Related papers: Inverse scattering theory for Schr\"odinger operat…
We analyse the scattering operator associated with the defocusing nonlinear Schr{\"o}dinger equation which captures the evolution of solutions over an infinite time-interval under the nonlinear flow of this equation. The asymptotic nature…
The inverse scattering problem of the three-dimensional Schroedinger equation is considered at fixed scattering energy with spherically symmetric potentials. The phase shifts determine the potential therefore a constructive scheme for…
An "exact discretization" of the Schroedinger operator is considered and its direct and inverse scattering problems are solved. It is shown that a differential-difference nonlinear evolution equation depending on two arbitrary constants can…
We study the nonlinear Schrodinger equations with a linear potential. A change of variables makes it possible to deduce results concerning finite time blow up and scattering theory from the case with no potential.
In this paper, we study the scattering theory of a class of continuum Schr\"{o}dinger operators with random sparse potentials. The existence and completeness of wave operators are proven by establishing the uniform boundedness of modified…
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…
Inverse scattering problem for the operator representing sum of the operator of the third derivative on semi-axis and of the operator of multiplication by a real function is studied in this paper. Properties of Jost solutions of such an…
The inverse scattering transform is extended to investigate the Tzitz\'{e}ica equation. A set of sectionally analytic eigenfunctions and auxiliary eigenfunctions are introduced. We note that in this procedure, the auxiliary eigenfunctions…
We propose a first order equation from which the Schrodinger equation can be derived. Matrices that obey certain properties are introduced for this purpose. We start by constructing the solutions of this equation in 1D and solve the problem…
In this paper, we prove the decay and scattering in the energy space for nonlinear Schr\"odinger equations with regular potentials in $\Bbb R^d$ namely, $i{\partial _t}u + \Delta u - V(x)u + \lambda |u|^{p - 1}u = 0$. We will prove decay…
We consider the inverse problem of recovering a potential by measuring the response at a point to a source located at the same point and then varying the point on the surface of a sphere. This is a similar to the inverse back-scattering…
An example of full solution of the inverse scattering problem on the half line is presented. For this purpose, a simple analytically solvable model system (Morse potential) is used, which is expected to be a reasonable approximation to a…
In this paper, we study the direct and inverse spectral problems for the Schrodinger operator with two generalized Regge boundary conditions. For the direct problem, we give the properties of the spectrum, including the asymptotic…
We prove existence of modified wave operators for one-dimensional Schr\"odinger equations with potential in $L^p(\reals)$, $p<2$. If in addition the potential is conditionally integrable, then the usual M\"oller wave operators exist. We…
We consider the scattering problem governed by the Helmholtz equation with inhomogeneity in both `conductivity' in the divergence form and `potential' in the lower order term. The support of the inhomogeneity is assumed to contain a convex…
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…
This paper investigates the inverse scattering problem for the magnetic Schr\"odinger equation. We first establish the well-posedness of the direct problem through a variational approach under physically meaningful assumptions on the…
We consider the cubic nonlinear Schr\"odinger equation with an exceptional potential. We obtain a sharp time decay for the global in time solution and we get the large time asymptotic profile of small solutions. We prove the existence of…
We study an inverse scattering problem for a pair of Hamiltonians $(H(h), H\_0 (h))$ on $L^2 (\r^n)$, where $H\_0 (h) = -h^2 \Delta$ and $H (h)= H\_0 (h) +V$, $V$ is a short-range potential with a regular behaviour at infinity and $h$ is…
For the direct problem, we give the asymptotic distribution of the (real and non-real) transmission eigenvalues for the Schrodinger operator on the half line. For the inverse problem, we prove that the potential can be uniquely determined…