Scattering transformation at fixed non-zero energy for the two-dimensional Schrödinger operator with potential decaying at infinity
Abstract
We study the problem of reconstructing the potential of the two-dimensional Schrödinger operator from scattering data measured at fixed energy. This problem, in contrast to the general multidimensional inverse problem, possesses an infinite-dimensional symmetry algebra generated by the Novikov-Veselov hierarchy and hence is "exactly soluble" in some sense; the complexity of the answer is approximately the same as in the one-dimensional problem. We make heavy use of methods developed in modern soliton theory. Since the quantum fixed-energy scattering problem is mathematically equivalent to the acoustic single-frequency scattering problem, we see that the results of the present paper apply in both cases.
- Publication:
-
Russian Mathematical Surveys
- Pub Date:
- December 2000
- DOI:
- 10.1070/RM2000v055n06ABEH000333
- Bibcode:
- 2000RuMaS..55.1015G