Double Discontinuous Inverse Problems for Sturm-Liouville Operator with Parameter-Dependent Conditions

and Applied Analysis 3 Proof. Consider a Hilbert SpaceH = L 2 (0, 1) ⊕C, equipped with the inner product


Introduction
Spectral problems of differential operators are studied in two main branches, namely, direct spectral problems and inverse spectral problems.Direct problems of spectral analysis consist in investigating the spectral properties of an operator.On the other hand, inverse problems aim at recovering operators from their spectral characteristics.Such problems often appear in mathematics, physics, mechanics, electronics, geophysics, and other branches of natural sciences.
Eigenvalue-dependent boundary conditions were studied extensively.The references [10,11] are well-known examples for problems with boundary conditions that depend linearly on the eigenvalue parameter.In [10,12], an operator-theoretic formulation of the problems with the spectral parameter contained in only one of the boundary conditions has been given.Inverse problems according to various spectral data for eigenparameter linearly dependent Sturm-Liouville operator were investigated in [13][14][15][16][17]. Boundary conditions that depend nonlinearly on the spectral parameter were also considered in [18][19][20][21][22][23].
It is proven that the coefficients of the problem can be uniquely determined by either Weyl function or given two different spectral sequences.The obtained results are generalizations of the similar results for the classical Sturm-Liouville operator on a finite interval.
The following asymptotics can be obtained from the integral equations given in the appendix: where  = Im √ .
The values of the parameter  for which the problem  has nonzero solutions are called eigenvalues, and the corresponding nontrivial solutions are called eigenfunctions.
The characteristic function Δ() and norming constants   of the problem  are defined as follows: It is obvious that Δ() is an entire function in  and the zeros, namely, {  } of Δ() coincide with the eigenvalues of the problem .Now, from ( 6) and ( 8), we can write Lemma 1. See the following.
(i) All eigenvalues of the problem  are real and algebraically simple; that is, Δ  (  ) ̸ = 0.

Main Results
We consider three statements of the inverse problem for the boundary value problem ; from the Weyl function, from the spectral data {  ,   } ≥0 , and from two spectra {  ,   } ≥0 .
For studying the inverse problem, we consider a boundary value problem L, together with , of the same form but with different coefficients q(), h, H, s ,  = 1, 2.
Hence, the proof is completed by Theorem 2.