Thierry Ramond

Semiclassical Study of Quantum Scattering on the Line

We study the well-known problem of 1-d quantum scattering by a potential barrier in the semiclassical limit. Using the so-called exact WKB method and semiclassical microlocal analysis techniques, we get a very precise and complete description of the scattering matrix, in particular when the energy is very close to a unique, quadratic maximum of the potential. In our one-dimensional setting, we also recover the Bohr-Sommerfeld quantization condition for the resonances generated by such a maximum.

Semiclassical scattering amplitude at the maximum of the potential

We study the scattering amplitude for Schrodinger operators at a critical energy level, which is a unique non- degenerate maximum of the potential. We do not assume that the maximum point is non-resonant and use results by Bony, Fujiie, Ramond and Zerzeri to analyze the contributions of the trapped trajectories. We prove a semiclassical expansion of the scattering amplitude and compute its leading term. We show that it has different orders of magnitude in specific regions of phase space. We also prove upper and lower bounds for the resolvent in this setting.

Breit–Wigner formula at barrier tops

For noncritical energies, the asymptotic behavior of the scattering phase and of the time-delay are known to be described by a Weyl type formula and the Breit–Wigner formula, respectively. We consider here the case of critical energy levels in dimension 1. We obtain the semiclassical asymptotics of the scattering phase and of the time-delay, uniformly with respect to the energy in a neighborhood of a critical value.

NON-REAL EIGENVALUES FOR PT -SYMMETRIC DOUBLE WELLS

We study small,

Microlocal kernel of pseudodifferential operators at a hyperbolic fixed point

{{\mathcal{PT}}}

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

PT-symmetric perturbations of self-adjoint double-well Schrödinger operators in dimension

Intervalles d'instabilité pour une équation de Hill à potentiel méromorphe

{n\ge 1}