The $ \bar\partial$-equation in the multidimensional inverse scattering problem

Abstract

CONTENTS Chapter I. A survey of the results § 1.1. The method of Faddeev in the inverse scattering problem for the Schrödinger equation § 1.2. The results of Newton, Ablowitz, and Nachman § 1.3. Necessary and sufficient properties of the scattering data. Generalized dispersion relations § 1.4. Methods of solution of the inverse problem with non-overdetermined data. The results of Moses and Prosser and their generalizations § 1.5. The inverse problem on a fixed energy level for the two-dimensional Schrödinger operator and non-linear equations. The methods of S. P. Novikov and Manakov and their further development § 1.6. The inverse problem on a fixed energy level for the three-dimensional Schrödinger operator. The results of Beals and Coifman. New results Chapter II. Necessary properties of the scattering data § 2.1. The Green-Faddeev function and its properties. Analysis of the integral equation § 2.2. Generalized scattering data . The non-linear -equation . Corollaries § 2.3. Properties of zeros of the Fredholm determinant for the equation § 2.4. Solution of the inverse problem on the basis of generalized dispersion relations Chapter III. Characterization of scattering data. Preliminary results § 3.1. Liouville's theorem for solutions of the non-linear -equation § 3.2. Formulae for solutions of the -equation in a concave domain of § 3.3. Estimates for the form § 3.4. Estimates for solutions of the -equation § 3.5. Proof of Liouville's theorem for the non-linear -equation Chapter IV. Characterization of scattering data. Final results § 4.1. Theorems concerning the characterization of scattering data in "physical" and "non-physical" domains § 4.2. A separate analyticity theorem for the non-linear -equation § 4.3. Continuability of solutions of the non-linear -equation from the "physical" to the "non-physical" domain § 4.4. Reconstruction of the Schrödinger operator from minimal scattering data § 4.5. Solution of the inverse scattering problem on a fixed energy level for the multidimensional Schrödinger operator Chapter V. Further results and problems § 5.1. The inverse scattering problem for the Schrödinger equation in a magnetic field § 5.2. Inverse scattering problems for the acoustic equation References