2000 Mathematics Subject Classification: Primary: 34L25; secondary: 47A40, 81Q10.In this article we prove that the wave operators describing the direct scattering of the defocusing matrix Zakharov-Shabat system with potentials having distinct nonzero values with the same modulus at ± ∞ exist, are asymptotically complete, and lead to a unitary scattering operator. We also prove that the free Hamiltonian operator is absolutely continuous.

In this short note, we apply the technique developed in [Math. Model. Nat. Phenom., 5 (2010), No. 4, 122-149] to study the long-time evolution for Schrödinger equation with slowly decaying potential.

We study semiclassical resonances in a box $\Omega \left(h\right)$ of height $h^N$, $N\gg 1$. We show that the semiclassical wave front set of the resonant states (including the “generalized eigenfunctions”) is contained in the set $𝒯$ of the trapped bicharacteristics. We also show that for a suitable self-adjoint reference operator $P^{\#}\left(h\right)$ with discrete spectrum the number of resonances in $\Omega \left(h\right)$ is bounded by the number of eigenvalues of $P^{\#}\left(h\right)$ in an interval a bit larger than the projection of $\Omega \left(h\right)$ on the real line. As an application, we prove a...