Related papers: Explicit formulas and global uniqueness for phasel…
We prove that the singularities of a potential in the two and three dimensional Schr\"odinger equation are the same as the singularities of the Born approximation (Diffraction Tomography), obtained from backscattering inverse data, with an…
We study an inverse source scattering problem for the Schr\"odinger equation with a quadratic nonlinearity. In general, uniqueness of inverse source problems can not be guaranteed at a fixed energy. Therefore, additional information is…
In this paper, we study the direct and inverse scattering of the Schr\"odinger equation in a three-dimensional planar waveguide. For the direct problem, we derive a resonance-free region and resolvent estimates for the resolvent of the…
We consider the cubic nonlinear Schr\"odinger equation with long-range linear potentials in one space dimension, and prove the modified scattering in the energy space for the associated final state problem with a prescribed small asymptotic…
We show global uniqueness in an inverse problem for the fractional Schr\"odinger equation: an unknown potential in a bounded domain is uniquely determined by exterior measurements of solutions. We also show global uniqueness in the partial…
We study the nonlinear Schr\"odinger equation with an inverse-square potential in dimensions $3\leq d \leq 6$. We consider both focusing and defocusing nonlinearities in the mass-supercritical and energy-subcritical regime. In the focusing…
We consider the defocusing energy-critical nonlinear Schr\"odinger equation with inverse-square potential $iu_t = -\Delta u + a|x|^{-2}u + |u|^4u$ in three space dimensions. We prove global well-posedness and scattering for $a>-\frac14…
In this paper, we study an inverse scattering problem associated with the time-harmonic Schr\"odinger equation where both the potential and the source terms are unknown. The source term is assumed to be a generalised Gaussian random…
We consider the 3D inverse scattering problem with non-over-determined scattering data. The data are the scattering amplitude $A(\beta, \alpha_0, k)$ for all $\beta \in S_\beta^2$, where $S_\beta^2$ is an open subset of the unit sphere…
We consider the spectral theory for discrete Schr\"odinger operators on the hexagonal lattice and their inverse scattering problem. We give a procedure for reconstructing the compactly supported potential from the scattering matrix for all…
In this article, we study the inverse scattering problem for the nonlinear fractional Helmholtz equation with cubic nonlinearity in three dimensions, where we recover a compactly supported potential from scattering amplitude.
In this paper,we show that spherical bounded energy solution of the defocusing 3D energy critical Schr\"odinger equation with harmonic potential, $(i\partial_t + \frac {\Delta}2+\frac {|x|^2}2)u=|u|^4u$, exits globally and scatters to free…
An inverse scattering problems for the 3-D generalized Helmholtz equation is considered. Only the modulus of the complex valued scattered wave field is assumed to be measured and the phase is not measured. Uniqueness theorem is proved.
We study the inverse scattering problem for electric potentials and magnetic fields in $\ere^d, d\geq 3$, that are asymptotic sums of homogeneous terms at infinity. The main result is that all these terms can be uniquely reconstructed from…
We study the threshold scattering problem for the energy-critical nonlinear Schr\"odinger equation with a repulsive inverse-square potential $\frac{a}{|x|^2} > 0$ in dimensions $d= 4, 5, 6$. On the energy level surface determined by the…
We define scattering data for the Newton equation in a potential $V\in C^2(\R^n,\R)$, $n\ge2$, that decays at infinity like $r^{-\alpha}$ for some $\alpha\in (0,1]$. We provide estimates on the scattering solutions and scattering data and…
We consider a 3d cubic focusing nonlinear Schr\"odinger equation with a potential $$i\partial_t u+\Delta u-Vu+|u|^2u=0,$$ where $V$ is a real-valued short-range potential having a small negative part. We find criteria for global…
A review of a recent method is presented to construct certain exact solutions to the focusing nonlinear Schr\"odinger equation on the line with a cubic nonlinearity. With motivation by the inverse scattering transform and help from the…
A solution of the scattering problem is obtained for the Schr\"odinger equation with the potential of induced dipole interaction, which decreases as the inverse square of the distance. Such a potential arises in the collision of an incident…
We present a conditionally exactly solvable singular potential for the one-dimensional Schr\"odinger equation which involves the exactly solvable inverse square root potential. Each of the two fundamental solutions that compose the general…