaves at infinity like a possibly high degree homogeneous function. Hypoellipticity techniques provide the well-posedness of the weak-Cauchy problem in both cases as well as instantaneous smoothing and exponential trend to equilibrium. Lower and upper bounds for the rate of convergence to equilibrium are obtained in terms of the lowest positive eigenvalue of the corresponding Witten laplacian,...

Spectral projection, residue of the scattering amplitude and Schrödinger group expansion for barrier-top resonances

Jean-François Bony, Setsuro Fujiié, Thierry Ramond, Maher Zerzeri (2011)

Annales de l’institut Fourier

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We study the spectral projection associated to a barrier-top resonance for the semiclassical Schrödinger operator. First, we prove a resolvent estimate for complex energies close to such a resonance. Using that estimate and an explicit representation of the resonant states, we show that the spectral projection has a semiclassical expansion in integer powers of h , and compute its leading term. We use this result to compute the residue of the scattering amplitude at such a resonance. Eventually,...