Related papers: Explicit formulas and global uniqueness for phasel…
We investigate scattering properties of a Moyal deformed version of the nonlinear Schr\"odinger equation in an even number of space dimensions. With rather weak conditions on the degree of nonlinearity, the Cauchy problem for general…
We consider the focusing cubic nonlinear Schr\"odinger equation with inverse-square potential in three space dimensions. We identify a sharp threshold between scattering and blowup, establishing a result analogous to that of Duyckaerts,…
We establish global well-posedness and scattering for solutions to the defocusing mass-critical (pseudoconformal) nonlinear Schr\"odinger equation $iu_t + \Delta u = |u|^{4/n} u$ for large spherically symmetric $L^2_x(\R^n)$ initial data in…
We introduce a perturbative formulation for a nonlinear extension of the J-matrix method of scattering in two dimensions. That is, we obtain the scattering matrix for the time-independent nonlinear Schr\"odinger equation in two dimensions…
Inverse scattering problem is discussed for the Maxwell's equations. A reduction of the Maxwell's system to a new Fredholm second-kind integral equation with a {\it scalar weakly singular kernel} is given for electromagnetic (EM) wave…
An inverse scattering method based on an auxiliary inverse Sturm-Liouville problem recently proposed by Horv\'ath and Apagyi [Mod. Phys. Lett. B 22, 2137 (2008)] is examined in various aspects and developed further to (re)construct…
We relax the regularity condition on potentials of the Schr\"odinger equation in uniqueness results on the inverse boundary value problem which were recently proved in [11] and [5].
Quantum scattering by a one-dimensional odd potential proportional to the square of the distance to the origin is considered. The Schr\"odinger equation is solved exactly and explicit algebraic expressions of the wavefunction are given. A…
We prove the scattering for a defocusing nonlinear Schr\"odinger equation with a sum of two repulsive potentials with strictly convex level surfaces, thus providing a scattering result in a trapped setting similar to the exterior of two…
We consider the defocusing energy-critical nonlinear Schr\"odinger equation of fourth order $iu_t+\Delta^2 u=-|u|^\frac{8}{d-4}u$. We prove that any finite energy solution is global and scatters both forward and backward in time in…
We present a refinement of the Spectral Method by incorporating an optimization method into it and generalize it to two space dimensions. We then apply this Refined Spectral Method as an extremely accurate technique for finding the bound…
In this article, we prove the global well-posedness and scattering of the cubic focusing infinite coupled nonlinear Schr\"odinger system on $\mathbb{R}^2$ below the threshold in $L_x^2h^1(\mathbb{R}^2\times \mathbb{Z})$. We first establish…
We develop the existence, uniqueness, continuity, stability, and scattering theory for energy-critical nonlinear Schr\"odinger equations in dimensions $n \geq 3$, for solutions which have large, but finite, energy and large, but finite,…
One-dimensional time-independent Schr\"odinger equation is solved for the asymmetric Hulth\'{e}n potential. Reflection and transmission coefficients and bound state solutions are obtained in terms of the hypergeometric functions. It is…
We consider nonlinear Schr\"odinger equations with either power-type or Hartree nonlinearity in the presence of an external potential. We show that for long-range nonlinearities, solutions cannot exhibit scattering to solitary waves or more…
We address the existence of global solutions to the initial value problem for the integrable nonlocal derivative nonlinear Schr\"{o}dinger equation in weighted Sobolev space $H^{2}(\mathbb{R})\cap H^{1,1}(\mathbb{R})$. The key to prove this…
We use the Tridiagonal Representation Approach (TRA) to obtain exact scattering and bound states solutions of the Schr\"odinger equation for short-range inverse-square singular hyperbolic potentials. The solutions are series of square…
We consider the focusing energy-critical nonlinear Schr\"odinger equation $iu_t+\Delta u = - |u|^{\frac4{d-2}}u$ in dimensions $d\geq 5$. We prove that if a maximal-lifespan solution $u:I\times\R^d\to \C$ obeys $\sup_{t\in I}\|\nabla…
We investigate the decay estimates of global solutions for a class of one-dimensional inhomogeneous nonlinear Schr\"odinger equations. While most existing results focus on spatial dimensions $d\geq2$, the decay properties in one dimension…
We prove global existence and scattering for a class of quadratic Schrodinger equations in dimension 2. The proof relies on the idea of space-time resonance.