New problems of the spectral theory of one-dimensional Schrodinger operators are formulated and solved: It is proved that the Banach algerbras generated by the generalized shift are isomorphic to the algebras with ordinary convolution. An asymptotic formula for spectral functions is obtained. It is proved that the spectral functions uniquely define the operators. The stability of the inverse problems of the spectral analysis is investigated. The inverse scattering problem is solved. The inverse problem for the operators with periodic potentials is solved (with I. V. Ostrovsky). Characteristic properties of the Weyl solutions are found. An asymptotic theory of the boundary problems with fine-grained boundary is constructed (with E. Ya. Khruslov). The theory gives the possibility to find the limits to which the solutions are converged under the unbounded fragmentation of the boundary and to evaluate the rate of convergence. The limits of the integral density of distribution of the eigenvalues of the random matrix ensembles in the limit of infinite matrix order, are obtained (with L. A. Pastur). A new method of solving the non-linear evolution equations, which allows one to solve the non-linear Cauchy problems with initial data which are not stabilizing at infinity, is developed.
Main publications:
Z. S. Agranovich, V. A. Marchenko. The Inverse Problem of Scattering Theory. Gordon and Breach, 1963.
