A New Look at the Multidimensional Inverse Scattering Problem
Volker Enss
备注：
LaTeX2e, 16 pages, to be published in: Understanding Physics, A.K. Richter ed., Copernicus Gesellschaft, Katlenburg-Lindau, 1998, pp. 31-; ISBN 3-9804862-2-2 (Proceedings Bonn 1996). For other formats see http://www.iram.rwth-aachen.de/~enss/ or ftp://work1.iram.rwth-aachen.de/pub/papers/enss/
作为发展方程的原型，我们考虑了Hilbert空间值函数φ（.）的薛定谔方程i（d/dt）φ（t）= H φ（t），H = H_0 + V（x）.其描述了在时间t时系统在至少2的空间维度上的状态。动能算符H_0可以与拉普拉斯算符（非相对论量子力学）成比例，H_0 = \sqrt{-Δ+ m^2}（相对论运动学，克莱因-戈登方程），狄拉克算符，或.，而当|x|趋于无穷大时，电势V（x）适当地趋于0。我们提出了一个几何方法的逆散射问题。对于给定的散射算子S，我们显示的潜力的唯一性，我们给出明确的限制的高能量行为的散射算子，我们给重建公式的潜力。我们的数学证明密切遵循物理直觉。一个关键的观察是，在高能量的波包的翻译占主导地位的传播期间的相互作用时间。该方法的扩展涵盖了例如薛定谔算子与磁场，多粒子系统和波动方程。
英文摘要：
As a prototype of an evolution equation we consider the Schrödinger equation i (d/dt) Ψ(t) = H Ψ(t), H = H_0 + V(x) for the Hilbert space valued function Ψ(.) which describes the state of the system at time t in space dimension at least 2. The kinetic energy operator H_0 may be propotional to the Laplacian (nonrelativistic quantum mechanics),
H_0 = \sqrt{-Δ+ m^2} (relativistic kinematics, Klein-Gordon equation), the Dirac operator, or ..., while the potential V(x) tends to 0 suitably as |x| to infinity.
We present a geometrical approach to the inverse scattering problem. For given scattering operator S we show uniqueness of the potential, we give explicit limits of the high-energy behavior of the scattering operator, and we give reconstruction formulas for the potential.
Our mathematical proofs closely follow physical intuition. A key observation is that at high energies translation of wave packets dominates over spreading during the interaction time. Extensions of the method cover e.g. Schrödinger operators with magnetic fields, multiparticle systems, and wave equations.
The paper is devoted to the development of the theory of inverse problems for evolution equations with terms rapidly oscillating in time. A new approach to setting such problems is developed for the case in which additional constraints are imposed only on several first terms of the asymptotics of the solution rather that on the whole solution. This approach is realized in the case of a multidimensional hyperbolic equation with unknown absolute term.
We consider the inverse scattering on the quantum graph associated with the hexagonal lattice. Assuming that the potentials on the edges are compactly supported, we show that the S-matrix for all energies in any open set in the continuous spectrum determines the potentials.
Enss (1983) proved a propagation estimate for the usual free Schroedinger operator that turned out later to be very useful for inverse scattering in the work of Enss--Weder (1995). Since then, this method has been called the Enss--Weder time-dependent method. We study the same type of propagation estimate for the fractional powers of the negative Laplacian and, as with the Enss--Weder method, we apply our estimate to inverse scattering. We find that the high-velocity limit of the scattering operator uniquely determines the short-range interactions.
In the context of the Cauchy problem for the Korteweg-de Vries equation we extend the inverse scattering transform to initial data that behave at plus infinity like a sum of Wigner-von Neumann type potentials with small coupling constants. Our arguments are based on the theory of Hankel operators.
We solve inverse scattering problem for Schrödinger operators with compactly supported potentials on the half line. We discretize S-matrix: we take the value of the S-matrix on some infinite sequence of positive real numbers. Using this sequence obtained from S-matrix we recover uniquely the potential by a new explicit formula, without the Gelfand-Levitan-Marchenko equation.
We consider an inverse $N$-body scattering problem of determining two potentials---an external potential acting on all particles and a pair interaction potential---from the scattering particles. This paper finds that the time-dependent Hartree-Fock approximation for a three-dimensional inverse $N$-body scattering in quantum mechanics enables us to recover the two potentials from the scattering states with high-velocity initial states. The main ingredient of mathematical analysis in this paper is based on the asymptotic analysis of the scattering operator defined in terms of a scattering soluti…
We study one of the multidimensional inverse scattering problems for quantum systems governed by the Stark Hamiltonians. By applying the time-dependent method developed by Enss and Weder in 1995, we prove that the high-velocity limit of the scattering operator determines uniquely the short-range interaction potentials. Moreover, we prove that, when a long-range interaction potential is given, the high-velocity limit of the Dollard-type modified scattering operator determines uniquely the short-range part of the interactions. We allow the potential functions to belong to very broad classes. The…
In a recent paper, we developed an inverse scattering approach to the Boussinesq equation in the case when no solitons are present. In this paper, we extend this approach to include solutions with solitons.
