相关论文: Schr\"{o}dinger Equations on normal real form Symm…
We prove Strichartz inequalities for the wave and Schr\"odinger equations on noncompact surfaces with ends of finite area, i.e. with ends isometric to $ \big( (r_0,\infty) \times {\mathbb S}^1 , dr^2 + e^{- 2 \phi (r)}d \theta^2 \big) $…
The primary objective in this paper is to give an answer to an open question posed by J. A. Barcel\'o, J. M. Bennett, A. Carbery, A. Ruiz and M. C. Vilela concerning the problem of determining the optimal range on $s\geq0$ and $p\geq1$ for…
We show that for a one-dimensional Schr\"odinger operator with a potential whose first moment is integrable the scattering matrix is in the unital Wiener algebra of functions with integrable Fourier transforms. Then we use this to derive…
A refinement of uniform resolvent estimate is given and several smoothing estimates for Schrodinger equations in the critical case are induced from it. The relation between this resolvent estimate and radiation condition is discussed. As an…
We consider the Cauchy problem for the defocusing cubic nonlinear Schr\"odinger equation in four space dimensions and establish almost sure local well-posedness and conditional almost sure scattering for random initial data in…
We prove a sharp, global-in-time Strichartz estimate for the Schr\"odinger equation on the cylinder $\mathbb{R}\times\mathbb{T}$.
We consider the nonlinear Schr\"odinger equation with periodic dispersion management. We first establish global-in-time Strichartz estimates for the underlying linear equation with suitable dispersion maps. As an application, we establish a…
We are mainly interested in extending the known results on ob-servability inequalities and stabilization for the Schr{\"o}dinger equation to the magnetic Schr{\"o}dinger equation. That is in presence of a magnetic potential. We establish…
In this article, we consider the infinite dimensional vector-valued resonant nonlinear Schr\"odinger system, which arises from the study of the asymptotic behavior of the defocusing nonlinear Schr\"{o}dinger equation on "wave guide"…
We revisit the local well-posedness theory of nonlinear Schr\"odinger and wave equations in Sobolev spaces $H^s$ and $\dot{H}^s$, $0< s\leq 1$. The theory has been well established over the past few decades under Sobolev initial data…
We derive dispersion estimates for solutions of a one-dimensional discrete Dirac equations with a potential. In particular, we improve our previous result, weakening the conditions on the potential. To this end we also provide new results…
We consider the scattering problem for the nonlinear Schr\"{o}dinger equation with a potential in two space dimensions. Appropriate resolvent estimates are proved and applied to estimate the operator $A(s)$ appearing in commutator…
Motivated by a recent work of Schippa (2022), we consider local smoothing estimates for Schr\"{o}dinger equations in modulation spaces. By using the C\'{o}rdoba-Fefferman type reverse square function inequality and the bilinear Strichartz…
We prove global well-posedness and scattering for the nonlinear Schr\"odinger equation with power-type nonlinearity \begin{equation*} \begin{cases} i u_t +\Delta u = |u|^p u, \quad \frac{4}{n}<p<\frac{4}{n-2}, u(0,x) = u_0(x)\in H^s(\R^n),…
We derive a dispersion estimate for one-dimensional perturbed radial Schr\"odinger operators. We also derive several new estimates for solutions of the underlying differential equation and investigate the behavior of the Jost function near…
We prove the optimal endpoint Strichartz estimates for Schr\"{o}dinger equation with charge transfer potentials and a general source term in $\mathbb{R}^n$ for $n\geq3$. The proof is based on using the projection on the scattering states…
We study the Schr\"odinger equation on a flat euclidean cone $\mathbb{R}_+ \times \mathbb{S}^1_\rho$ of cross-sectional radius $\rho > 0$, developing asymptotics for the fundamental solution both in the regime near the cone point and at…
We prove decay with respect to some Lebesgue norms for a class of Schr\"odinger equations with non-local nonlinearities by showing new Morawetz inequalities and estimates. As a byproduct, we obtain large-data scattering in the energy space…
We prove local Strichartz estimates on compact manifolds with boundary. Our results also apply more generally to compact manifolds with Lipschitz metrics.
In this article, we consider the nonlinear Schr\"odinger equation on the cylinder $\mathbb{R}^d\times \mathbb{T}$. In the long range case, we show there is no linear scattering state of the nonlinear Schr\"odinger equation on $\mathbb{R}^d…