Related papers: Modified scattering operator for nonlinear Schr\"o…
This work deals with Schr\"odinger equations with quadratic and sub-quadratic Hamiltonians perturbed by a potential. In particular we shall focus on bounded, but not necessarily smooth perturbations. We shall give a representation of such…
We consider non linear elliptic equations of the form $\Delta u = f(u,\nabla u)$ for suitable analytic nonlinearity $f$, in the vinicity of infinity in $\mathbb{R}^d$, that is on the complement of a compact set.We show that there is a…
We consider a long-range scattering theory for discrete Schr\"odinger operators on the hexagonal lattice, which describe tight-binding Hamiltonians on the graphene sheet. We construct Isozaki-Kitada modifiers for a pair of the difference…
We study an inverse scattering problem for the discrete Schr\"{o}dinger operator on the multi-dimensional square lattice, with compactly supported potential. We show that the potential is uniquely reconstructed from a scattering matrix for…
We consider the nonlinear Schr\"odinger equation in three space dimensions with a focusing cubic nonlinearity and defocusing quintic nonlinearity and in the presence of an external inverse-square potential. We establish scattering in the…
This paper establishes the $L^p$ boundedness of wave operators for linear Schr\"odinger equations in $\mathbb{R}^3$ with time-dependent potentials. The approach to the proof is based on new cancellation lemmas. As a typical application…
We prove a global well-posedness result for defocusing nonlinear Schrodinger equations with time dependent potential. We then focus on time dependent harmonic potentials. This aspect is motivated by Physics (Bose--Einstein condensation),…
Starting with the Dirac equation outside the event horizon of a non-extreme Kerr black hole, we develop a time-dependent scattering theory for massive Dirac particles. The explicit computation of the modified wave operators at infinity is…
We present a new approach to solve a Schr\"odinger Equation autonomous at infinity, by identifying the relation between the arrangement of the spectrum of the concerned operator and the behavior of the nonlinearity at zero and at infinity.…
We consider scattering for the discrete Schr\"{o}dinger operator on the square lattice $\mathbb{Z}^d$, $d \ge 1$, with compactly supported potential. We give formulas for finding the phased scattering amplitude from phaseless near-field…
A propagation method for the time dependent Schr\"odinger equation was studied leading to a general scheme of solving ode type equations. Standard space discretization of time-dependent pde's usually results in system of ode's of the form…
We undertake a comprehensive study of the nonlinear Schr\"odinger equation $$ i u_t +\Delta u = \lambda_1|u|^{p_1} u+ \lambda_2 |u|^{p_2} u, $$ where $u(t,x)$ is a complex-valued function in spacetime $\R_t\times\R^n_x$, $\lambda_1$ and…
We consider a class of stationary Schr\"{o}dinger-Poisson systems with a general nonlinearity $f(u)$ and coercive sign-changing potential $V$ so that the Schr\"{o}dinger operator $-\Delta+V$ is indefinite. Previous results in this framework…
We show that for a one-dimensional Schr\"odinger operator with a potential whose (j+1)'th moment is integrable the j'th derivative of the scattering matrix is in the Wiener algebra of functions with integrable Fourier transforms. We use…
A one-dimensional generalized nonlinear Schroedinger equation is considered, and the corresponding inverse scattering problem is analyzed when the potential is compactly supported and depends on the wave function. The unique recovery of the…
The time evolution of the scattering and spectral data is obtained for the differential operator $\displaystyle\frac{d^4}{dx^4} +\displaystyle\frac{d}{dx} u(x,t)\displaystyle\frac{d}{dx}+v(x,t),$ where $u(x,t)$ and $v(x,t)$ are real-valued…
We give an explicit formula for the wave operators for Schroedinger operators on the half-line with a potential decaying strictly faster than the polynomial of degree minus two. The formula consists of the main term given by the scattering…
An ordinary differential operator of the fourth order with coefficients converging at infinity sufficiently rapidly to constant limits is considered. Scattering theory for this operator is developed in terms of special solutions of the…
Spectral components of one-dimensional Schr\"odinger operator with complex potential are investigated. An effective upper bound for the total number of eigenvalues and spectral singularities is established. For dissipative Schr\"odinger…
In this paper, we study large time behavior of complex-valued solutions to nonlinear Klein-Gordon equation with a gauge invariant quadratic nonlinearity in two spatial dimensions. To find a possible asymptotic behavior, we consider the…