arXiv:2010.03181v1 [math-ph] 7 Oct 2020 EQUIVALENT CLASSES OF STURM-LIOUVILLE PROBLEMS EVGENY L. KOROTYAEV Abstract. We consider Sturm-Liouville problems on the finite interval. We show that spec- tral data for the case of Dirichlet boundary conditions are equivalent to spectral data for Neumann boundary conditions. In particular, the solution of the inverse problem for the first one is equivalent to the solution of the inverse problem for the second one. Moreover, we discuss similar results for other Sturm-Liouville problems, including a periodic case. 1. Introduction and main results 1.1. Introduction. In this paper we discuss inverse spectral theory for Sturm-Liouville prob- lems on the unit interval, under the different boundary conditions. We shortly describe well- known results about it. Consider the Sturm-Liouville problem on the interval [0, 1] with the Dirichlet boundary conditions: −f ′′ + qf = λf, f (0) = f (1) = 0, where the real potential q ∈ L 2 (0, 1). Let µ n and y n ,n  1 be the corresponding eigenvalues and real eigenfunctions, such that y ′ n (0) = 1. All these eigenvalues are simple and satisfy µ 1 <µ 2 < ... and µ n =(πn) 2 + o(1) as n →∞. We recall only some important steps mainly on the characterization problem, i.e., the complete description of spectral data that correspond to some fixed class of potentials. Borg obtained the first uniqueness result for Sturm-Liouville problems. Later on, Marchenko [27] proved that a spectral function (constructed from the eigenvalues µ n plus so-called normalizing constants c n =  1 0 y 2 n dx) determine the potential uniquely (see also Krein [24], [25] about it). Gel’fand and Levitan [7] created a basic method to reconstruct the potential q from its spectral function: they determined an integral equation and expressed q (x) in terms of the solution of this equation. Unfortunately, there was a gap between necessary and sufficient conditions for the spectral functions corresponding to fixed classes of q (x). This gap was closed by Marchenko and Ostrovski [28]. They gave the complete solution of the inverse problem in terms of two spectra, for large class of potentials. Trubowitz and co-authors [5], [11], [12] suggested an analytic approach, see also the nice book [29] and references therein. It is based on analytic properties of the mapping q → (µ n ,h s,n ) ∞ 1 ∈{spectral data} and the explicit solutions when only a finite number of spectral parameters from (µ n ,h s,n ) ∞ n=1 changed. Here h s,n = log |y ′ n (1)| are the norming constants, which differ from the normalizing constants c n , but the characterizations are equivalent (see Appendix in [2]). Also, this approach was applied to other inverse problems with purely discrete spectrum: (for an impedance [3], [4], singular Sturm-Liouville operators on a finite interval [8]; periodic potentials [6], [14], perturbed harmonic oscillators [1] and vector-valued operators [2]. Moreover, it was used to construct action-angel variables for periodic KdV [13]. Date : October 8, 2020. 2020 Mathematics Subject Classification. 34A55 (34B24 47E05 47N50 81Q10). Key words and phrases. inverse problem, eigenvalues, Sturm-Liouville problem. 1