Preface to the revised edition

In the first edition of this book the main attention was focused on the methods of solving the inverse problem of spectral analysis and on the conditions (necessary and sufficient) which the spectral data must satisfy in order to make it possible to reconstruct the potential of the corresponding Sturm-Liouville operator. These conditions imply that the spectral data (e.g. spectral function or scattering data) must be known for all values of spectral parameter which belong to the spectrum of the operator. But from the physical meaning of the inverse problem it is obvious that the values of spectral data on the whole spectrum are impossible to obtain from any observations. For example, in the inverse problem of quantum scattering theory the energy of the particles acts as the spectral parameter, and in order to find the values of scattering data on the whole spectrum one has to conduct an experiment with the particles of infinitely large energy. But for big enough values of energy the scattering process is not any more described by Schrödinger equation with potential q(x). Therefore, even allowing, ideally, the possibility to experiment with particles of arbitrarily large energies, we would obtain, starting from a certain energy, data relevant to process, which has certainly nothing to do with the equation that we want to reconstruct. Hence, a principal question is as follows: what information about the potential q(x) can be obtained, if the spectral function or scattering data are known (generally speaking, approximately) only on a finite interval of values of the spectral parameter? The new Chapter 5, devoted to solving this problem, was added to this edition. The convenient formulae are obtained, which allow to estimate the precision with which the eigenfunctions and potentials of Schrödinger operator can be restored when the scattering data or spectral function are known only on a finite interval of values of spectral parameter.