Time Decay for Nonlinear Wave Equations in Two Space Dimensions.
On geometrically finite hyperbolic manifolds , including those with non-maximal rank cusps, we give upper bounds on the number of resonances of the Laplacian in disks of size as . In particular, if the parabolic subgroups of satisfy a certain Diophantine condition, the bound is .
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 of height , . We show that the semiclassical wave front set of the resonant states (including the “generalized eigenfunctions”) is contained in the set of the trapped bicharacteristics. We also show that for a suitable self-adjoint reference operator with discrete spectrum the number of resonances in is bounded by the number of eigenvalues of in an interval a bit larger than the projection of on the real line. As an application, we prove a...