Related papers: Smoothing effect for the regularized Schr\"odinger…
We prove exact controllability for quasi-linear Hamiltonian Schr\"odinger equations on tori of dimension greater or equal then two. The result holds true for sufficiently small initial conditions satisfying natural minimal regularity…
We study fractional hypoelliptic Ornstein-Uhlenbeck operators acting on $L^2(\mathbb{R}^n)$ satisfying the Kalman rank condition. We prove that the semigroups generated by these operators enjoy Gevrey regularizing effects. Two byproducts…
In this paper, the classical Schr\"odinger equation, which allows the study of classical dynamics in terms of wave functions, is analyzed theoretically and numerically. First, departing from classical (Newtonian) mechanics, and assuming an…
Existence of homoclinic orbits in the cubic nonlinear Schr\"odinger equation under singular perturbations is proved. Emphasis is placed upon the regularity of the semigroup $e^{\e t \pa_x^2}$ at $\e = 0$. This article is a substantial…
New Strichartz estimates for the modulated cubic nonlinear Schr\"{o}dinger equation are proved. These Strichartz estimates allow us to show that this equation is pathwise locally well-posed. We also show that improved Strichartz estimates…
This paper concerns the homogenization of Schrodinger equations for non-crystalline matter, that is to say the coefficients are given by the composition of stationary functions with stochastic deformations. Two rigorous results of so-called…
We consider the damped nonlinear Schr\''{o}dinger equation with saturation: i.e., the complex evolution equation contains in its left hand side, besides the potential term $V(x)u,$ a nonlinear term of the form $\mathrm{i}\mu…
We study the orbital stablity and instability of solitary wave solutions for nonlinear Schr\"odinger equations of derivative type.
The paper describes a new approach to global smoothing problems for dispersive and non-dispersive evolution equations based on the global canonical transforms and the underlying global microlocal analysis. For this purpose, the Egorov-type…
Self-adjoint Schr\"odinger operators with $\delta$ and $\delta'$-potentials supported on a smooth compact hypersurface are defined explicitly via boundary conditions. The spectral properties of these operators are investigated, regularity…
This paper continues the investigations from [7] and is concerned with the derivation of first-order conditions for a control constrained optimization problem governed by a non-smooth elliptic PDE. The control enters the state equation not…
We consider a family of surfaces of revolution, each with a single periodic geodesic which is degenerately unstable. We prove a local smoothing estimate for solutions to the linear Schr\"odinger equation with a loss that depends on the…
Via minimization arguments and Concentration Compactness Principle, we prove the orbital stability of standing wave solutions for a class of quasilinear Schr\"{o}dinger equation arising from physics.
We study the orbital instability of solitary waves for a generalized derivative nonlinear Schr\"odinger equation. We give sufficient conditions for instability of a two-parameter family of solitary waves in a degenerate case.
For the one-dimensional mass-critical/supercritical pseudo-relativistic nonlinear Schrodinger equation, a stationary solution can be constructed as an energy minimizer under an additional kinetic energy constraint and the set of energy…
This paper shows how abstract resolvent estimates imply local smoothing for solutions to the Schr\"odinger equation. If the resolvent estimate has a loss when compared to the optimal, non-trapping estimate, there is a corresponding loss in…
We consider the nonlinear Schrodinger equation under a partial quadratic confinement. We show that the global dispersion corresponding to the direction(s) with no potential is enough to prove global in time Strichartz estimates, from which…
We establish global bounds for solutions to stationary and time-dependent Schr\"odinger equations associated with the sublaplacian $\mathcal L$ on the Heisenberg group, as well as its pure fractional power $\mathcal L^s$ and conformally…
We prove scattering for the 2D cubic derivative Schr\"odinger equation with small data in the critical Besov space with one degree angular regularity. The main new ingredient is that we prove a spherically averaged maximal function estimate…
We prove the local energy decay and the smoothing effect for the damped Schr{\"o}dinger equation on R^d. The self-adjoint part is a Laplacian associated to a long-range perturbation of the flat metric. The proofs are based on uniform…