On families of isospectral Sturm - Liouville boundary value problems
Аннотация
Nowadays, there are various methods for solving inverse spectral problems: the method of transformation operator, that is, Gelfand-Levitan method, the method of spectral mappings, the method of etalon models and others. V.A. Marchenko showed, that the Sturm-Liouville operator on a finite segment is determined uniquely by its eigenvalues and a sequence of normalizing constants, that is, by its spectral function. I.M. Gelfand and B.M. Levitan found necessary and sufficient conditions on recovering boundary value Sturm-Liouville problems by their spectral functions. This method is based on recovering a potential and boundary conditions by spectral data by means of a Fredholm integral equation of a second kind with parameters. While constructing isospectral boundary value Sturm-Liouville problems with a prescribed spectrum 2 , 0, we have employed the Gelfand-Levitan method. The main result of the work is an algorithm for recovering a family of boundary value Sturm-Liouville problems = ((), , ), whose spectra satisfy the condition () = { 2 , 0}. At that, the found coefficients = (, 1 , 2 , . . .), = ( 1 , 2 , . . .), = ( 1 , 2 , . . .) depend on infinitely many parameters , = 1, .