On families of isospectral Sturm–Liouville boundary value problems

Abstract

The work is devoted to describing all boundary value Sturm–Liouville problems on a finite segment with the same spectrum. Such problems are called isospectral and they were studied in works by E.L. Isaacson, H.P. McKean, B.E. Dahlberg, E. Trubowitz, M. Jodeit, B.M. Levitan, Y.A. Ashrafyan, T.N. Harutyunyan. 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 $n^{2}$, $n \ge 0$, 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 $L=L(q(x),h, H)$, whose spectra satisfy the condition $\sigma(L)=\{n^2,n\ge 0\}$. At that, the found coefficients $ q=q(x, \gamma_1, \gamma_2, \ldots)$, $h=h(\gamma_1, \gamma_2, \ldots)$, $H=H(\gamma_1, \gamma_2, \ldots)$ depend on infinitely many parameters $\gamma_j$, $j= \overline{1,\infty}$.