Reconstruction of the Differential Operator with Spectral Parameter in the Boundary Condition

Solvability of the inverse spectral problem for Sturm-Liouville operator with spectral parameter in the boundary condition

“…Taking this equality into account and arguing as in the proof of Theorem 2 in [39], we can verify that the function ∆ (λ) has two zeros between λ −1 and λ 1 with different signs, and between λ k−1 and λ k with k ≤ −1, as well as between λ k and λ k+1 with k ≥ 1 exactly one zero. Therefore, (8) holds.…”

“…Using the second and third conditions of the theorem as in [19] (see also [24] and [39]), we can show that ν 2 m < λ 2 m < ν 2 m+1 (m = 1, 2, ...). Thus, the zeros of the functions √ λs √ λ and s 1 √ λ are interleaved and satisfy the asymptotic formulas ( 29) and (33), whence it follows (see Theorem 3.4.1 in [35]) that there exists a unique real function q (x) ∈ L 2 [0, π] such that the sequences of zeros under consideration are the spectra of boundary value problems L 0 and L 1 with the function q (x) and equalities s (λ) = s (π, λ) , s 1 (λ) = s ′ (π, λ) are true.…”

“…It was also proved in [39] that these eigenvalues are real, nonzero, and simple under the following condition (T ): for all functions y (x) ∈ W 2 2 [0, π] , y (x) ̸ ≡ 0 satisfying conditions (2), the inequality…”

“…The spectrum of one boundary value problem, some sequence of signs, and some number are used as spectral data. Note that in [39] a similar problem was completely solved in the case of other boundary conditions.…”

Inverse Nodal Problems for Pencils of Singular Sturm-Liouville Operators

“…In studies [11], [13], [18], the spectral properties of the operator produced by the regular differential equation given with non-separated boundary conditions containing the spectral parameter were examined and the uniqueness theorems related to the solution of the spectral inverse problem were proved. In studies [5]- [8], the spectral properties of the operator produced by the Schrödinger equation with the singular coefficient given with the boundary conditions depending on the spectral parameter were examined and the solution of the inverse spectral problems according to different spectral data was given.…”