Related papers: Maximal estimates for Schroedinger equation with i…
The endpoint Strichartz estimates for two-dimensional Schrodinger equations were recovered by averaging the solutions in L^2 in the angular variable by Tao. For Schrodinger equations with defocusing inverse square potential, we proved that…
For the one-dimensional Schr\"odinger equation, we obtain sharp maximal-in-time and maximal-in-space estimates for systems of orthonormal initial data. The maximal-in-time estimates generalize a classical result of Kenig--Ponce--Vega and…
In this paper, we studied the space-time estimates for the solution to the Schr\"odinger equation. By polynomial partitioning, induction arguments, bilinear to linear arguments and broad norm estimates, we set up several maximal estimates…
In this paper, we study the Strichartz-type estimates of the solution for the linear wave equation with inverse square potential. Assuming the initial data possesses additional angular regularity, especially the radial initial data, the…
The purpose of this paper is to study the validity of global-in-time Strichartz estimates for the Schr\"odinger equation on $\mathbb{R}^n$, $n\ge3$, with the negative inverse-square potential $-\sigma|x|^{-2}$ in the critical case…
For $\alpha >1$ we consider the initial value problem for the dispersive equation $i\partial_t u +(-\Delta)^{\alpha/2} u= 0$. We prove an endpoint $L^p$ inequality for the maximal function $\sup_{t\in[0,1]}|u(\cdot,t)|$ with initial values…
We consider non-linear Schr\"odinger equations with a potential, and non-local non-linearities, that are models in mesoscopic physics, for example of a quantum capacitor, and that also are models of molecular structure. We study in detail…
We prove spacetime weighted-L^2 estimates for the Schrodinger and wave equation with an inverse-square potential. We then deduce Strichartz estimates for these equations.
In this work, we study inverse resonance problems for the Schr\"odinger operator on the real line with the potential supported in $[0,1]$. In general, all eigenvalues and resonances can not uniquely determine the potential. (i) It is shown…
A long standing conjecture for the linear Schroedinger equation states that 1/4 of derivative in $L^2$, in the sense of Sobolev spaces, suffices in any dimension for the solution to that equation to converge almost everywhere to the initial…
The Schroedinger equation is considered on the line when the potential is real valued, compactly supported, and square integrable. The nonuniqueness is analyzed in the recovery of such a potential from the data consisting of the ratio of a…
In this paper, we consider the energy critical nonlinear Schr\"odinger equation with a repulsive inverse square potential. In particular, we deal with radial initial data, whose energy is equal to the energy of static solution to the…
In this paper, we study forward problem and inverse problem for the fractional magnetic Schrodinger equation with nonlinear electric potential. We first investigate the maximum principle for the linearized equation and apply it to show that…
The initial value problem for the homogeneous Schr\"odinger equation is investigated for radially symmetric initial data with slow decay rates and not too wild oscillations. Our global wellposedness results apply to initial data for which…
We study an initial value problem for the one-dimensional non-stationary linear Schr\"odinger equation with a point singular potential. In our approach, the problem is considered as a system of coupled initial-boundary value (IBV) problems…
We study pointwise convergence of the solutions to Schr\"odinger equations with initial datum $f\in H^s(\mathbb R^n)$. The conjecture is that the solution $e^{it\Delta}f$ converges to $f$ almost everywhere for all $f\in H^s(\mathbb R^n)$ if…
In the present paper we consider Schr\"odinger equations with variable coefficients and potentials, where the principal part is a long-range perturbation of the flat Laplacian and potentials have at most linear growth at spatial infinity.…
An inverse problem is considered for an inhomogeneous Schr\"odinger equation. Assuming that the potential vanishes outside a finite interval and satisfies some other technical assumptions, one proves the uniqueness of the recovery of this…
In this paper, we combine the argument of [12] and [27] to prove the maximal estimates for fractional Schr\"odinger equations $(i\partial_t+\mathcal{L}_{\mathbf{A}}^{\frac\alpha 2})u=0$ in the purely magnetic fields which includes the…
We consider the one-dimensional Schr\"odinger equation with a potential satisfying the standard assumptions of the inverse scattering theory and supported on the half-line $x\ge 0$. For this equation at fixed positive energy we give…