Direct and inverse scattering problems for the Schrödinger operator in dimension one are investigated. Relations between the scattering problems defined by regular and zero range potentials are investigated.

Correct mathematical definition of the «^'-interaction is introduced and investigated in detail. Relations with the complex scaling procedure are analyzed. The inverse scattering problem for long range oscillating potentials is investigated. Nonuniqueness of the solution of this problem is observed. It leads to a new soliton like solution of the KdV equation. Potentials, corresponding to the same scattering matrix as certain zero range potentials, are calculated. A model scattering theory for three one dimensional particles is constructed using the theory of selfadjoint extensions for symmetric operators. The scattering matrix is calculated in terms of elementary functions. Decay operators are introduced for the Schrödinger evolution in the framework of the Lax-Phillips scattering theory.