Recovering differential pencils with spectral boundary conditions and spectral jump conditions

In this work, we discuss the inverse problem for second order differential pencils with boundary and jump conditions dependent on the spectral parameter. We establish the following uniqueness theorems: (i)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(i)$\end{document} the potentials qk(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$q_{k}(x)$\end{document} and boundary conditions of such a problem can be uniquely established by some information on eigenfunctions at some internal point b∈(π2,π)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$b\in (\frac{\pi }{2},\pi )$\end{document} and parts of two spectra; (ii)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(ii)$\end{document} if one boundary condition and the potentials qk(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$q_{k}(x)$\end{document} are prescribed on the interval [π/2(1−α),π]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$[\pi /2(1-\alpha ),\pi ]$\end{document} for some α∈(0,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\alpha \in (0, 1)$\end{document}, then parts of spectra S⊆σ(L)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$S\subseteq \sigma (L)$\end{document} are enough to determine the potentials qk(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$q_{k}(x)$\end{document} on the whole interval [0,π]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$[0, \pi ]$\end{document} and another boundary condition.


Introduction
Inverse spectral problems are a branch of science that seeks to determine the coefficients of the boundary value problems from their spectral characteristics. This kind of problems often arise in mathematics, mechanics, physics, electronics, geophysics, and various branches of natural sciences and engineering [2,3,14,17,18,24]. First studies and results of inverse problems for classical Sturm-Liouville operators were given by Ambartsumyan in 1929 [1], and this field of science has been developed by many researchers in the next years [8,21,29,33,37,38]. In particular, some aspects of the inverse problem theory for differential pencils and for spectral jump and boundary conditions have been investigated in [4,6,23,[29][30][31][32][33][34][35].
To recover the potential on all interval and all coefficients in boundary conditions in the usual case, it is necessary to know two spectra of the boundary value problem with different boundary conditions [36]. Indeed, if a finite number of eigenvalues are deleted, the potential is not uniquely determined by one full spectrum and one partial spectrum. Mochizuki and Trooshin [20] showed that the spectral data of parts of two spectra and a set of values of eigenfunctions at some internal point suffice to determine the potential, and they addressed the interior inverse problem of Sturm-Liouville operators on the finite interval [0,1]. Afterwards, this technique has been used by some authors to survey the inverse problem of Sturm-Liouville operators in various forms [10,22,25,29,33]. Alongside this method, in [9], Hochstadt and Lieberman found the half inverse problem method and showed that if the potential is prescribed on [1/2, 1], one spectrum can uniquely determine the potential on the whole interval [0, 1]. Hochstadt-Lieberman type theorem was also investigated by many scholars for differential operators in the next years [4,11,12,30,31]. Then Gesztesy and Simon generalized Hochstadt-Lieberman type theorem and recovered the Sturm-Liouville operator with Robin boundary conditions from parts of one spectrum and partial information on the potential [7]. They showed that if the coefficient h 0 and the potential are provided on [0, 1/2 + α/2] for some α ∈ (0, 1), then parts of one spectrum can give the coefficient h 1 and the potential on all of [0, 1]. Insofar as we know, Mochizuki-Trooshin and Hochstadt-Lieberman type theorems for differential pencils with spectral boundary and jump conditions have not been considered before. The target of this work is to investigate two inverse problems to L by these two techniques. We would like to investigate the inverse problems for L by some information on eigenfunctions at some internal point and parts of two spectra taking Mochizuki-Trooshin type theorem as well as from partial information on the potentials and parts of a finite number of spectra by Gesztesy-Simon type theorem. This brings certain difficulties, and the results obtained are a generalization of the classical Sturm-Liouville problems.
The main goal of this paper is to present the potentials q k (x) and the parameters h k , H k , k = 0, 1 by developing the ideas of the Mochizuki-Trooshin and Gesztesy-Simon methods [7,20]. The present paper is organized as follows. Section 2 is devoted to some preliminaries. In Sect. 3, two uniqueness theorems for boundary value problem (1.1)-(1.4) are proved.

Preliminaries
In the first part of the paper, we provide the solution of the boundary value problem L and its spectral characteristics. At first, we remind the following notation from [5].
The values of the parameter λ = ρ 2 for which L has nonzero solutions are called eigenvalues, and the corresponding nontrivial solutions are called eigenfunctions. The set of eigenvalues is called the spectrum of L.
Let y(x, ρ) be the solution of differential pencil (1.1) under the initial conditions y(0, ρ) = 1, y (0, ρ) = h 1 ρ + h 0 and jump conditions (1.4). From [23,34], this solution can be obtained in the following form for sufficiently large ρ: and analogously we have formulae (2.1) and (2.2) as follows for sufficiently large ρ: Moreover, we know that these functions and derivatives with respect to x are entire in ρ of exponential type. We denote by (ρ) := V (y(x, ρ)) the characteristic function for L. This relation together with (1.3) and (2.4) gives for sufficiently large ρ: The roots of this characteristic function are the eigenvalues of L [5], and these eigenvalues have the following asymptotic formula for sufficiently large n: where ω = 1 π Q(π) + 1 2π i ln i-H 1 i+H 1 + 1 2π i ln i-A 1 i+A 1 . Also, using the known method [5], one gets for large enough ρ ∈ G δ := {ρ ∈ C; | ρρ n |≥ δ, ∀n, } and some positive constant C δ . In virtue of Ref. [7,15], we bring the following lemma which is momentous to demonstrating the main results of our article.

Inverse problem with partial information
In this section, two uniqueness theorems are brought that are the main results of this work. We would like to recover the differential pencil and the coefficients used in boundary conditions from some information on eigenfunctions at some interior point and parts of two spectra in Theorem 3.1. Taking partial information on the potentials and a subset of eigenvalues, we also study this inverse problem in Theorem 3.3. So, we consider a boundary value problem L := L( q 1 , q 0 , h 1 , h 0 , H 1 , H 0 , α, β, γ ) beside L : = L(q 1 , q 0 , h 1 , h 0 , H 1 , H 0 , α, β, γ ). We note that the parameters α, β, and γ are known a priori. Also, if a symbol shows an object in L, then the same symbol with tilde shows the corresponding object in L.
Let l(n) and r(n) be two sequences of the natural numbers such that and let μ n be the eigenvalues of L 1 := L(q 0 , q 1 , h 0 , h 1 , H 0 , H 1 , α, β, γ ), H k = H k , H k ∈ R for k = 0, 1.
It is needed to express the following lemma to prove Theorem 3.1.