Related papers: Renormalized two-body low-energy scattering
We develop a complete stationary scattering theory for Schr\"odinger operators on $\mathbb R^d$, $d\ge 2$, with $C^2$ long-range potentials. This extends former results in the literature, in particular [Is1, Is2, II, GY], which all require…
Exact solutions of the Schr\"odinger equation for the Coulomb potential are used in the scope of both stationary and time-dependent scattering theories in order to find the parameters which define regularization of the Rutherford…
We apply renormalisation-group methods to two-body scattering by a combination of known long-range and unknown short-range potentials. We impose a cut-off in the basis of distorted waves of the long-range potential and identify possible…
We study and develop the stationary scattering theory for a class of one-body Stark Hamiltonians with short-range potentials, including the Coulomb potential, continuing our study in [AIIS1,AIIS2]. The classical scattering orbits are…
We study the stationary scattering for $(-\Delta)^{\frac 12} + V(x)$ on $\mathbb{R}^3$. For poly-homogeneous potentials decaying at infinity, we prove that the asymptotics of the potential can be recovered from the scattering matrix at a…
A distorted-wave version of the renormalisation group is applied to scattering by an inverse-square potential and to three-body systems. In attractive three-body systems, the short-distance wave function satisfies a Schroedinger equation…
In this paper, we study the well-posedness theory and the scattering asymptotics for the energy-critical, Schr\"odinger equation with indefinite potential \begin{equation*} \left\{\begin{array}{l} i \partial_t u+\Delta u-V(x)u…
We study the stationary scattering theory for the matrix Schr\"odinger equation on the half line, with the most general boundary condition at the origin, and with integrable selfadjoint matrix potentials. We prove the limiting absorption…
Spectral and scattering theory at low energy for the relativistic Schroedinger operator are investigated. Some striking properties at thresholds of this operator are exhibited, as for example the absence of 0-energy resonance. Low energy…
We study the scattering theory for the Schr\"odinger and wave equations with rough potentials in a scale of homogeneous Sobolev spaces. The first half of the paper concerns with an inverse-square potential in both of subcritical and…
For a general class of $N$-body Schr\"odinger operators with short-range pair-potentials the wave and scattering matrices as well as the restricted wave operators are all defined at any non-threshold energy. This holds without imposing any…
We apply the renormalisation-group to two-body scattering by a combination of known long-range and unknown short-range forces. A crucial feature is that the low-energy effective theory is regulated by applying a cut-off in the basis of…
We study the scattering properties of Schr\"{o}dinger operators with potentials that have short-range decay along a collection of rays in $\bbR^d$. This generalizes the classical setting of short-range scattering in which the potential is…
Based on our previous study [IS2] we develop fully the stationary scattering theory for the Schrodinger operator on a manifold possessing an escape function. A particular class of examples are manifolds with Euclidean and/or hyperbolic…
Basing on analogy between the three-body scattering problem and the diffraction problem of the plane wave (for the case of the short range pair potentials) by the system of six half transparent screens, we presented a new approach to the…
We consider one-dimensional random Schr\"odinger operators with a background potential, arising in the inverse problem of scattering. We study the influence of the background potential on the essential spectrum of the random Schr\"odinger…
The transfer matrix ${\mathbf{M}}$ of a short-range potential may be expressed in terms of the time-evolution operator for an effective two-level quantum system with a time-dependent non-Hermitian Hamiltonian. This leads to a dynamical…
We study the long-time behavior of solutions to nonlinear Schroedinger equations with some critical rough potential of inverse square type. The new ingredients are the interaction Morawetz-type inequalities and Sobolev norm property…
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…
The purpose of this paper is to illustrate the I-method by studying low-regularity solutions of the nonlinear Schr\'[o]dinger equation in two space dimensions. By applying this method, together with the interaction Morawetz estimate, (see…