The geometrical approach to multidimensional inverse scattering

Abstract

We prove that in multidimensional short-range potential scattering the high velocity limit of the scattering operator of an N-body system determines uniquely the potential. For a given long-range potential the short-range potential of the N-body system is uniquely determined by the high velocity limit of the modified Dollard scattering operator. Moreover, we prove that any one of the Dollard scattering operators determines uniquely the total potential. We obtain as well a reconstruction formula with an error term. Our simple proof uses a geometrical time-dependent method.

Recovering the mass and the charge of a Reissner–Nordström black hole by an inverse scattering experiment

“…The strategy we adopt to prove this Theorem is based on a high-energy asymptotic expansion of S out , a well-known technique initially developed in the case of multidimensional Schrödinger operators by Enss, Weder [11]. This method can be used to study Hamiltonians with electric and magnetic potentials [1], the Dirac equation [21] and Stark Hamiltonians [31,27].…”

“…The strategy we adopt to prove our main result is based on a high-energy asymptotic expansion of the scattering operator S. Such a technique was introduced by Enss and Weder in [11] and used sucessfully to recover the potential of multidimensional Schrödinger operators (note that the case of multidimensional Dirac operators in flat spacetime was treated later by Jung in [21]). They showed that the first term of the high-energy asymptotics is exactly the Radon transform of the potential they are looking for.…”

An inverse scattering problem for short-range systems in a time-periodic electric field

“…We prove Theorem 3 by studying the high energy limit of [S(s), p], (Enss-Weder's approach [4]). We need n ≥ 3 in the case E 0 = 0 in order to use the inversion of the Radon transform [6] on the orthogonal hyperplane to E 0 .…”

“…This method can be used to study Hamiltonians with electric and magnetic potentials on L 2 (R n ) [1], the Dirac equation [9], the N-body case [4], the Stark effect ( [15], [17]), the AharonovBohm effect [18].…”

Low-energy inverse problems in three-body scattering

“…In addition, the leading term in the asymptotics depends linearly on them. This problem was studied, under various assumptions, by Enss and Weder [3,4], Novikov [13] and Wang [20].…”