Inverse Spectral problem for the Sturm-Liouville Operator with Eigenvalue Parameter Dependent Boundary Conditions

Necessary and sufficient conditions for the solvability of the inverse problem for non-self-adjoint pencils of Sturm-Liouville operators on the half-line

“…In (18) we replace the second derivatives using equation (1), and then we replace ϕ(x, λ) using (12). This yields…”

“…We prove now the sufficiency. Let a function M (λ) ∈ V N , satisfying the hypothesis of Theorem 4, be given, and let ϕ(x, λ) be the solution of the main equation (12). Then (12) gives us the analytic continuation of ϕ (x, λ) to the whole λ-plane, and for each fixed x ≥ 0, the function ϕ(x, λ) is entire in λ of order 1/2.…”

“…Let a function M (λ) ∈ V N , satisfying the hypothesis of Theorem 4, be given, and let ϕ(x, λ) be the solution of the main equation (12). Then (12) gives us the analytic continuation of ϕ (x, λ) to the whole λ-plane, and for each fixed x ≥ 0, the function ϕ(x, λ) is entire in λ of order 1/2. Using Lemma 1.5.1 from [8], by the standard technique, one can show that the functions ϕ (ν) (x, λ), ν = 0, 1, 2, are absolutely continuous with respect to x on compact sets, and…”

Inverse spectral problem for non-selfadjoint Dirac operator with boundary and jump conditions dependent on the spectral parameter

“…Inverse problems for Sturm-Liouville operators either with the spectral parameter λ in the boundary conditions, or with interior jump point conditions, have been studied fairly completely (see [12,15,18,19] for not containing spectral parameter, [2][3][4][5][6][7]10] for depending on the spectral parameter, and the references therein).…”

Recovering singular Sturm-Liouville differential pencils from spectral data

“…Inverse problems for differential operators with boundary conditions dependent on the spectral parameter are more difficult to investigate, and nowadays there are only a number of papers in this direction (see [11][12][13][14][15][16][17]). In particular, [11][12][13][14][15][16] study such problems on a finite interval. Inverse spectral problems for the non-self-adjoint Sturm-Liouville pencil (1) and (2) on the half-line were considered in [17], where recovering L from the Weyl function was studied.…”