International Scholarly Research Notices

International Scholarly Research Notices / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 754718 | https://doi.org/10.5402/2011/754718

I. Dehghani Tazehkand, A. Jodayree Akbarfam, "On Inverse Sturm-Liouville Problems with Spectral Parameter Linearly Contained in the Boundary Conditions", International Scholarly Research Notices, vol. 2011, Article ID 754718, 23 pages, 2011. https://doi.org/10.5402/2011/754718

On Inverse Sturm-Liouville Problems with Spectral Parameter Linearly Contained in the Boundary Conditions

Academic Editor: O. Miyagaki
Received08 Oct 2011
Accepted30 Oct 2011
Published25 Dec 2011

Abstract

In this paper, we study Sturm-Liouville problems with spectral parameter linearly contained in one of the boundary conditions. We prove uniqueness theorems for the solution of the inverse problems according to the Weyl function, spectral data, and two spectra. Then, we recover the potential function and coefficients of boundary conditions from the spectral data by the method of spectral mappings.

1. Introduction

We consider the Sturm-Liouville boundary value problem 𝐿=𝐿(π‘ž(π‘₯),β„Ž,𝐻,𝐻1,𝐻2):β„“π‘¦βˆΆ=βˆ’π‘¦ξ…žξ…ž+π‘ž(π‘₯)𝑦=πœ†π‘¦,0<π‘₯<πœ‹,(1.1)π‘ˆ(𝑦)∢=π‘¦ξ…žξ€·(0)βˆ’β„Žπ‘¦(0)=0,(1.2)𝑉(𝑦)∢=πœ†βˆ’π»1ξ€Έπ‘¦ξ…ž(ξ€·πœ‹)+πœ†π»βˆ’π»2𝑦(πœ‹)=0,(1.3) where the potential π‘ž(π‘₯)∈𝐿2(0,πœ‹) is a real-valued function, β„Ž,𝐻,𝐻1,𝐻2βˆˆβ„, π‘ŸβˆΆ=𝐻𝐻1βˆ’π»2>0 and πœ† is the spectral parameter.

Sturm-Liouville problems with the spectral parameter linearly contained in the boundary conditions have been investigated extensively. Fulton [1] and Walter [2] have given an operator-theoretic formulation of the problems of the form (1.1)–(1.3). It has been shown that one can associate a self-adjoint operator in adequate Hilbert space with such problems whenever π‘Ÿ>0. In the case π‘Ÿ<0 the problem (1.1)–(1.3) can be associated with a self-adjoint operator in Pontryagin space and not all eigenvalues are necessarily real (see [3, 4]). Binding et al. [5] have discussed the oscillation theory for these problems under a variety of assumptions on the coefficients. Basic properties and eigenfunction expansions have been studied in [6–9].

Inverse problems with parameter-dependent boundary conditions have also been studied. Browne and Sleeman [10] have shown that the eigenvalues and appropriately defined norming constants of the problems of the form (1.1)–(1.3) determine the potential π‘ž in the sense that two such problem with identical spectra and norming constants must have the same potential. Moreover in [11] Guliyev has proved that the kernel of the operator transforming the function √cosπœ†π‘› to the corresponding solution of (1.1) satisfies the Gelfand-Levitan-Marchenko-type integral equation. Then by using this equation, he has shown that the boundary value problem (1.1)–(1.3) can uniquely be determined from its spectrum and norming constants. There, he has used a method analogous to that of Gel’fand and Levitan [12] to reconstruction of the problem from these spectral characteristics.

In this paper we define the Weyl function for this problem and then we prove uniqueness theorems for the solution of the inverse problem of recovering 𝐿 from the Weyl function, spectral data, and two spectra and show connections between the different spectral characteristics. For solving inverse problem we use the method of spectral mappings, which has evolved from the contour integral method. Note that the contour integral method was first used for the study of inverse spectral problems by Levinson [13] and Leibenzon [14]. Yurko [15] later developed the ideas of the contour integral method. This method is an effective tool for investigating a wide class of inverse problems not only for Sturm-Liouville operators, but also for other more complicated classes of operators such as differential operators with singularities and/or turning points, pencils of operators, and others (see [16–18]). In the method of spectral mappings, at first, we use Cauchy’s integral formula in the complex plane of the spectral parameter for specially constructed analytic functions having singularities connected with the given spectral characteristics. Then by this we reduce the inverse problem to the so-called main equation which is a linear equation in a corresponding Banach space of sequences. In Section 4 we give a derivation of the main equation and prove its unique solvability. Using the solution of the main equation, we provide an algorithm for the solution of the inverse problem.

2. Preliminaries

Let πœ‘(π‘₯,πœ†), πœ“(π‘₯,πœ†), and 𝑆(π‘₯,πœ†) be the solutions of (1.1) satisfying the initial conditionsπœ‘(0,πœ†)=1,πœ‘ξ…ž(0,πœ†)=β„Ž,πœ“(πœ‹,πœ†)=πœ†βˆ’π»1,πœ“ξ…ž(πœ‹,πœ†)=βˆ’πœ†π»+𝐻2.(2.1)𝑆(0,πœ†)=0,π‘†ξ…ž(0,πœ†)=1.(2.2) We defineΞ”(πœ†)∢=βŸ¨πœ“(π‘₯,πœ†),πœ‘(π‘₯,πœ†)⟩,(2.3) whereβŸ¨π‘¦(π‘₯),𝑧(π‘₯)⟩∢=𝑦(π‘₯)π‘§ξ…ž(π‘₯)βˆ’π‘¦ξ…ž(π‘₯)𝑧(π‘₯)(2.4) is the Wronskian of 𝑦 and 𝑧. We note that if 𝑦(π‘₯,πœ†) and 𝑧(π‘₯,πœ‡) are solutions of ℓ𝑦=πœ†π‘¦ and ℓ𝑧=πœ‡π‘§, respectively, then𝑑𝑑π‘₯βŸ¨π‘¦,π‘§βŸ©=(πœ†βˆ’πœ‡)𝑦𝑧.(2.5) By virtue of the Liouville’s formula for the Wronskian (see [19, page 83]), Ξ”(πœ†) does not depend on π‘₯. The function Ξ”(πœ†) is called the characteristic function of 𝐿. Substituting π‘₯=0 and π‘₯=πœ‹ into (2.3) we getΞ”(πœ†)=𝑉(πœ‘)=βˆ’π‘ˆ(πœ“).(2.6) The function Ξ”(πœ†) is entire in πœ† and it has an at most countable set of zeroes {πœ†π‘›}.

Lemma 2.1. The zeros {πœ†π‘›} of the characteristic function coincide with the eigenvalues of the boundary value problem 𝐿. The functions πœ‘(π‘₯,πœ†π‘›) and πœ“(π‘₯,πœ†π‘›) are eigenfunctions, and there exists a sequence {π‘˜π‘›} such that πœ“ξ€·π‘₯,πœ†π‘›ξ€Έ=π‘˜π‘›πœ‘ξ€·π‘₯,πœ†π‘›ξ€Έ,π‘˜π‘›β‰ 0.(2.7)

Proof. Let πœ†βˆ˜ be a zero of Ξ”(πœ†). Then by virtue of (2.3) we have πœ“(π‘₯,πœ†βˆ˜)=π‘˜βˆ˜πœ‘(π‘₯,πœ†βˆ˜) and π‘˜βˆ˜=πœ“(0,πœ†βˆ˜). Since 𝐻𝐻1βˆ’π»2>0, we have π‘˜0β‰ 0, and the functions πœ‘(π‘₯,πœ†βˆ˜) and πœ“(π‘₯,πœ†βˆ˜) satisfy the boundary conditions (1.2) and (1.3). Hence πœ†βˆ˜ is an eigenvalue, and πœ‘(π‘₯,πœ†βˆ˜), πœ“(π‘₯,πœ†βˆ˜) are eigenfunctions related to πœ†βˆ˜.
Conversely, let πœ†βˆ˜ be an eigenvalue of 𝐿, and let π‘¦βˆ˜ be a corresponding eigenfunction. Then π‘ˆ(π‘¦βˆ˜)=𝑉(π‘¦βˆ˜)=0. Clearly π‘¦βˆ˜(0)β‰ 0, otherwise, π‘¦ξ…žβˆ˜(0)=0 and by uniqueness theorem for (1.1) (see [19, Chapter  1]), π‘¦βˆ˜β‰‘0. Without loss of generality, we put π‘¦βˆ˜(0)=1. Then π‘¦ξ…žβˆ˜(0)=β„Ž. Hence π‘¦βˆ˜(π‘₯)=πœ‘(π‘₯,πœ†βˆ˜). Thus, from (2.6), Ξ”(πœ†βˆ˜)=𝑉(πœ‘(π‘₯,πœ†βˆ˜))=𝑉(π‘¦βˆ˜(π‘₯))=0 is obtained.

Let the inner product in the Hilbert space β„‹=𝐿2(0,πœ‹)βŠ•β„‚ be defined byξ€œ(𝐹,𝐺)∢=πœ‹π‘œπΉ1(π‘₯)𝐺11(π‘₯)𝑑π‘₯+π‘ŸπΉ2𝐺2,(2.8) whereβŽ›βŽœβŽœβŽπΉπΉ=1𝐹(π‘₯)2βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπΊ,𝐺=1𝐺(π‘₯)2βŽžβŽŸβŽŸβŽ βˆˆβ„‹.(2.9) We define an operator 𝑇 acting in β„‹ such thatβŽ›βŽœβŽœβŽπ‘‡(𝐹)∢=βˆ’πΉ1ξ…žξ…ž(π‘₯)+π‘ž(π‘₯)𝐹1𝐻(π‘₯)1πΉξ…ž1(πœ‹)+𝐻2𝐹1(βŽžβŽŸβŽŸβŽ πœ‹),(2.10) with𝐷(𝑇)=πΉβˆˆβ„‹βˆ£πΉ1(π‘₯),πΉξ…ž1[],(π‘₯)∈𝐴𝐢0,πœ‹β„“πΉ1∈𝐿2(0,πœ‹),πΉξ…ž1(0)βˆ’β„ŽπΉ1(0)=0,𝐹2=πΉξ…ž1(πœ‹)+𝐻𝐹1(ξ€Ύ.πœ‹)(2.11) It is easy to show that {πœ†π‘›} are the eigenvalues of 𝑇 andΞ¦π‘›βŽ›βŽœβŽœβŽπœ‘ξ€·βˆΆ=π‘₯,πœ†π‘›ξ€Έπœ‘ξ…žξ€·πœ‹,πœ†π‘›ξ€Έξ€·+π»πœ‘πœ‹,πœ†π‘›ξ€ΈβŽžβŽŸβŽŸβŽ (2.12) are eigenelements of 𝑇.

Lemma 2.2. The eigenvalues {πœ†π‘›} and the eigenfunctions πœ‘(π‘₯,πœ†π‘›) and πœ“(π‘₯,πœ†π‘›) are real.

Proof. Let πœ†βˆ˜=𝑒+𝑖𝑣, 𝑣≠0 be a nonreal eigenvalue with an eigenfunction π‘¦βˆ˜(π‘₯). Since π‘ž(π‘₯), β„Ž, 𝐻, 𝐻1, and 𝐻2 are real, we get that πœ†βˆ˜ is also the eigenvalue with the eigenfunction π‘¦βˆ˜(π‘₯)β‰ 0. By vitue of (2.5) we get 𝑑𝑑π‘₯𝑦(π‘₯),π‘¦βˆ˜ξ‚­=ξ‚€πœ†(π‘₯)βˆ˜βˆ’πœ†βˆ˜ξ‚π‘¦βˆ˜(π‘₯)π‘¦βˆ˜(π‘₯),(2.13) and hence with the help of (1.2) ξ‚€πœ†βˆ˜βˆ’πœ†βˆ˜ξ‚ξ€œπœ‹0π‘¦βˆ˜(π‘₯)π‘¦βˆ˜(π‘₯)=π‘¦βˆ˜(πœ‹)π‘¦ξ…žβˆ˜(πœ‹)βˆ’π‘¦ξ…žβˆ˜(πœ‹)π‘¦βˆ˜ξ‚¬(πœ‹)=𝑦(π‘₯),π‘¦βˆ˜ξ‚­|||(π‘₯)π‘₯=πœ‹.(2.14) Also, by virtue of (1.3) we can write ξ‚€πœ†βˆ˜βˆ’πœ†βˆ˜ξ‚||π‘¦ξ…žβˆ˜(πœ‹)+π»π‘¦βˆ˜||(πœ‹)2=𝐻2βˆ’π»π»1𝑦(π‘₯),π‘¦βˆ˜ξ‚­|||(π‘₯)π‘₯=πœ‹.(2.15) Now, from (2.14) and (2.15) we get ξ‚€πœ†βˆ˜βˆ’πœ†βˆ˜ξ‚ξƒ©ξ€œπœ‹0||π‘¦βˆ˜||(π‘₯)2||𝑦𝑑π‘₯+ξ…žβˆ˜(πœ‹)+π»π‘¦βˆ˜||(πœ‹)2π‘Ÿξƒͺ=0,(2.16) hence π‘¦βˆ˜(π‘₯)≑0, which is a contradiction. Thus, all eigenvalues {πœ†π‘›} are real, and consequently the eigenfunctions πœ‘(π‘₯,πœ†π‘›) and πœ“(π‘₯,πœ†π‘›) are real too.

Lemma 2.3. The eigenelements of 𝑇 related to different eigenvalues are orthogonal in β„‹.

Proof. Let π‘Œπ‘›βŽ›βŽœβŽœβŽπ‘¦βˆΆ=𝑛𝑧(π‘₯)π‘›βŽžβŽŸβŽŸβŽ ,π‘Œπ‘šβŽ›βŽœβŽœβŽπ‘¦βˆΆ=π‘šπ‘§(π‘₯)π‘šβŽžβŽŸβŽŸβŽ (2.17) be eigenelements of 𝑇 related to eigenvalues πœ†π‘›,πœ†π‘š(πœ†π‘›β‰ πœ†π‘š), respectively. Since 𝑇(π‘Œπ‘›)=πœ†π‘›π‘Œπ‘› and 𝑇(π‘Œπ‘š)=πœ†π‘šπ‘Œπ‘š, we have πœ†π‘›π‘§π‘›=πœ†π‘›ξ€·π‘¦ξ…žπ‘›(πœ‹)+𝐻𝑦𝑛(πœ‹)=𝐻1π‘¦ξ…žπ‘›(πœ‹)+𝐻2π‘¦π‘›πœ†(πœ‹),π‘šπ‘§π‘š=πœ†π‘šξ€·π‘¦ξ…žπ‘š(πœ‹)+π»π‘¦π‘š(ξ€Έπœ‹)=𝐻1π‘¦ξ…žπ‘š(πœ‹)+𝐻2π‘¦π‘š(πœ‹),(2.18) and hence ξ€·πœ†π‘›βˆ’πœ†π‘šξ€Έπ‘§π‘›π‘§π‘šξ‚€π‘¦=π‘Ÿξ…žπ‘›(πœ‹)π‘¦π‘š(πœ‹)βˆ’π‘¦π‘›(πœ‹)π‘¦ξ…žπ‘šξ‚(πœ‹).(2.19) Also, since ℓ𝑦𝑛=πœ†π‘›π‘¦π‘› and β„“π‘¦π‘š=πœ†π‘šπ‘¦π‘š, from (2.5) we get ξ€·πœ†π‘›βˆ’πœ†π‘šξ€Έξ€œπœ‹0𝑦𝑛(π‘₯)π‘¦π‘š(π‘₯)𝑑π‘₯=𝑦𝑛(πœ‹)π‘¦ξ…žπ‘š(πœ‹)βˆ’π‘¦ξ…žπ‘›(πœ‹)π‘¦π‘š(πœ‹).(2.20) Thus, ξ€·π‘Œπ‘›,π‘Œπ‘šξ€Έ=ξ€œπœ‹0𝑦𝑛(π‘₯)π‘¦π‘š1(π‘₯)𝑑π‘₯+π‘Ÿπ‘§π‘›π‘§π‘š=0.(2.21)

Here we define norming constants byπ›Ύπ‘›β€–β€–Ξ¦βˆΆ=𝑛‖‖2=ξ€œπœ‹0πœ‘2ξ€·π‘₯,πœ†π‘›ξ€Έ1𝑑π‘₯+π‘Ÿξ€·πœ‘ξ…žξ€·πœ‹,πœ†π‘›ξ€Έξ€·+π»πœ‘πœ‹,πœ†π‘›ξ€Έξ€Έ2.(2.22) The numbers {πœ†π‘›,𝛾𝑛}𝑛β‰₯0 are called the spectral data of the problem (1.1)–(1.3).

Lemma 2.4. The following relation holds: Μ‡Ξ”ξ€·πœ†π‘›ξ€Έ=βˆ’π‘˜π‘›π›Ύπ‘›,(2.23) where the numbers π‘˜π‘› are defined by (2.7) and Μ‡Ξ”(πœ†)=(𝑑/π‘‘πœ†)Ξ”(πœ†).

Proof. By virtue of (2.5) we have 𝑑𝑑π‘₯πœ“(π‘₯,πœ†),πœ‘π‘₯,πœ†π‘›=ξ€·ξ€Έξ¬πœ†βˆ’πœ†π‘›ξ€Έξ€·πœ“(π‘₯,πœ†)πœ‘π‘₯,πœ†π‘›ξ€Έ,(2.24) and hence with the help of (2.6) and (2.7), ξ€œπœ‹0ξ€·πœ“(π‘₯,πœ†)πœ‘π‘₯,πœ†π‘›ξ€Έ=ξ«ξ€·πœ“(π‘₯,πœ†),πœ‘π‘₯,πœ†π‘›||ξ€Έξ¬πœ‹0πœ†βˆ’πœ†π‘›π‘Ÿ=βˆ’π‘˜π‘›βˆ’Ξ”(πœ†)πœ†βˆ’πœ†π‘›.(2.25) For πœ†β†’πœ†π‘›, this yields Μ‡Ξ”ξ€·πœ†π‘›ξ€Έ=βˆ’π‘˜π‘›ξ€œπœ‹0πœ‘2ξ€·π‘₯,πœ†π‘›ξ€Έπ‘Ÿπ‘‘π‘₯βˆ’π‘˜π‘›=βˆ’π‘˜π‘›ξƒ©π›Ύπ‘›βˆ’ξ€·πœ‘ξ…žξ€·πœ‹,πœ†π‘›ξ€Έξ€·+π»πœ‘π‘₯,πœ†π‘›ξ€Έξ€Έ2π‘Ÿξƒͺβˆ’π‘Ÿπ‘˜π‘›=βˆ’π‘˜π‘›ξƒ©π›Ύπ‘›βˆ’ξ€·πœ“ξ…žξ€·πœ‹,πœ†π‘›ξ€Έξ€·+π»πœ“π‘₯,πœ†π‘›ξ€Έξ€Έ2π‘˜2π‘›π‘Ÿξƒͺβˆ’π‘Ÿπ‘˜π‘›=βˆ’π‘˜π‘›ξ‚΅π›Ύπ‘›βˆ’π‘Ÿπ‘˜2π‘›ξ‚Άβˆ’π‘Ÿπ‘˜π‘›=βˆ’π‘˜π‘›π›Ύπ‘›.(2.26)

Remark 2.5. From this lemma we get Μ‡Ξ”(πœ†π‘›)β‰ 0. Thus, the eigenvalues of the boundary value problem (1.1)–(1.3) are simple.

Lemma 2.6. For, |𝜌|β†’βˆž, the following asymptotic formulae hold: ξ‚΅π‘’πœ‘(π‘₯,πœ†)=cos𝜌π‘₯+𝑂|𝜏|π‘₯||𝜌||𝑒=𝑂|𝜏|π‘₯ξ€Έ,πœ‘ξ…žξ€·π‘’(π‘₯,πœ†)=βˆ’πœŒsin𝜌π‘₯+𝑂|𝜏|π‘₯ξ€Έξ€·||𝜌||𝑒=𝑂|𝜏|π‘₯ξ€Έ,(2.27)πœ“(π‘₯,πœ†)=𝜌2ξ€·||𝜌||𝑒cos𝜌(πœ‹βˆ’π‘₯)+𝑂|𝜏|(πœ‹βˆ’π‘₯)ξ€Έξ€·||𝜌=𝑂2||𝑒|𝜏|(πœ‹βˆ’π‘₯)ξ€Έ,πœ“ξ…ž(π‘₯,πœ†)=𝜌3ξ€·||𝜌sin𝜌(πœ‹βˆ’π‘₯)+𝑂2||𝑒|𝜏|(πœ‹βˆ’π‘₯)ξ€Έξ€·||𝜌=𝑂3||𝑒|𝜏|(πœ‹βˆ’π‘₯)ξ€Έ,(2.28) uniformly with respect to π‘₯∈[0,πœ‹]. Here and in the sequel πœ†=𝜌2 and 𝜏=Im𝜌.

Proof. The asymptotic formulae (2.27) have been proved in [20, Lemma 1.1.2]. We prove (2.28). Let us show that ξ€·πœ“(π‘₯,πœ†)=πœ†βˆ’π»1ξ€Έξ€·cos𝜌(πœ‹βˆ’π‘₯)+πœ†π»βˆ’π»2ξ€Έsin𝜌(πœ‹βˆ’π‘₯)𝜌+ξ€œπœ‹π‘₯sin𝜌(π‘‘βˆ’π‘₯)πœŒπ‘ž(𝑑)πœ“(𝑑,πœ†)𝑑𝑑.(2.29)
Since πœ“(𝑑,πœ†) satisfies (1.1), we haveπ‘ž(𝑑)πœ“(𝑑,πœ†)=πœ“ξ…žξ…ž(𝑑,πœ†)+𝜌2πœ“(𝑑,πœ†).(2.30) Substituting this in the following integral ξ€œπœ‹π‘₯sin𝜌(π‘‘βˆ’π‘₯)πœŒπ‘ž(𝑑)πœ“(𝑑,πœ†)𝑑𝑑,(2.31) and twice integrating by parts the term involving πœ“ξ…žξ…ž(𝑑,πœ†), we obtain (2.29).
Differentiating (2.29), we calculateπœ“ξ…žξ€·(π‘₯,πœ†)=πœŒπœ†βˆ’π»1𝐻sin𝜌(πœ‹βˆ’π‘₯)+2ξ€Έβˆ’ξ€œβˆ’πœ†π»cos𝜌(πœ‹βˆ’π‘₯)πœ‹π‘₯cos𝜌(π‘‘βˆ’π‘₯)π‘ž(𝑑)πœ“(𝑑,πœ†)𝑑𝑑.(2.32) In (2.29), we put πœ“(π‘₯,πœ†)=𝑒|𝜏|(πœ‹βˆ’π‘₯)𝑓(π‘₯,πœ†). Then 𝑓(π‘₯,πœ†)=πœ†βˆ’π»1ξ€Έcos𝜌(πœ‹βˆ’π‘₯)π‘’βˆ’|𝜏|(πœ‹βˆ’π‘₯)+ξ€·πœ†π»βˆ’π»2ξ€Έsin𝜌(πœ‹βˆ’π‘₯)πœŒπ‘’βˆ’|𝜏|(πœ‹βˆ’π‘₯)+ξ€œπœ‹π‘₯sin𝜌(π‘‘βˆ’π‘₯)πœŒπ‘’βˆ’|𝜏|(π‘‘βˆ’π‘₯)π‘ž(𝑑)𝑓(𝑑,πœ†)𝑑𝑑.(2.33) Let πœ‡(πœ†)=max0≀π‘₯β‰€πœ‹|𝑓(π‘₯,πœ†)|. Then using the inequalities ||||cos𝜌(πœ‹βˆ’π‘₯)≀𝑒|𝜏|(πœ‹βˆ’π‘₯),||||sin𝜌(πœ‹βˆ’π‘₯)≀𝑒|𝜏|(πœ‹βˆ’π‘₯),(2.34) we obtain ||𝜌||πœ‡(πœ†)≀2||𝐻||1+||𝜌||+||𝐻1||||𝜌||2+||𝐻2||||𝜌||3ξƒͺ+1||𝜌||ξ€œπœ‡(πœ†)πœ‹0||||π‘ž(𝑑)𝑑.(2.35) For sufficiently larg |𝜌|, this gives ||𝜌||πœ‡(πœ†)≀𝐢2ξƒ©βˆ«1βˆ’πœ‹0π‘ž(𝑑)𝑑𝑑||𝜌||ξƒͺβˆ’1.(2.36) Hence |𝑓(π‘₯,πœ†)|β‰€πœ‡(πœ†)=𝑂(|𝜌|2), as |𝜌|β†’βˆž, and therefore πœ“(π‘₯,πœ†)=𝑂(|𝜌|2𝑒|𝜏|(πœ‹βˆ’π‘₯)), uniformly with respect to π‘₯∈[0,πœ‹] as |𝜌|β†’βˆž. Substituting this estimate into right-hand sides of (2.29) and (2.32), we get (2.28).

Theorem 2.7. The boundary value problem 𝐿 has a countable set of eigenvalues {πœ†π‘›}𝑛β‰₯0. Moreover, for 𝑛β‰₯0, πœŒπ‘›βˆšβˆΆ=πœ†π‘›πœ”=π‘›βˆ’1++πœπœ‹π‘›π‘›π‘›,ξ€½πœπ‘›ξ€Ύβˆˆπ‘™2,πœ‘ξ€·(2.37)π‘₯,πœ†π‘›ξ€Έπœ‰=cos(π‘›βˆ’1)π‘₯+𝑛(π‘₯)𝑛,||πœ‰π‘›(||𝛾π‘₯)≀𝐢,(2.38)𝑛=πœ‹2+πœξ…žπ‘›π‘›,ξ€½πœξ…žπ‘›ξ€Ύβˆˆπ‘™2,(2.39) where 1πœ”=β„Ž+𝐻+2ξ€œπœ‹0π‘ž(𝑑)𝑑𝑑.(2.40)

Proof. (1) By virtue of the following asymptotic formulae (see [20, page 7]) πœ‘(π‘₯,πœ†)=cos𝜌π‘₯+π‘ž1(π‘₯)sin𝜌π‘₯𝜌+1ξ€œ2𝜌π‘₯0ξ‚΅π‘’π‘ž(𝑑)sin𝜌(π‘₯βˆ’2𝑑)𝑑𝑑+𝑂|𝜏|π‘₯𝜌2ξ‚Ά,πœ‘ξ…ž(π‘₯,πœ†)=βˆ’πœŒsin𝜌π‘₯+π‘ž11(π‘₯)cos𝜌π‘₯+2ξ€œπ‘₯0ξ‚΅π‘’π‘ž(𝑑)cos𝜌(π‘₯βˆ’2𝑑)𝑑𝑑+𝑂|𝜏|π‘₯πœŒξ‚Ά,(2.41) where π‘ž11(π‘₯)=β„Ž+2ξ€œπ‘₯0π‘ž(𝑑)𝑑𝑑,(2.42) and using (2.6) we get Ξ”(πœ†)=βˆ’πœŒ3sinπœŒπœ‹+πœ”πœŒ2cosπœŒπœ‹+𝜌2𝐼(𝜌),(2.43) where 1𝐼(𝜌)=2ξ€œπœ‹0ξ‚΅1π‘ž(𝑑)cos𝜌(πœ‹βˆ’2𝑑)𝑑𝑑+π‘‚πœŒπ‘’|𝜏|πœ‹ξ‚Ά.(2.44) Denote 𝐺𝛿={𝜌∢|πœŒβˆ’π‘˜|β‰₯𝛿,π‘˜=0,Β±1,Β±2,…}, 𝛿>0. Using (2.43) we get for πœŒβˆˆπΊπ›ΏΞ”(πœ†)=βˆ’πœŒ3ξ‚΅ξ‚΅1sinπœŒπœ‹1+π‘‚πœŒξ‚Άξ‚Ά.(2.45) Now, since (see [20, page 6]) ||||sinπœŒπœ‹β‰₯𝐢𝛿𝑒|𝜏|πœ‹,πœŒβˆˆπΊπ›Ώ,(2.46) we have ||||Ξ”(πœ†)β‰₯𝐢𝛿||𝜌||3𝑒|𝜏|πœ‹,πœŒβˆˆπΊπ›Ώ,||𝜌||β‰₯πœŒβˆ—,(2.47) for sufficiently large πœŒβˆ—=πœŒβˆ—(𝛿).
Let Γ𝑛={πœ†βˆΆ|πœ†|=(π‘›βˆ’1/2)2}. By virtue of (2.43),Ξ”(πœ†)=𝑓(πœ†)+𝑔(πœ†),𝑓(πœ†)=βˆ’πœŒ3||||||𝜌||sinπœŒπœ‹,𝑔(πœ†)≀𝐢2𝑒|𝜏|πœ‹.(2.48) It follows from (2.46) that |𝑓(πœ†)|>|𝑔(πœ†)|, πœ†βˆˆΞ“π‘›, for sufficiently large 𝑛(𝑛β‰₯π‘›βˆ—). Hence by Rouché’s theorem [21, page 125], the number of zeros of Ξ”(πœ†) inside Γ𝑛 coincides with the number of zeros of 𝑓(πœ†), that is, it equals 𝑛+1. Thus, in the circle |πœ†|<(π‘›βˆ’1/2)2 there exist exactly 𝑛+1 eigenvalues of πΏβˆΆπœ†0,…,πœ†π‘›. Analogously, by using Rouché’s theorem one can prove that for sufficiently large vales of 𝑛, every circle πœŽπ‘›(𝛿)={𝜌∢|πœŒβˆ’(π‘›βˆ’1)|≀𝛿} contains exactly one zero of Ξ”(𝜌2), namely, πœŒπ‘›=βˆšπœ†π‘›. Since 𝛿>0 is arbitrary, we must have πœŒπ‘›=π‘›βˆ’1+πœ€π‘›,πœ€π‘›=π‘œ(1),π‘›βŸΆβˆž.(2.49) Since πœŒπ‘› are zeros of Ξ”(𝜌2), from (2.43) we get Ξ”ξ€·πœŒ2𝑛=βˆ’π‘›βˆ’1+πœ€π‘›ξ€Έ3ξ€·sinπ‘›βˆ’1+πœ€π‘›ξ€Έξ€·πœ‹+πœ”π‘›βˆ’1+πœ€π‘›ξ€Έ2ξ€·cosπ‘›βˆ’1+πœ€π‘›ξ€Έπœ‹+ξ€·π‘›βˆ’1+πœ€π‘›ξ€Έ2πΌξ€·π‘›βˆ’1+πœ€π‘›ξ€Έ=0,(2.50) and consequently βˆ’π‘›sinπœ€π‘›πœ‹+πœ”cosπœ€π‘›πœ‹+πœ—π‘›=0,(2.51) where πœ—π‘›=(1βˆ’πœ€π‘›)sinπœ€π‘›πœ‹+(βˆ’1)π‘›βˆ’1𝐼(π‘›βˆ’1+πœ€π‘›). Hence sinπœ€π‘›πœ‹=𝑂(1/𝑛), that is, πœ€π‘›=𝑂(1/𝑛). Thus, {πœ—π‘›}βˆˆπ‘™2. Using (2.51) once more we get more precisely πœ€π‘›=πœ”/πœ‹π‘›+πœπ‘›/𝑛, where {πœπ‘›}βˆˆπ‘™2.
(2) Substituting (2.37) into (2.41), we get (2.38) whereπœ‰π‘›ξ‚€π‘ž(π‘₯)=1πœ”(π‘₯)βˆ’π‘₯πœ‹βˆ’π‘₯πœπ‘›ξ‚1sin(π‘›βˆ’1)π‘₯+2ξ€œπ‘₯0ξ‚€1π‘ž(𝑑)sin(π‘›βˆ’1)(π‘₯βˆ’2𝑑)𝑑𝑑+𝑂𝑛.(2.52) Therefore, |πœ‰π‘›(π‘₯)|≀𝐢, and (2.38) is proved.
(3) From (2.38) we calculateξ€œπœ‹0πœ‘2ξ€·π‘₯,πœ†π‘›ξ€Έξ€œπ‘‘π‘₯=πœ‹0cos2ξ€œ(π‘›βˆ’1)π‘₯𝑑π‘₯+2πœ‹0cos(π‘›βˆ’1)π‘₯πœ‰π‘›(π‘₯)π‘›ξ€œπ‘‘π‘₯+πœ‹0πœ‰2𝑛(π‘₯)𝑛2=πœ‹π‘‘π‘₯2+πœξ…žπ‘›π‘›,ξ€½πœξ…žπ‘›ξ€Ύβˆˆπ‘™2.(2.53) Also, using (1.3), (2.37), and (2.38) we obtain 1π‘Ÿξ€·πœ‘ξ…žξ€·πœ‹,πœ†π‘›ξ€Έξ€·+π»πœ‘πœ‹,πœ†π‘›ξ€Έξ€Έ2=ξ‚΅πœ‘(πœ‹,πœ†π‘›)𝐻1βˆ’πœ†π‘›ξ‚Ά2ξ‚€1=𝑂𝑛4.(2.54) Thus, 𝛾𝑛=πœ‹2+πœξ…žπ‘›π‘›,ξ€½πœξ…žπ‘›ξ€Ύβˆˆπ‘™2,(2.55) and Theorem 2.7 is proved.

Theorem 2.8. The specification of the spectrum {πœ†π‘›}𝑛β‰₯0 uniquely determines the characteristic function Ξ”(πœ†) by the formula ξ€·Ξ”(πœ†)=πœ‹πœ†βˆ’πœ†0πœ†ξ€Έξ€·1ξ€Έβˆ’πœ†βˆžξ‘π‘›=2πœ†π‘›βˆ’πœ†(π‘›βˆ’1)2.(2.56)

Proof. It follows from (2.6), (2.27), and (2.28) that Ξ”(πœ†) is an entire function of πœ† of order 1/2, and hence by Hadamard’s factorization theorem [21, page 289], Ξ”(πœ†) is uniquely determined up to a multiplicative constant by its zeros: Ξ”(πœ†)=πΆβˆžξ‘π‘›=0ξ‚΅πœ†1βˆ’πœ†π‘›ξ‚Ά.(2.57) The case Ξ”(0)=0 requires minor modifications. We consider the function Δ(πœ†)∢=βˆ’πœŒ3sinπœŒπœ‹=βˆ’πœ†2πœ‹βˆžξ‘π‘›=1ξ‚€πœ†1βˆ’π‘›2.(2.58) Then Ξ”(πœ†)Δ(πœ†)=πΆπœ†βˆ’πœ†0πœ†ξ€Έξ€·1ξ€Έβˆ’πœ†πœ‹πœ†0πœ†1πœ†2βˆžξ‘π‘›=1𝑛2πœ†βˆžπ‘›+1𝑛=1ξƒ©πœ†1+𝑛+1βˆ’π‘›2𝑛2ξƒͺ.βˆ’πœ†(2.59) With the help of (2.37) and (2.43), we calculate limπœ†β†’βˆ’βˆžΞ”(πœ†)Δ(πœ†)=1,limβˆžπœ†β†’βˆ’βˆžξ‘π‘›=1ξƒ©πœ†1+𝑛+1βˆ’π‘›2𝑛2ξƒͺβˆ’πœ†=1,(2.60) and hence 𝐢=βˆ’πœ‹πœ†0πœ†1βˆžξ‘π‘›=1πœ†π‘›+1𝑛2.(2.61) Substituting this into (2.57), we get (2.56).

Remark 2.9. Analogous results are valid for boundary value problems with other types of spectral parameter-dependent boundary conditions. Let us state some of these results for one of them which will be used below.
Consider the boundary value problem 𝐿0=𝐿0(π‘ž(π‘₯),𝐻,𝐻1,𝐻2) for (1.1) with the boundary conditions 𝑦(0)=𝑉(𝑦)=0. The eigenvalues {πœ†0𝑛}𝑛β‰₯0 of 𝐿0 are simple and coincide with the zeros of the characteristic function Ξ”0(πœ†)∢=πœ“(0,πœ†)=𝑉(𝑆) andΞ”0ξ€·(πœ†)=πœ†βˆ’πœ†0ξ€Έβˆžξ‘π‘›=1πœ†π‘›βˆ’πœ†(π‘›βˆ’1/2)2,𝜌(2.62)0π‘›ξ”βˆΆ=πœ†0𝑛1=π‘›βˆ’2+πœ”0+πœπœ‹π‘›0𝑛𝑛,ξ€½πœ0π‘›ξ€Ύβˆˆπ‘™2,(2.63) where πœ”01=𝐻+2ξ€œπœ‹0π‘ž(𝑑)𝑑𝑑.(2.64)

3. Inverse Problems

In this section, we study three inverse problems of recovering 𝐿 from its spectral characteristics, namely,(i)from the Weyl function,(ii)from the so-called spectral data,(iii)from two spectra.

For each class of inverse problems we prove the corresponding uniqueness theorems and show connection between the different spectral characteristics.

3.1. The Inverse Problem from the Weyl Function

Let Ξ¦(π‘₯,πœ†) be the solution of (1.1) under the conditions π‘ˆ(Ξ¦)=1 and 𝑉(Ξ¦)=0. We set 𝑀(πœ†)∢=Ξ¦(0,πœ†). The functions Ξ¦(π‘₯,πœ†) and 𝑀(πœ†) are called the Weyl solution and the Weyl function for the boundary value problem 𝐿, respectively. The notion of the Weyl function introduced here is a generalization of the Weyl function for the classical Sturm-Liouville operators (see [20, 22]). ClearlyΞ¦(π‘₯,πœ†)=βˆ’πœ“(π‘₯,πœ†)ΔΔ(πœ†)=𝑆(π‘₯,πœ†)+𝑀(πœ†)πœ‘(π‘₯,πœ†),(3.1)𝑀(πœ†)=βˆ’0(πœ†)Ξ”(πœ†),(3.2)βŸ¨πœ‘(π‘₯,πœ†),Ξ¦(π‘₯,πœ†)βŸ©β‰‘1.(3.3) Since Ξ”(πœ†) and Ξ”0(πœ†) have no common zeros, it follows from (3.2) that 𝑀(πœ†) is a meromorphic function with poles {πœ†π‘›}𝑛β‰₯0 and zeros {πœ†0𝑛}𝑛β‰₯0. We consider the following inverse problem.

Inverse Problem 1
Given the Weyl function 𝑀(πœ†), construct π‘ž(π‘₯), β„Ž, 𝐻, 𝐻1, and 𝐻2.

Let us prove the uniqueness theorem for the solution of the Inverse Problem 1. For this purpose we agree that together with 𝐿 we consider a boundary value problem 𝐿 of the same form but with different coefficients Μƒπ‘ž(π‘₯), ξ‚β„Ž, 𝐻, 𝐻1, and 𝐻2. Everywhere below if a certain symbol 𝛼 denotes an object related to 𝐿, then the corresponding symbol 𝛼 with tilde denotes the analogous object related to 𝐿.

Theorem 3.1. If 𝑀(πœ†)=𝑀(πœ†), then π‘ž(π‘₯)=Μƒπ‘ž(π‘₯) a.e. on (0,πœ‹), ξ‚β„Žβ„Ž=, 𝐻𝐻=, 𝐻1=𝐻1 and 𝐻2=𝐻2. Thus, the specification of the Weyl function 𝑀(πœ†) uniquely determines 𝐿.

Proof. Let us define the matrix 𝑃(π‘₯,πœ†)=[π‘ƒπ‘—π‘˜(π‘₯,πœ†)]𝑗,π‘˜=1,2 by the formula βŽ‘βŽ’βŽ’βŽ£ξ‚π‘ƒ(π‘₯,πœ†)ξ‚πœ‘(π‘₯,πœ†)Ξ¦(π‘₯,πœ†)ξ‚πœ‘ξ…ž(Φπ‘₯,πœ†)ξ…ž(⎀βŽ₯βŽ₯⎦=βŽ‘βŽ’βŽ’βŽ£πœ‘π‘₯,πœ†)πœ‘(π‘₯,πœ†)Ξ¦(π‘₯,πœ†)ξ…ž(π‘₯,πœ†)Ξ¦ξ…ž(⎀βŽ₯βŽ₯⎦π‘₯,πœ†).(3.4) Using (3.3) and (3.4) we calculate for 𝑗=1,2: 𝑃𝑗1(π‘₯,πœ†)=πœ‘(π‘—βˆ’1)(Φπ‘₯,πœ†)ξ…ž(π‘₯,πœ†)βˆ’Ξ¦(π‘—βˆ’1)(π‘₯,πœ†)ξ‚πœ‘ξ…ž(𝑃π‘₯,πœ†),𝑗2(π‘₯,πœ†)=Ξ¦(π‘—βˆ’1)(π‘₯,πœ†)ξ‚πœ‘(π‘₯,πœ†)βˆ’πœ‘(π‘—βˆ’1)(π‘₯,πœ†)Ξ¦(π‘₯,πœ†),(3.5)πœ‘(π‘₯,πœ†)=𝑃11(π‘₯,πœ†)ξ‚πœ‘(π‘₯,πœ†)+𝑃12(π‘₯,πœ†)ξ‚πœ‘ξ…ž(π‘₯,πœ†),Ξ¦(π‘₯,πœ†)=𝑃11(π‘₯,πœ†)Ξ¦(π‘₯,πœ†)+𝑃12Φ(π‘₯,πœ†)ξ…ž(π‘₯,πœ†).(3.6) It follows from (3.1), (3.3), and (3.5) that 𝑃111(π‘₯,πœ†)=1+Ξ”ξ€·ξ€·(πœ†)πœ“(π‘₯,πœ†)ξ‚πœ‘ξ…ž(π‘₯,πœ†)βˆ’πœ‘ξ…žξ€Έξ€·(π‘₯,πœ†)βˆ’πœ‘(π‘₯,πœ†)ξ‚πœ“ξ…ž(π‘₯,πœ†)βˆ’πœ“ξ…ž,𝑃(π‘₯,πœ†)ξ€Έξ€Έ121(π‘₯,πœ†)=Ξ”ξ€·ξ€Έ.(πœ†)πœ‘(π‘₯,πœ†)ξ‚πœ“(π‘₯,πœ†)βˆ’πœ“(π‘₯,πœ†)ξ‚πœ‘(π‘₯,πœ†)(3.7) By virtue of (2.27), (2.28), and (2.47), this yields ||𝑃11||≀𝐢(π‘₯,πœ†)βˆ’1𝛿||𝜌||,||𝑃12||≀𝐢(π‘₯,πœ†)𝛿||𝜌||,πœŒβˆˆπΊπ›Ώ,||𝜌||β‰₯πœŒβˆ—.(3.8) Uniformly with respect to π‘₯∈[0,πœ‹]. Similarly, we have ||𝑃22||≀𝐢(π‘₯,πœ†)βˆ’1𝛿||𝜌||,||𝑃21||(π‘₯,πœ†)≀𝐢𝛿,πœŒβˆˆπΊπ›Ώ,||𝜌||β‰₯πœŒβˆ—,(3.9) Uniformly with respect to π‘₯∈[0,πœ‹]. On the other hand according to (3.1) and (3.5), 𝑃11𝑆(π‘₯,πœ†)=πœ‘(π‘₯,πœ†)ξ…ž(π‘₯,πœ†)βˆ’π‘†(π‘₯,πœ†)ξ‚πœ‘ξ…žξ‚€ξ‚‹ξ‚(π‘₯,πœ†)+𝑀(πœ†)βˆ’π‘€(πœ†)πœ‘(π‘₯,πœ†)ξ‚πœ‘ξ…žπ‘ƒ(π‘₯,πœ†),12(π‘₯,πœ†)=𝑆(π‘₯,πœ†)ξ‚πœ‘(π‘₯,πœ†)βˆ’πœ‘(π‘₯,πœ†)𝑆(π‘₯,πœ†)+𝑀(πœ†)βˆ’π‘€(πœ†)πœ‘(π‘₯,πœ†)ξ‚πœ‘(π‘₯,πœ†).(3.10) Since 𝑀(πœ†)≑𝑀(πœ†), it follows that for each fixed π‘₯, the functions 𝑃11(π‘₯,πœ†) and 𝑃12(π‘₯,πœ†) are entire in πœ†. With the help of (3.8), this yields 𝑃11(π‘₯,πœ†)≑1, 𝑃12(π‘₯,πœ†)≑0. Substituting into (3.6), we get πœ‘(π‘₯,πœ†)β‰‘ξ‚πœ‘(π‘₯,πœ†), Φ(π‘₯,πœ†)≑Φ(π‘₯,πœ†) for all π‘₯ and πœ†. Hence from (1.1) and (2.1) we get π‘ž(π‘₯)=Μƒπ‘ž(π‘₯) a.e. on (0,πœ‹), ξ‚β„Žβ„Ž=, 𝐻𝐻=, 𝐻1=𝐻1, and 𝐻2=𝐻2. Consequently, 𝐿𝐿=.

3.2. The Inverse Problem from the Spectral Data

Let {πœ†π‘›}𝑛β‰₯0 and {𝛾𝑛}𝑛β‰₯0 be the eigenvalues and norming constants of 𝐿, respectively. We consider the following inverse problem.

Inverse Problem 2
Given the spectral data {πœ†π‘›,𝛾𝑛}𝑛β‰₯0, construct π‘ž(π‘₯),β„Ž,𝐻,𝐻1, and 𝐻2.

Lemma 3.2. The following representation holds: 𝑀(πœ†)=βˆžξ“π‘›=01π›Ύπ‘›ξ€·πœ†βˆ’πœ†π‘›ξ€Έ.(3.11)

Proof. Consider the contour integral 𝐽𝑁(1πœ†)=ξ€œ2πœ‹π‘–Ξ“π‘π‘€(πœ‡)πœ†βˆ’πœ‡π‘‘πœ‡,πœ†βˆˆintΓ𝑁,(3.12) where the contour Γ𝑁 is assumed to have the counterclockwise circuit. Since Ξ”0(πœ†)=πœ“(0,πœ†), it follows from (2.28) that |Ξ”0(πœ†)|≀𝐢|𝜌|2𝑒|𝜏|πœ‹. Then, using (3.1) and (2.47), we get for sufficiently large πœŒβˆ—>0, ||𝑀||≀𝐢(πœ†)𝛿||𝜌||,πœŒβˆˆπΊπ›Ώ,||𝜌||β‰₯πœŒβˆ—.(3.13) Moreover, using (3.2) and (2.23), we calculate Resπœ†=πœ†π‘›Ξ”π‘€(πœ†)=βˆ’0ξ€·πœ†π‘›ξ€ΈΜ‡Ξ”ξ€·πœ†π‘›ξ€Έπ‘˜=βˆ’π‘›Μ‡Ξ”ξ€·πœ†π‘›ξ€Έ=1𝛾𝑛.(3.14) In view of (3.13), limπ‘β†’βˆžπ½π‘(πœ†)=0. By virtue of (3.14) and residue theorem [21, page 112], we have 𝐽𝑁(πœ†)=βˆ’π‘€(πœ†)+𝑁𝑛=01π›Ύπ‘›ξ€·πœ†βˆ’πœ†π‘›ξ€Έ,(3.15) and consequently (3.11) is proved.

Let us prove a uniqueness theorem for the solution of Inverse Problem 2.

Theorem 3.3. If πœ†π‘›=Μƒπœ†π‘› and 𝛾𝑛=̃𝛾𝑛 for all 𝑛β‰₯0, then π‘ž(π‘₯)=Μƒπ‘ž(π‘₯) a.e. on (0,πœ‹), ξ‚β„Žβ„Ž=, 𝐻𝐻=, 𝐻1=𝐻1 and 𝐻2=𝐻2. Thus, the specification of the spectral data {πœ†π‘›,𝛾𝑛}𝑛β‰₯0, uniquely determines 𝐿.

Proof. If πœ†π‘›=Μƒπœ†π‘› and 𝛾𝑛=̃𝛾𝑛 for all 𝑛β‰₯0, then from Lemma 3.2, we get 𝑀(πœ†)=𝑀(πœ†). Hence By virtue of Theorem 3.1, this implies π‘ž(π‘₯)=Μƒπ‘ž(π‘₯) a.e. on (0,πœ‹), ξ‚β„Žβ„Ž=, 𝐻𝐻=, 𝐻1=𝐻1 and 𝐻2=𝐻2. Thus, Theorem 3.3 is proved.

Remark 3.4. By virtue of (3.11), the specification of the Weyl function 𝑀(πœ†) is equivalent to the specification of the spectral data {πœ†π‘›,𝛾𝑛}𝑛β‰₯0, that is, the Inverse Problem 1 is equivalent to the Inverse Problem 2.

3.3. The Inverse Problem from Two Spectra

Let {πœ†π‘›}𝑛β‰₯0 and {πœ†0𝑛}𝑛β‰₯0 be the eigenvalues of the problems 𝐿 and 𝐿0, respectively. We consider the following inverse problem.

Inverse Problem 3
Given two spectra {πœ†π‘›,πœ†0𝑛}𝑛β‰₯0, construct π‘ž(π‘₯), β„Ž, 𝐻, 𝐻1, and 𝐻2.

Let us prove a uniqueness theorem for the solution of Inverse Problem 3.

Theorem 3.5. If πœ†π‘›=Μƒπœ†π‘› and πœ†0𝑛=Μƒπœ†0𝑛, then π‘ž(π‘₯)=Μƒπ‘ž(π‘₯) a.e. on (0,πœ‹), ξ‚β„Žβ„Ž=, 𝐻𝐻=, 𝐻1=𝐻1, and 𝐻2=𝐻2. Thus, the specification of two spectra {πœ†π‘›,πœ†0𝑛}𝑛β‰₯0, uniquely determines 𝐿.

Proof. According to Lemma 2.1 and Remark 2.9 the sets {πœ†π‘›}𝑛β‰₯0 and {πœ†0𝑛}𝑛β‰₯0 coincide with the set of zeros of the functions Ξ”(πœ†) and Ξ”0(πœ†), respectively. Using (2.56) and (2.62), we get Δ(πœ†)=Ξ”(πœ†) and Ξ”0Δ(πœ†)=0(πœ†). Together with (3.2) this yields 𝑀(πœ†)=𝑀(πœ†). By Theorem 3.1 we get π‘ž(π‘₯)=Μƒπ‘ž(π‘₯) a.e. on (0,πœ‹), ξ‚β„Žβ„Ž=, 𝐻𝐻=, 𝐻1=𝐻1, and 𝐻2=𝐻2.

Remark 3.6. It follows from Theorems 3.1 and 3.5 that the specification of Weyl function 𝑀(πœ†) is equivalent to the specification of two spectra {πœ†π‘›,πœ†0𝑛}𝑛β‰₯0, that is, the Inverse Problem 1 is equivalent to the Inverse Problem 3.

4. Solution of the Inverse Problem

In this section, we give a constructive procedure for the solution of the inverse problem of recovering 𝐿 from the given spectral data {πœ†π‘›,𝛾𝑛}𝑛β‰₯0 by the method of spectral mappings and state necessary and sufficient solvability conditions.

Let {πœ†π‘›,𝛾𝑛}𝑛β‰₯0 be the spectral data of 𝐿. Denote𝐷(π‘₯,πœ†,πœ‡)∢=βŸ¨πœ‘(π‘₯,πœ†),πœ‘(π‘₯,πœ‡)⟩=ξ€œπœ†βˆ’πœ‡π‘₯0πœ‘(𝑑,πœ†)πœ‘(𝑑,πœ‡)𝑑𝑑.(4.1)

Let us choose a model boundary value problem 𝐻𝐿=𝐿(Μƒπ‘ž(π‘₯),β„Ž,𝐻,1,𝐻2) with real Μƒπ‘ž(π‘₯)∈𝐿2(0,πœ‹), ξ‚β„Ž, 𝐻, 𝐻1, 𝐻2, and ξ‚π»ξ‚π»Μƒπ‘ŸβˆΆ=1βˆ’ξ‚π»2>0 such that ξ‚πœ”=πœ” (take, e.g., Μƒπ‘ž(π‘₯)≑0, ξ‚β„Ž=0, 𝐻=πœ”). Let {Μƒπœ†π‘›,̃𝛾𝑛}𝑛β‰₯0 be the spectral data of 𝐿. Denoteπœ‰π‘›||𝜌∢=π‘›βˆ’ΜƒπœŒπ‘›||+||π›Ύπ‘›βˆ’Μƒπ›Ύπ‘›||.(4.2) Since πœ”=ξ‚πœ”, it follows from (2.37), (2.39), and analogous formulae for ΜƒπœŒπ‘› and ̃𝛾𝑛 thatβˆžξ“π‘›=0πœ‰π‘›<∞.(4.3) Denoteπœ†π‘›0=πœ†π‘›,πœ†π‘›1=Μƒπœ†π‘›,𝛾𝑛0=𝛾𝑛,𝛾𝑛1=̃𝛾𝑛,πœ‘π‘›π‘–ξ€·(π‘₯)=πœ‘π‘₯,πœ†π‘›π‘–ξ€Έ,ξ‚πœ‘π‘›π‘–ξ€·(π‘₯)=ξ‚πœ‘π‘₯,πœ†π‘›π‘–ξ€Έ,𝑃𝑛𝑖,π‘˜π‘—1(π‘₯)=π›Ύπ‘˜π‘—π·ξ€·π‘₯,πœ†π‘›π‘–,πœ†π‘˜π‘—ξ€Έ,𝑃𝑛𝑖,π‘˜π‘—1(π‘₯)=π›Ύπ‘˜π‘—ξ‚π·ξ€·π‘₯,πœ†π‘›π‘–,πœ†π‘˜π‘—ξ€Έ,(4.4) where 𝑖,𝑗=0,1, 𝑛,π‘˜β‰₯0. Let π‘₯∈[0,πœ‹], 𝑛β‰₯0, 𝑖,𝜈=0,1. It follows from (2.27) and (2.37) that||πœ‘(𝜈)𝑛𝑖||(π‘₯)≀𝐢(𝑛+1)𝜈.(4.5) Moreover, for a fixed π‘Ž>0||πœ‘(𝜈)||(π‘₯,πœ†)≀𝐢(𝑛+1)𝜈,||πœŒβˆ’πœŒπ‘›1||β‰€π‘Ž.(4.6) Applying Schwarz’s lemma [21, page 130] in the 𝜌-plane to the circle |πœŒβˆ’πœŒπ‘›1|β‰€π‘Ž and to the function 𝑓(𝜌)∢=πœ‘(𝜈)(π‘₯,πœ†)βˆ’πœ‘(𝜈)(π‘₯,πœ†π‘›1) with fixed 𝜈, 𝑛, π‘₯, and π‘Ž, we get||πœ‘(𝜈)(π‘₯,πœ†)βˆ’πœ‘(𝜈)ξ€·π‘₯,πœ†π‘›1ξ€Έ||≀𝐢(𝑛+1)𝜈||πœŒβˆ’πœŒπ‘›1||,||πœŒβˆ’πœŒπ‘›1||β‰€π‘Ž.(4.7) In particular, this yields||πœ‘(𝜈)𝑛0(π‘₯)βˆ’πœ‘(𝜈)𝑛1||(π‘₯)β‰€πΆπœ‰π‘›(𝑛+1)𝜈.(4.8) By similar arguments (see [20, page 48]) one gets that the following estimates are valid for π‘₯∈[0,πœ‹], 𝑛,π‘˜β‰₯0, 𝑖,𝑗,𝜈=0,1:||𝑃𝑛𝑖,π‘˜π‘—(||≀𝐢π‘₯)||||,||π‘ƒπ‘›βˆ’π‘˜+1𝑛𝑖,π‘˜0(π‘₯)βˆ’π‘ƒπ‘›π‘–,π‘˜1||≀(π‘₯)πΆπœ‰π‘˜||||,||π‘ƒπ‘›βˆ’π‘˜+1𝑛0,π‘˜π‘—(π‘₯)βˆ’π‘ƒπ‘›1,π‘˜π‘—||≀(π‘₯)πΆπœ‰π‘›||||,||π‘ƒπ‘›βˆ’π‘˜+1𝑛0,π‘˜0(π‘₯)βˆ’π‘ƒπ‘›1,π‘˜0(π‘₯)βˆ’π‘ƒπ‘›0,π‘˜1(π‘₯)+𝑃𝑛1,π‘˜1(||≀π‘₯)πΆπœ‰π‘›πœ‰π‘˜||||.π‘›βˆ’π‘˜+1(4.9) The analogous estimates are also valid for ξ‚πœ‘π‘›π‘–(π‘₯) and 𝑃𝑛𝑖,π‘˜π‘—(π‘₯).

Lemma 4.1. The following relations hold: ξ‚πœ‘(π‘₯,πœ†)=πœ‘(π‘₯,πœ†)+βˆžξ“π‘˜=0ξƒ©βŸ¨ξ‚πœ‘(π‘₯,πœ†),ξ‚πœ‘π‘˜0(π‘₯)βŸ©π›Ύπ‘˜0ξ€·πœ†βˆ’πœ†π‘˜0ξ€Έπœ‘π‘˜0(π‘₯)βˆ’βŸ¨ξ‚πœ‘(π‘₯,πœ†),ξ‚πœ‘π‘˜1(π‘₯)βŸ©π›Ύπ‘˜1ξ€·πœ†βˆ’πœ†π‘˜1ξ€Έπœ‘π‘˜1ξƒͺ,(π‘₯)(4.10)βŸ¨πœ‘(π‘₯,πœ†),πœ‘(π‘₯,πœ‡)βŸ©βˆ’πœ†βˆ’πœ‡βŸ¨ξ‚πœ‘(π‘₯,πœ†),ξ‚πœ‘(π‘₯,πœ‡)⟩+πœ†βˆ’πœ‡βˆžξ“π‘˜=0ξƒ©βŸ¨ξ‚πœ‘(π‘₯,πœ†),ξ‚πœ‘π‘˜0(π‘₯)βŸ©π›Ύπ‘˜0ξ€·πœ†βˆ’πœ†π‘˜0ξ€ΈβŸ¨πœ‘π‘˜0(π‘₯),πœ‘(π‘₯,πœ‡)βŸ©ξ€·πœ†π‘˜0ξ€Έβˆ’βˆ’πœ‡βŸ¨ξ‚πœ‘(π‘₯,πœ†),ξ‚πœ‘π‘˜1(π‘₯)βŸ©π›Ύπ‘˜1ξ€·πœ†βˆ’πœ†π‘˜1ξ€ΈβŸ¨πœ‘π‘˜1(π‘₯),πœ‘(π‘₯,πœ‡)βŸ©ξ€·πœ†π‘˜1ξ€Έξƒͺβˆ’πœ‡=0.(4.11) Both series converge absolutely and uniformly with respect to π‘₯∈[0,πœ‹] and πœ†, πœ‡ on compact sets.

Proof. (1) Denote πœ†ξ…ž=minπ‘›π‘–πœ†π‘›π‘– and take a fixed 𝛿>0.
Let 𝐼∢={πœ†βˆΆ|Imπœ†|≀𝛿,Reπœ†β‰₯πœ†ξ…žβˆ’π›Ώ}, and let πœƒβˆΆ=πœ•πΌ be the boundary of 𝐼. Denote Ξ“ξ…žπ‘=Ξ“π‘βˆ©πΌ, Ξ“π‘ξ…žξ…ž=Ξ“π‘β§΅Ξ“ξ…žπ‘, πœƒξ…žπ‘=πœƒβˆ©intΓ𝑁. In the πœ†-plane we consider closed contours πœƒπ‘=πœƒξ…žπ‘βˆͺΞ“ξ…žπ‘ (with counterclockwise circuit), πœƒ0𝑁=πœƒξ…žπ‘βˆͺΞ“π‘ξ…žξ…ž (with clockwise circuit). Let 𝑃(π‘₯,πœ†)=[π‘π‘—π‘˜(π‘₯,πœ†)]𝑗,π‘˜=1,2 be the matrix defined by (3.4). It follows from (3.5) and (3.1) that for each fixed π‘₯, the functions π‘ƒπ‘—π‘˜(π‘₯,πœ†) are meromorphic in πœ† with simple poles {πœ†π‘›}𝑛β‰₯0 and {Μƒπœ†π‘›}𝑛β‰₯0. By Cauchy’s integral formula [21, page 84],𝑃1π‘˜(π‘₯,πœ†)βˆ’π›Ώ1π‘˜=1ξ€œ2πœ‹π‘–πœƒ0𝑁𝑃1π‘˜(π‘₯,πœ‰)βˆ’π›Ώ1π‘˜πœ†βˆ’πœ‰π‘‘πœ‰,π‘˜=1,2,πœ†βˆˆintπœƒ0𝑁,(4.12) where π›Ώπ‘—π‘˜ is the Kronecker delta. Hence 𝑃1π‘˜(π‘₯,πœ†)βˆ’π›Ώ1π‘˜=1ξ€œ2πœ‹π‘–πœƒπ‘π‘ƒ1π‘˜(π‘₯,πœ‰)1πœ†βˆ’πœ‰π‘‘πœ‰βˆ’ξ€œ2πœ‹π‘–Ξ“π‘π‘ƒ1π‘˜(π‘₯,πœ‰)βˆ’π›Ώ1π‘˜πœ†βˆ’πœ‰π‘‘πœ‰,(4.13) where Γ𝑁 is used with counterclockwise circuit. Substituting into (3.6) we obtain 1πœ‘(π‘₯,πœ†)=ξ‚πœ‘(π‘₯,πœ†)+ξ€œ2πœ‹π‘–πœƒπ‘ξ‚πœ‘(π‘₯,πœ†)𝑃11(π‘₯,πœ‰)+ξ‚πœ‘ξ…ž(π‘₯,πœ†)𝑃12(π‘₯,πœ‰)πœ†βˆ’πœ‰π‘‘πœ‰+πœ€π‘(π‘₯,πœ†),(4.14) where πœ€π‘1(π‘₯,πœ†)=βˆ’ξ€œ2πœ‹π‘–Ξ“π‘ξ€·π‘ƒξ‚πœ‘(π‘₯,πœ†)11(ξ€Έπ‘₯,πœ‰)βˆ’1+ξ‚πœ‘ξ…ž(π‘₯,πœ†)𝑃12(π‘₯,πœ‰)πœ†βˆ’πœ‰π‘‘πœ‰.(4.15) By virtue of (3.8) we have that limπ‘β†’βˆžπœ€π‘(π‘₯,πœ†)=0(4.16) uniformly with respect to π‘₯∈[0,πœ‹] and πœ† on compact sets. Using (3.5) we calculate +1πœ‘(π‘₯,πœ†)=ξ‚πœ‘(π‘₯,πœ†)ξ€œ2πœ‹π‘–πœƒπ‘ξ‚€ξ‚€πœ‘ξ‚Ξ¦ξ‚πœ‘(π‘₯,πœ†)(π‘₯,πœ‰)ξ…ž(π‘₯,πœ‰)βˆ’Ξ¦(π‘₯,πœ‰)ξ‚πœ‘ξ…žξ‚(π‘₯,πœ‰)+ξ‚πœ‘ξ…ž(π‘₯,πœ†)Ξ¦(π‘₯,πœ‰)ξ‚πœ‘(π‘₯,πœ‰)βˆ’πœ‘(π‘₯,πœ‰)Ξ¦(π‘₯,πœ‰)ξ‚ξ‚π‘‘πœ‰πœ†βˆ’πœ‰+πœ€π‘(π‘₯,πœ†).(4.17) This, in combination with (3.1), implies that 1ξ‚πœ‘(π‘₯,πœ†)=πœ‘(π‘₯,πœ†)+ξ€œ2πœ‹π‘–πœƒπ‘βŸ¨ξ‚πœ‘(π‘₯,πœ†),ξ‚πœ‘(π‘₯,πœ‰)βŸ©ξ‚Šπœ†βˆ’πœ‰π‘€(πœ‰)πœ‘(π‘₯,πœ‰)π‘‘πœ‰βˆ’πœ€π‘(π‘₯,πœ†),(4.18) where ξ‚Šξ‚‹π‘€(πœ†)=𝑀(πœ†)βˆ’π‘€(πœ†), since the terms with 𝑆(π‘₯,πœ‰) vanish by Cauchy’s theorem [21, page 85]. It follows from (3.14) that Resπœ‰=πœ†π‘˜π‘—βŸ¨ξ‚πœ‘(π‘₯,πœ†),ξ‚πœ‘(π‘₯,πœ‰)βŸ©ξ‚Šπœ†βˆ’πœ‰π‘€(πœ‰)πœ‘(π‘₯,πœ‰)=(βˆ’1)π‘—ξ«ξ‚πœ‘(π‘₯,πœ†),ξ‚πœ‘π‘˜π‘—ξ¬(π‘₯)π›Ύπ‘˜π‘—ξ€·πœ†βˆ’πœ†π‘˜π‘—ξ€Έπœ‘π‘˜π‘—(π‘₯).(4.19) Applying residue theorem to the integral in (4.18) and using (4.16) we obtain (4.10).
(2) Since1ξ‚΅1πœ†βˆ’πœ‡βˆ’1πœ†βˆ’πœ‰ξ‚Ά=1πœ‡βˆ’πœ‰(πœ†βˆ’πœ‰)(πœ‰βˆ’πœ‡),(4.20) we have by Cauchy’s integral formula π‘ƒπ‘—π‘˜(π‘₯,πœ†)βˆ’π‘ƒπ‘—π‘˜(π‘₯,πœ‡)=1πœ†βˆ’πœ‡ξ€œ2πœ‹π‘–πœƒ0π‘π‘ƒπ‘—π‘˜(π‘₯,πœ‰)(πœ†βˆ’πœ‰)(πœ‰βˆ’πœ‡)π‘‘πœ‰,𝑗,π‘˜=1,2,πœ†,πœ‡βˆˆintπœƒ0𝑁.(4.21) Acting in the same way as above and using (3.8) and (3.9), we obtain π‘ƒπ‘—π‘˜(π‘₯,πœ†)βˆ’π‘ƒπ‘—π‘˜(π‘₯,πœ‡)=1πœ†βˆ’πœ‡ξ€œ2πœ‹π‘–πœƒπ‘π‘ƒπ‘—π‘˜(π‘₯,πœ‰)(πœ†βˆ’πœ‰)(πœ‰βˆ’πœ‡)π‘‘πœ‰+πœ€π‘π‘—π‘˜(π‘₯,πœ†,πœ‡),(4.22) where limπ‘β†’βˆžπœ€π‘π‘—π‘˜(π‘₯,πœ†,πœ‡)=0, 𝑗,π‘˜=1,2.
It follows from (3.5) and (3.3) that𝑃11(π‘₯,πœ†)πœ‘ξ…ž(π‘₯,πœ†)βˆ’π‘ƒ21(π‘₯,πœ†)πœ‘(π‘₯,πœ†)=ξ‚πœ‘ξ…ž(𝑃π‘₯,πœ†),22(π‘₯,πœ†)πœ‘(π‘₯,πœ†)βˆ’π‘ƒ12(π‘₯,πœ†)πœ‘ξ…žβŽ‘βŽ’βŽ’βŽ£π‘¦π‘¦(π‘₯,πœ†)=ξ‚πœ‘(π‘₯,πœ†),(4.23)𝑃(π‘₯,πœ†)(π‘₯)ξ…žβŽ€βŽ₯βŽ₯⎦=ξ‚¬ξ‚ξ‚­βŽ‘βŽ’βŽ’βŽ£πœ‘πœ‘(π‘₯)𝑦(π‘₯),Ξ¦(π‘₯,πœ†)(π‘₯,πœ†)ξ…žβŽ€βŽ₯βŽ₯⎦⎑⎒⎒⎣ΦΦ(π‘₯,πœ†)βˆ’βŸ¨π‘¦(π‘₯),ξ‚πœ‘(π‘₯,πœ†)⟩(π‘₯,πœ†)ξ…žβŽ€βŽ₯βŽ₯⎦(π‘₯,πœ†),(4.24) for any 𝑦(π‘₯)∈𝐢1[0,πœ‹]. Using (4.22) and (4.24), we get 𝑃(π‘₯,πœ†)βˆ’π‘ƒ(π‘₯,πœ‡)βŽ‘βŽ’βŽ’βŽ£π‘¦πœ†βˆ’πœ‡π‘¦(π‘₯)ξ…ž(⎀βŽ₯βŽ₯⎦=1π‘₯)ξ€œ2πœ‹π‘–πœƒπ‘βŽ›βŽœβŽœβŽξ‚¬ξ‚ξ‚­βŽ‘βŽ’βŽ’βŽ£πœ‘π‘¦(π‘₯),Ξ¦(π‘₯,πœ‰)πœ‘(π‘₯,πœ‰)ξ…ž(⎀βŽ₯βŽ₯⎦⎑⎒⎒⎣Φπ‘₯,πœ‰)βˆ’βŸ¨π‘¦(π‘₯),ξ‚πœ‘(π‘₯,πœ‰)⟩Φ(π‘₯,πœ‰)ξ…ž(⎀βŽ₯βŽ₯βŽ¦βŽžβŽŸβŽŸβŽ Γ—π‘₯,πœ‰)π‘‘πœ‰(πœ†βˆ’πœ‰)(πœ‰βˆ’πœ‡)+πœ€0𝑁(π‘₯,πœ†,πœ‡),(4.25) where limπ‘β†’βˆžπœ€0𝑁(π‘₯,πœ†,πœ‡)=0. From (3.4) and (4.23), we get βŽ›βŽœβŽœβŽβŽ‘βŽ’βŽ’βŽ£det(𝑃(π‘₯,πœ†)βˆ’π‘ƒ(π‘₯,πœ‡))ξ‚πœ‘(π‘₯,πœ†)ξ‚πœ‘ξ…ž(⎀βŽ₯βŽ₯⎦,βŽ‘βŽ’βŽ’βŽ£πœ‘π‘₯,πœ†)πœ‘(π‘₯,πœ‡)ξ…ž(⎀βŽ₯βŽ₯⎦⎞⎟⎟⎠π‘₯,πœ‡)=βŸ¨πœ‘(π‘₯,πœ†),πœ‘(π‘₯,πœ‡)βŸ©βˆ’βŸ¨ξ‚πœ‘(π‘₯,πœ†),ξ‚πœ‘(π‘₯,πœ‡)⟩.(4.26) Hence, for 𝑦(π‘₯)=ξ‚πœ‘(π‘₯,πœ†), (4.25) gives βŸ¨πœ‘(π‘₯,πœ†),πœ‘(π‘₯,πœ‡)βŸ©βˆ’πœ†βˆ’πœ‡βŸ¨ξ‚πœ‘(π‘₯,πœ†),ξ‚πœ‘(π‘₯,πœ‡)⟩=1πœ†βˆ’πœ‡ξ€œ2πœ‹π‘–πœƒπ‘βŽ›βŽœβŽœβŽξ‚¬ξ‚ξ‚­ξ‚πœ‘(π‘₯,πœ†),Ξ¦(π‘₯,πœ‰)βŸ¨πœ‘(π‘₯,πœ‰),πœ‘(π‘₯,πœ‡)βŸ©βˆ’(πœ†βˆ’πœ‰)(πœ‰βˆ’πœ‡)βŸ¨ξ‚πœ‘(π‘₯,πœ†),ξ‚πœ‘(π‘₯,πœ‰)⟩⟨Φ(π‘₯,πœ‰),πœ‘(π‘₯,πœ‡)⟩⎞⎟⎟⎠(πœ†βˆ’πœ‰)(πœ‰βˆ’πœ‡)π‘‘πœ‰+πœ€1𝑁(π‘₯,πœ†,πœ‡),(4.27) where limπ‘β†’βˆžπœ€1𝑁(π‘₯,πœ†,πœ‡)=0. By virtue of (3.1), (3.14), and the residue theorem, we get (4.11).

It follows from the definition of 𝑃𝑛𝑖,π‘˜π‘—(π‘₯), 𝑃𝑛𝑖,π‘˜π‘—(π‘₯), and from (4.10) and (4.11) thatξ‚πœ‘π‘›π‘–(π‘₯)=πœ‘π‘›π‘–(π‘₯)+βˆžξ“π‘˜=0𝑃𝑛𝑖,π‘˜0(π‘₯)πœ‘π‘˜0(𝑃π‘₯)βˆ’π‘›π‘–,π‘˜1(π‘₯)πœ‘π‘˜1(,𝑃π‘₯)(4.28)𝑛𝑖,ℓ𝑗𝑃(π‘₯)βˆ’π‘›π‘–,ℓ𝑗(π‘₯)+βˆžξ“π‘˜=0𝑃𝑛𝑖,π‘˜0(π‘₯)π‘ƒπ‘˜0,ℓ𝑗𝑃(π‘₯)βˆ’π‘›π‘–,π‘˜1(π‘₯)π‘ƒπ‘˜1,ℓ𝑗(π‘₯)=0.(4.29) For each π‘₯∈[0,πœ‹], the relation (4.28) can be considered as a system of linear equations with respect to πœ‘π‘›π‘–(π‘₯), 𝑛β‰₯0, 𝑖=0,1. But the series in (4.28) converges only β€œwith brackets.” Therefore, it is not convenient to use (4.28) as a main equation of the inverse problem. Below we will transfer (4.28) to a linear equations in corresponding Banach space of sequences (see (4.38) or (4.40)).

Let 𝑉 be a set of indices 𝑒=(𝑛,𝑖), 𝑛β‰₯0, 𝑖=0,1. For each fixed π‘₯∈[0,πœ‹], we define the vectorπœ“(π‘₯)=[πœ“π‘’(π‘₯)]π‘’βˆˆπ‘‰=βŽ‘βŽ’βŽ’βŽ£πœ“π‘›0πœ“(π‘₯)𝑛1⎀βŽ₯βŽ₯⎦(π‘₯)𝑛β‰₯0=ξ€Ίπœ“00,πœ“01,πœ“10,πœ“11ξ€»,…𝑇(4.30) by the formulaeπœ“π‘›0(π‘₯)=πœ’π‘›ξ€·πœ‘π‘›0(π‘₯)βˆ’πœ‘π‘›1ξ€Έ(π‘₯),πœ“π‘›1(π‘₯)=πœ‘π‘›1(π‘₯),(4.31) whereπœ’π‘›=ξ‚»πœ‰π‘›βˆ’1,πœ‰π‘›β‰ 0,0,πœ‰π‘›=0.(4.32) We also define the block matrix𝐻𝐻(π‘₯)=𝑒,𝑣(π‘₯)𝑒,π‘£βˆˆπ‘‰=βŽ‘βŽ’βŽ’βŽ£π»π‘›0,π‘˜0(π‘₯)𝐻𝑛0,π‘˜1𝐻(π‘₯)𝑛1,π‘˜0(π‘₯)𝐻𝑛1,π‘˜1⎀βŽ₯βŽ₯⎦(π‘₯)𝑛,π‘˜β‰₯0,𝑒=(𝑛,𝑖),𝑣=(π‘˜,𝑗),(4.33) by the formulae𝐻𝑛0,π‘˜0(π‘₯)=πœ‰π‘˜πœ’π‘›ξ€·π‘ƒπ‘›0,π‘˜0(π‘₯)βˆ’π‘ƒπ‘›1,π‘˜0ξ€Έ,𝐻(π‘₯)𝑛0,π‘˜1(π‘₯)=πœ’π‘›ξ€·π‘ƒπ‘›0,π‘˜0(π‘₯)βˆ’π‘ƒπ‘›0,π‘˜1(π‘₯)βˆ’π‘ƒπ‘›1,π‘˜0(π‘₯)+𝑃𝑛1,π‘˜1(ξ€Έ,𝐻π‘₯)𝑛1,π‘˜0(π‘₯)=πœ‰π‘˜π‘ƒπ‘›1,π‘˜0𝐻(π‘₯),𝑛1,π‘˜1(π‘₯)=𝑃𝑛1,π‘˜0(π‘₯)βˆ’π‘ƒπ‘›1,π‘˜1(π‘₯).(4.34) Analogously we define ξ‚πœ“(π‘₯), 𝐻(π‘₯) by replacing in the previous definitions, πœ‘π‘›π‘–(π‘₯) by ξ‚πœ‘π‘›π‘–(π‘₯) and 𝑃𝑛𝑖,π‘˜π‘—(π‘₯) by 𝑃𝑛𝑖,π‘˜π‘—(π‘₯). It follows from (4.5)–(4.9) that||πœ“(𝜈)𝑛𝑖||(π‘₯)≀𝐢(𝑛+1)𝜈,||𝐻𝑛𝑖,π‘˜π‘—||≀(π‘₯)πΆπœ‰π‘˜||||.π‘›βˆ’π‘˜+1(4.35) Similarly||ξ‚πœ“(𝜈)𝑛𝑖||(π‘₯)≀𝐢(𝑛+1)𝜈,||𝐻𝑛𝑖,π‘˜π‘—||≀(π‘₯)πΆπœ‰π‘˜||||.π‘›βˆ’π‘˜+1(4.36)

Let us consider the Banach space π‘™βˆž(𝑉) of bounded sequences 𝛼=[𝛼𝑒]π‘’βˆˆπ‘‰ with the norm β€–π›Όβ€–βˆž=supπ‘’βˆˆπ‘‰|𝛼𝑒