Équation de Hill à potentiel méromorphe
Intervalles d'instabilité pour une équation de Hill à potentiel méromorphe
Resolvent and Scattering Matrix at the Maximum of the Potential
2000 Mathematics Subject Classification: 35P25, 81U20, 35S30, 47A10, 35B38. We study the microlocal structure of the resolvent of the semiclassical Schrödinger operator with short range potential at an energy which is a unique non-degenerate global maximum of the potential. We prove that it is a semiclassical Fourier integral operator quantizing the incoming and outgoing Lagrangian submanifolds associated to the fixed hyperbolic point. We then discuss two applications of this result...
Semiclassical expansion for the thermodynamic limit of the ground state energy of Kac's operator
We continue the study started by the first author of the semiclassical Kac Operator. This kind of operator has been obtained for example by M. Kac as he was studying a 2D spin lattice by the so-called “transfer operator method”. We are interested here in the thermodynamical limit of the ground state energy of this operator. For Kac’s spin model, is the free energy per spin, and the semiclassical regime corresponds to the mean-field approximation. Under suitable assumptions, which are satisfied...
Matrice de scattering et résonances associées à une orbite hétérocline
Spectral projection, residue of the scattering amplitude and Schrödinger group expansion for barrier-top resonances
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 , and compute its leading term. We use this result to compute the residue of the scattering amplitude at such a resonance. Eventually, we...
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