02-127 Ricardo Weder

The Time-Dependent Approach to Inverse scattering (66K, Latex) Mar 16, 02

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Abstract. In these lectures I give an introduction to the time-dependent approach to inverse scattering, that has been developed during the last few years. The aim of this approach is to solve various inverse scattering problems with time-dependent methods that closely follow the physical (and geometrical) intuition of the scattering phenomena. This method has been applied to many linear and nonlinear scattering problems. We first discuss the case of quantum mechanical potential scattering. We give explicit limits for the high-energy behaviour of the scattering operator that offer us formulae for the unique reconstruction of the potential. Then, we consider the case of the Aharonov-Bohm effect ( Schr\"{o}dinger operators with singular magnetic potentials and exterior domains). This is a particularly interesting inverse scattering problem that shows that in quantum mechanics a magnetic field acts on a charged particle -by means of the magnetic potential- even in regions where it is identically zero. The key issue for these two problems is that at high-energies translation of the wave packet dominates over spreading during the interaction time. In fact, in this limit it is sufficient for the calculation of the scattering operator to consider translation of wave packets rather than their correct free evolution. Finally, we study the nonlinear Schr\"{o}dinger equation with a potential. In this case, from the scattering operator we uniquely reconstruct the potential and the nonlinearity. For this purpose, we observe that in the small amplitude limit the nonlinear effects become negligible and scattering is dominated by the linear term. Using this idea we prove that the derivative at zero of the nonlinear scattering operator is the linear one. With the aid of this fact we first uniquely reconstruct the potential from the associated linear inverse scattering problem and in a second step we uniquely reconstruct the nonlinearity.

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