An inverse problem for the Sturm-Liouville operator with nonseparable self-adjoint boundary conditions

Spectral properties of the Sturm-Liouville operator with a spectral parameter quadratically included in the boundary condition

“…The problem of reconstruction of a class of similar boundary value problems by spectral data was completely studied in [33] by another method. The classes of non-similar (self-adjoint) boundary value problems were considered in the papers of M. G. Gasymov, I. M. Guseinov, and I. M. Nabiev [3], I.M. Guseinov and I. M. Nabiev [10], [11].…”

Solvability theorems for an inverse nonself-adjoint Sturm-Liouville problem with nonseparated boundary conditions

“…Moreover, these spectral data were used essentially [6]. Later, there were attempts to choose the problem to be reconstructed or auxiliary problems so as to use less spectral data for the reconstruction [7][8][9][10][11]. In particular, in [8,9] a nonself-adjoint problem was replaced by a self-adjoint one, and it was shown that, for its unique reconstruction, as spectral data it suffices to use three spectra, some sequence of signs, and some real number.…”

“…In particular, in [8,9] a nonself-adjoint problem was replaced by a self-adjoint one, and it was shown that, for its unique reconstruction, as spectral data it suffices to use three spectra, some sequence of signs, and some real number. In [10], an auxiliary problem was chosen so as to reduce the number of spectral data required for the reconstruction of a self-adjoint problem by one spectrum; i.e., only two spectra, some sequence of signs, and some real number were used as spectral data. In the present paper, we consider a nonself-adjoint Sturm-Liouville problem with nonseparated boundary conditions.…”

“…In the present paper, we consider a nonself-adjoint Sturm-Liouville problem with nonseparated boundary conditions. We show that, for its unique reconstruction, one can use even less spectral data as compared with the reconstruction of a self-adjoint problem in [8][9][10]; more precisely, we need two spectra and, in addition, three eigenvalues. Moreover, we show that the result obtained in the present paper generalizes the Levitan-Gasymov criterion [12].…”

Inverse Problem for Quasi-Periodic Differential Pencils with Jump Conditions Inside the Interval

“…m n > 1, ω n = 0 ). Then it follows from (7) that (Q(ρ)Q(ρ)) (ν−1) |ρ=νn = (Ḋ(ρ)D(ρ)) (ν−1) |ρ=νn , ν = 1, m n − 1.…”