Schrödinger equation on Cartan-Hadamard manifolds with oscillating nonlinearities

On the Schrödinger Equation with Dissipative Nonlinearities of Derivative Type

which suggests dissipativity if Imλ < 0. In fact, it is proved in [17] that the solution decays like O((t log t)−1/2) in Lx as t → +∞ if Imλ < 0 and u0 is small enough. Since the non-trivial free solution (i.e., the solution to (1) for N ≡ 0, u0 = 0) only decays like O(t−1/2), this gain of additional logarithmic time decay reflects a disspative character. Now we turn our attentions to the gener...

Concentration of the Brownian bridge on Cartan-Hadamard manifolds with pinched negative sectional curvature

We study the rate of concentration of a Brownian bridge in time one around the corresponding geodesical segment on a Cartan-Hadamard manifold with pinched negative sectional curvature, when the distance between the two extremities tends to infinity. This improves on previous results by A. Eberle [7], and one of us [21]. Along the way, we derive a new asymptotic estimate for the logarithmic deri...

Schrödinger operators with oscillating potentials ∗

Schrödinger operators H with oscillating potentials such as cos x are considered. Such potentials are not relatively compact with respect to the free Hamiltonian. But we show that they do not change the essential spectrum. Moreover we derive upper bounds for negative eigenvalue sums of H.

Nonlinear Schrödinger Equation on Four - Dimensional Compact Manifolds

— We prove two new results about the Cauchy problem in the energy space for nonlinear Schrödinger equations on four-dimensional compact manifolds. The first one concerns global well-posedness for Hartree-type nonlinearities and includes approximations of cubic NLS on the sphere as a particular case. The second one provides, in the case of zonal data on the sphere, local well-posedness for quadr...

Cartan Spinor Bundles on Manifolds. *