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

Authors: I. Dehghani Tazehkand; A. Jodayree Akbarfam
Journal: ISRN Mathematical Analysis
 (2011)

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## 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.


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## Body

## 1. Introduction

We consider the Sturm-Liouville boundary value problemL=L(q(x),h,H,H1,H2):(1.1)ly:=-y′′+q(x)y=λy,0<x<π,(1.2)U(y):=y′(0)-hy(0)=0,(1.3)V(y):=(λ-H1)y′(π)+(λH-H2)y(π)=0,
where the potential q(x)∈L2(0,π) is a real-valued function, h,H,H1,H2∈ℝ, r:=HH1-H2>0 and λ 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 r>0. In the case r<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 q 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 cosλn 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 recoveringL 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

Letφ(x,λ), ψ(x,λ), and S(x,λ) be the solutions of (1.1) satisfying the initial conditions(2.1)φ(0,λ)=1,φ′(0,λ)=h,ψ(π,λ)=λ-H1,ψ′(π,λ)=-λH+H2.(2.2)S(0,λ)=0,S′(0,λ)=1.
We define(2.3)Δ(λ):=〈ψ(x,λ),φ(x,λ)〉,
where(2.4)〈y(x),z(x)〉:=y(x)z′(x)-y′(x)z(x)
is the Wronskian of y and z. We note that if y(x,λ) and z(x,μ) are solutions of ℓy=λy and ℓz=μz, respectively, then(2.5)ddx〈y,z〉=(λ-μ)yz.
By virtue of the Liouville’s formula for the Wronskian (see [19, page 83]), Δ(λ) does not depend on x. The function Δ(λ) is called the characteristic function of L. Substituting x=0 and x=π into (2.3) we get(2.6)Δ(λ)=V(φ)=-U(ψ).
The function Δ(λ) is entire in λ and it has an at most countable set of zeroes {λn}.Lemma 2.1.
The zeros{λn} of the characteristic function coincide with the eigenvalues of the boundary value problem L. The functions φ(x,λn) and ψ(x,λn) are eigenfunctions, and there exists a sequence {kn} such that
(2.7)ψ(x,λn)=knφ(x,λn),kn≠0.Proof.
Letλ∘ be a zero of Δ(λ). Then by virtue of (2.3) we have ψ(x,λ∘)=k∘φ(x,λ∘) and k∘=ψ(0,λ∘). Since HH1-H2>0, we have k0≠0, and the functions φ(x,λ∘) and ψ(x,λ∘) satisfy the boundary conditions (1.2) and (1.3). Hence λ∘ is an eigenvalue, and φ(x,λ∘), ψ(x,λ∘) are eigenfunctions related to λ∘.
Conversely, letλ∘ be an eigenvalue of L, and let y∘ be a corresponding eigenfunction. Then U(y∘)=V(y∘)=0. Clearly y∘(0)≠0, otherwise, y∘′(0)=0 and by uniqueness theorem for (1.1) (see [19, Chapter  1]), y∘≡0. Without loss of generality, we put y∘(0)=1. Then y∘′(0)=h. Hence y∘(x)=φ(x,λ∘). Thus, from (2.6), Δ(λ∘)=V(φ(x,λ∘))=V(y∘(x))=0 is obtained.Let the inner product in the Hilbert spaceℋ=L2(0,π)⊕ℂ be defined by(2.8)(F,G):=∫oπF1(x)G1(x)¯dx+1rF2G2¯,
where(2.9)F=(F1(x)F2),G=(G1(x)G2)∈H.
We define an operator T acting in ℋ such that(2.10)T(F):=(-F1′′(x)+q(x)F1(x)H1F1′(π)+H2F1(π)),
with(2.11)D(T)={F∈H∣F1(x),F1′(x)∈AC[0,π],lF1∈L2(0,π),F1′(0)-hF1(0)=0,F2=F1′(π)+HF1(π)}.
It is easy to show that {λn} are the eigenvalues of T and(2.12)Φn:=(φ(x,λn)φ′(π,λn)+Hφ(π,λn))
are eigenelements of T.Lemma 2.2.
The eigenvalues{λn} and the eigenfunctions φ(x,λn) and ψ(x,λn) are real.Proof.
Letλ∘=u+iv, v≠0 be a nonreal eigenvalue with an eigenfunction y∘(x). Since q(x), h, H, H1, and H2 are real, we get that λ∘¯ is also the eigenvalue with the eigenfunction y∘(x)¯≠0. By vitue of (2.5) we get
(2.13)ddx〈y(x),y∘(x)¯〉=(λ∘-λ∘¯)y∘(x)y∘(x)¯,
and hence with the help of (1.2)
(2.14)(λ∘-λ∘¯)∫0πy∘(x)y∘(x)¯=y∘(π)y∘′(π)¯-y∘′(π)y∘(π)¯=〈y(x),y∘(x)¯〉|x=π.
Also, by virtue of (1.3) we can write
(2.15)(λ∘-λ∘¯)|y∘′(π)+Hy∘(π)|2=(H2-HH1)〈y(x),y∘(x)¯〉|x=π.
Now, from (2.14) and (2.15) we get
(2.16)(λ∘-λ∘¯)(∫0π|y∘(x)|2dx+|y∘′(π)+Hy∘(π)|2r)=0,
hence y∘(x)≡0, which is a contradiction. Thus, all eigenvalues {λn} are real, and consequently the eigenfunctions φ(x,λn) and ψ(x,λn) are real too.Lemma 2.3.
The eigenelements ofT related to different eigenvalues are orthogonal in ℋ.Proof.
Let(2.17)Yn:=(yn(x)zn),Ym:=(ym(x)zm)
be eigenelements of T related to eigenvalues λn,λm(λn≠λm), respectively. Since T(Yn)=λnYn and T(Ym)=λmYm, we have
(2.18)λnzn=λn(yn′(π)+Hyn(π))=H1yn′(π)+H2yn(π),λmzm=λm(ym′(π)+Hym(π))=H1ym′(π)+H2ym(π),
and hence
(2.19)(λn-λm)znzm¯=r(yn′(π)ym(π)¯-yn(π)ym′(π)¯).
Also, since ℓyn=λnyn and ℓym=λmym, from (2.5) we get
(2.20)(λn-λm)∫0πyn(x)ym(x)¯dx=yn(π)ym′(π)¯-yn′(π)ym(π)¯.
Thus,
(2.21)(Yn,Ym)=∫0πyn(x)ym(x)dx+1rznzm¯=0.Here we define norming constants by(2.22)γn:=‖Φn‖2=∫0πφ2(x,λn)dx+1r(φ′(π,λn)+Hφ(π,λn))2.
The numbers {λn,γn}n≥0 are called the spectral data of the problem (1.1)–(1.3).Lemma 2.4.
The following relation holds:(2.23)Δ̇(λn)=-knγn,
where the numbers kn are defined by (2.7) and Δ̇(λ)=(d/dλ)Δ(λ).Proof.
By virtue of (2.5) we have
(2.24)ddx〈ψ(x,λ),φ(x,λn)〉=(λ-λn)ψ(x,λ)φ(x,λn),
and hence with the help of (2.6) and (2.7),
(2.25)∫0πψ(x,λ)φ(x,λn)=〈ψ(x,λ),φ(x,λn)〉|0πλ-λn=-rkn-Δ(λ)λ-λn.
For λ→λn, this yields
(2.26)Δ̇(λn)=-kn∫0πφ2(x,λn)dx-rkn=-kn(γn-(φ′(π,λn)+Hφ(x,λn))2r)-rkn=-kn(γn-(ψ′(π,λn)+Hψ(x,λn))2kn2r)-rkn=-kn(γn-rkn2)-rkn=-knγn.Remark 2.5.
From this lemma we getΔ̇(λn)≠0. Thus, the eigenvalues of the boundary value problem (1.1)–(1.3) are simple.Lemma 2.6.
For,|ρ|→∞, the following asymptotic formulae hold:
(2.27)φ(x,λ)=cos⁡ρx+O(e|τ|x|ρ|)=O(e|τ|x),φ′(x,λ)=-ρsin⁡ρx+O(e|τ|x)=O(|ρ|e|τ|x),(2.28)ψ(x,λ)=ρ2cos⁡ρ(π-x)+O(|ρ|e|τ|(π-x))=O(|ρ2|e|τ|(π-x)),ψ′(x,λ)=ρ3sin⁡ρ(π-x)+O(|ρ2|e|τ|(π-x))=O(|ρ3|e|τ|(π-x)),
uniformly with respect to x∈[0,π]. Here and in the sequel λ=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
(2.29)ψ(x,λ)=(λ-H1)cos⁡ρ(π-x)+(λH-H2)sin⁡ρ(π-x)ρ+∫xπsin⁡ρ(t-x)ρq(t)ψ(t,λ)dt.
Sinceψ(t,λ) satisfies (1.1), we have(2.30)q(t)ψ(t,λ)=ψ′′(t,λ)+ρ2ψ(t,λ).
Substituting this in the following integral
(2.31)∫xπsin⁡ρ(t-x)ρq(t)ψ(t,λ)dt,
and twice integrating by parts the term involving ψ′′(t,λ), we obtain (2.29).
Differentiating (2.29), we calculate(2.32)ψ′(x,λ)=ρ(λ-H1)sin⁡ρ(π-x)+(H2-λH)cos⁡ρ(π-x)-∫xπcos⁡ρ(t-x)q(t)ψ(t,λ)dt.
In (2.29), we put ψ(x,λ)=e|τ|(π-x)f(x,λ). Then
(2.33)f(x,λ)=(λ-H1)cos⁡ρ(π-x)e-|τ|(π-x)+(λH-H2)sin⁡ρ(π-x)ρe-|τ|(π-x)+∫xπsin⁡ρ(t-x)ρe-|τ|(t-x)q(t)f(t,λ)dt.
Let μ(λ)=max0≤x≤π|f(x,λ)|. Then using the inequalities
(2.34)|cos⁡ρ(π-x)|≤e|τ|(π-x),|sin⁡ρ(π-x)|≤e|τ|(π-x),
we obtain
(2.35)μ(λ)≤|ρ|2(1+|H||ρ|+|H1||ρ|2+|H2||ρ|3)+1|ρ|μ(λ)∫0π|q(t)|d.
For sufficiently  larg |ρ|, this gives
(2.36)μ(λ)≤C|ρ|2(1-∫0πq(t)dt|ρ|)-1.
Hence |f(x,λ)|≤μ(λ)=O(|ρ|2), as |ρ|→∞, and therefore ψ(x,λ)=O(|ρ|2e|τ|(π-x)), uniformly with respect to x∈[0,π] as |ρ|→∞. Substituting this estimate into right-hand sides of (2.29) and (2.32), we get (2.28).Theorem 2.7.
The boundary value problemL has a countable set of eigenvalues {λn}n≥0. Moreover, for n≥0,
(2.37)ρn:=λn=n-1+ωπn+ζnn,{ζn}∈l2,(2.38)φ(x,λn)=cos⁡(n-1)x+ξn(x)n,|ξn(x)|≤C,(2.39)γn=π2+ζn′n,{ζn′}∈l2,
where
(2.40)ω=h+H+12∫0πq(t)dt.Proof.
(1) By virtue of the following asymptotic formulae (see [20, page 7])
(2.41)φ(x,λ)=cos⁡ρx+q1(x)sin⁡ρxρ+12ρ∫0xq(t)sin⁡ρ(x-2t)dt+O(e|τ|xρ2),φ′(x,λ)=-ρsin⁡ρx+q1(x)cos⁡ρx+12∫0xq(t)cos⁡ρ(x-2t)dt+O(e|τ|xρ),
where
(2.42)q1(x)=h+12∫0xq(t)dt,
and using (2.6) we get
(2.43)Δ(λ)=-ρ3sin⁡ρπ+ωρ2cos⁡ρπ+ρ2I(ρ),
where
(2.44)I(ρ)=12∫0πq(t)cos⁡ρ(π-2t)dt+O(1ρe|τ|π).
Denote Gδ={ρ:|ρ-k|≥δ,k=0,±1,±2,…}, δ>0. Using (2.43) we get for ρ∈Gδ(2.45)Δ(λ)=-ρ3sin⁡ρπ(1+O(1ρ)).
Now, since (see [20, page 6])
(2.46)|sin⁡ρπ|≥Cδe|τ|π,ρ∈Gδ,
we have
(2.47)|Δ(λ)|≥Cδ|ρ|3e|τ|π,ρ∈Gδ,|ρ|≥ρ*,
for sufficiently large ρ*=ρ*(δ).
LetΓn={λ:|λ|=(n-1/2)2}. By virtue of (2.43),(2.48)Δ(λ)=f(λ)+g(λ),f(λ)=-ρ3sin⁡ρπ,|g(λ)|≤C|ρ|2e|τ|π.
It follows from (2.46) that |f(λ)|>|g(λ)|, λ∈Γn, for sufficiently large n(n≥n*). Hence by Rouché’s theorem [21, page 125], the number of zeros of Δ(λ) inside Γn coincides with the number of zeros of f(λ), that is, it equals n+1. Thus, in the circle |λ|<(n-1/2)2 there exist exactly n+1 eigenvalues of L:λ0,…,λn. Analogously, by using Rouché’s theorem one can prove that for sufficiently large vales of n, every circle σn(δ)={ρ:|ρ-(n-1)|≤δ} contains exactly one zero of Δ(ρ2), namely, ρn=λn. Since δ>0 is arbitrary, we must have
(2.49)ρn=n-1+ɛn,ɛn=o(1),n⟶∞.
Since ρn are zeros of Δ(ρ2), from (2.43) we get
(2.50)Δ(ρn2)=-(n-1+ɛn)3sin⁡(n-1+ɛn)π+ω(n-1+ɛn)2cos⁡(n-1+ɛn)π+(n-1+ɛn)2I(n-1+ɛn)=0,
and consequently
(2.51)-nsin⁡ɛnπ+ωcos⁡ɛnπ+ϑn=0,
where ϑn=(1-ɛn)sinɛnπ+(-1)n-1I(n-1+ɛn). Hence sinɛnπ=O(1/n), that is, ɛn=O(1/n). Thus, {ϑn}∈l2. Using (2.51) once more we get more precisely ɛn=ω/πn+ζn/n, where {ζn}∈l2.
(2) Substituting (2.37) into (2.41), we get (2.38) where(2.52)ξn(x)=(q1(x)-xωπ-xζn)sin⁡(n-1)x+12∫0xq(t)sin⁡(n-1)(x-2t)dt+O(1n).
Therefore, |ξn(x)|≤C, and (2.38) is proved.
(3) From (2.38) we calculate(2.53)∫0πφ2(x,λn)dx=∫0πcos⁡2(n-1)xdx+2∫0πcos⁡(n-1)xξn(x)ndx+∫0πξn2(x)n2dx=π2+ζn′n,{ζn′}∈l2.
Also, using (1.3), (2.37), and (2.38) we obtain
(2.54)1r(φ′(π,λn)+Hφ(π,λn))2=(φ(π,λn)H1-λn)2=O(1n4).
Thus,
(2.55)γn=π2+ζn′n,{ζn′}∈l2,
and Theorem 2.7 is proved.Theorem 2.8.
The specification of the spectrum{λn}n≥0 uniquely determines the characteristic function Δ(λ) by the formula
(2.56)Δ(λ)=π(λ-λ0)(λ1-λ)∏n=2∞λn-λ(n-1)2.Proof.
It follows from (2.6), (2.27), and (2.28) that Δ(λ) is an entire function of λ of order 1/2, and hence by Hadamard’s factorization theorem [21, page 289], Δ(λ) is uniquely determined up to a multiplicative constant by its zeros:
(2.57)Δ(λ)=C∏n=0∞(1-λλn).
The case Δ(0)=0 requires minor modifications. We consider the function
(2.58)Δ̃(λ):=-ρ3sin⁡ρπ=-λ2π∏n=1∞(1-λn2).
Then
(2.59)Δ(λ)Δ̃(λ)=C(λ-λ0)(λ1-λ)πλ0λ1λ2∏n=1∞n2λn+1∏n=1∞(1+λn+1-n2n2-λ).
With the help of (2.37) and (2.43), we calculate
(2.60)lim⁡λ→-∞Δ(λ)Δ̃(λ)=1,lim⁡λ→-∞∏n=1∞(1+λn+1-n2n2-λ)=1,
and hence
(2.61)C=-πλ0λ1∏n=1∞λn+1n2.
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 problemL0=L0(q(x),H,H1,H2) for (1.1) with the boundary conditions y(0)=V(y)=0. The eigenvalues {λn0}n≥0 of L0 are simple and coincide with the zeros of the characteristic function Δ0(λ):=ψ(0,λ)=V(S) and(2.62)Δ0(λ)=(λ-λ0)∏n=1∞λn-λ(n-1/2)2,(2.63)ρn0:=λn0=n-12+ω0πn+ζn0n,{ζn0}∈l2,
where
(2.64)ω0=H+12∫0πq(t)dt.


## 3. Inverse Problems

In this section, we study three inverse problems of recoveringL 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

LetΦ(x,λ) be the solution of (1.1) under the conditions U(Φ)=1 and V(Φ)=0. We set M(λ):=Φ(0,λ). The functions Φ(x,λ) and M(λ) 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]). Clearly(3.1)Φ(x,λ)=-ψ(x,λ)Δ(λ)=S(x,λ)+M(λ)φ(x,λ),(3.2)M(λ)=-Δ0(λ)Δ(λ),(3.3)〈φ(x,λ),Φ(x,λ)〉≡1.
Since Δ(λ) and Δ0(λ) have no common zeros, it follows from (3.2) that M(λ) is a meromorphic function with poles {λn}n≥0 and zeros {λn0}n≥0. We consider the following inverse problem.Inverse Problem 1
Given the Weyl functionM(λ), construct q(x), h, H, H1, and H2.Let us prove the uniqueness theorem for the solution of the Inverse Problem1. For this purpose we agree that together with L we consider a boundary value problem L̃ of the same form but with different coefficients q̃(x), h̃, H̃, H̃1, and H̃2. Everywhere below if a certain symbol α denotes an object related to L, then the corresponding symbol α̃ with tilde denotes the analogous object related to L̃.Theorem 3.1.
IfM(λ)=M̃(λ), then q(x)=q̃(x) a.e. on (0,π), h=h̃, H=H̃, H1=H̃1 and H2=H̃2. Thus, the specification of the Weyl function M(λ) uniquely determines L.Proof.
Let us define the matrixP(x,λ)=[Pjk(x,λ)]j,k=1,2 by the formula
(3.4)P(x,λ)[φ̃(x,λ)Φ̃(x,λ)φ̃′(x,λ)Φ̃′(x,λ)]=[φ(x,λ)Φ(x,λ)φ′(x,λ)Φ′(x,λ)].
Using (3.3) and (3.4) we calculate for j=1,2:
(3.5)Pj1(x,λ)=φ(j-1)(x,λ)Φ̃′(x,λ)-Φ(j-1)(x,λ)φ̃′(x,λ),Pj2(x,λ)=Φ(j-1)(x,λ)φ̃(x,λ)-φ(j-1)(x,λ)Φ̃(x,λ),(3.6)φ(x,λ)=P11(x,λ)φ̃(x,λ)+P12(x,λ)φ̃′(x,λ),Φ(x,λ)=P11(x,λ)Φ̃(x,λ)+P12(x,λ)Φ̃′(x,λ).
It follows from (3.1), (3.3), and (3.5) that
(3.7)P11(x,λ)=1+1Δ(λ)(ψ(x,λ)(φ̃′(x,λ)-φ′(x,λ))-φ(x,λ)(ψ̃′(x,λ)-ψ′(x,λ))),P12(x,λ)=1Δ(λ)(φ(x,λ)ψ̃(x,λ)-ψ(x,λ)φ̃(x,λ)).
By virtue of (2.27), (2.28), and (2.47), this yields
(3.8)|P11(x,λ)-1|≤Cδ|ρ|,|P12(x,λ)|≤Cδ|ρ|,ρ∈Gδ,|ρ|≥ρ*.
Uniformly with respect to x∈[0,π]. Similarly, we have
(3.9)|P22(x,λ)-1|≤Cδ|ρ|,|P21(x,λ)|≤Cδ,ρ∈Gδ,|ρ|≥ρ*,
Uniformly with respect to x∈[0,π]. On the other hand according to (3.1) and (3.5),
(3.10)P11(x,λ)=φ(x,λ)S̃′(x,λ)-S(x,λ)φ̃′(x,λ)+(M̃(λ)-M(λ))φ(x,λ)φ̃′(x,λ),P12(x,λ)=S(x,λ)φ̃(x,λ)-φ(x,λ)S̃(x,λ)+(M(λ)-M̃(λ))φ(x,λ)φ̃(x,λ).
Since M(λ)≡M̃(λ), it follows that for each fixed x, the functions P11(x,λ) and P12(x,λ) are entire in λ. With the help of (3.8), this yields P11(x,λ)≡1, P12(x,λ)≡0. Substituting into (3.6), we get φ(x,λ)≡φ̃(x,λ),  Φ(x,λ)≡Φ̃(x,λ) for all x and λ. Hence from (1.1) and (2.1) we get q(x)=q̃(x) a.e. on (0,π), h=h̃, H=H̃, H1=H̃1, and H2=H̃2. Consequently, L=L̃.


### 3.2. The Inverse Problem from the Spectral Data

Let{λn}n≥0 and {γn}n≥0 be the eigenvalues and norming constants of L, respectively. We consider the following inverse problem.Inverse Problem 2
Given the spectral data{λn,γn}n≥0, construct q(x),h,H,H1, and H2.Lemma 3.2.
The following representation holds:(3.11)M(λ)=∑n=0∞1γn(λ-λn).Proof.
Consider the contour integral(3.12)JN(λ)=12πi∫ΓNM(μ)λ-μdμ,λ∈intΓN,
where the contour ΓN is assumed to have the counterclockwise circuit. Since Δ0(λ)=ψ(0,λ), it follows from (2.28) that |Δ0(λ)|≤C|ρ|2e|τ|π. Then, using (3.1) and (2.47), we get for sufficiently large ρ*>0,
(3.13)|M(λ)|≤Cδ|ρ|,ρ∈Gδ,|ρ|≥ρ*.
Moreover, using (3.2) and (2.23), we calculate
(3.14)Resλ=λnM(λ)=-Δ0(λn)Δ̇(λn)=-knΔ̇(λn)=1γn.
In view of (3.13), limN→∞JN(λ)=0. By virtue of (3.14) and residue theorem [21, page 112], we have
(3.15)JN(λ)=-M(λ)+∑n=0N1γn(λ-λn),
and consequently (3.11) is proved.Let us prove a uniqueness theorem for the solution of Inverse Problem2.Theorem 3.3.
Ifλn=λ̃n and γn=γ̃n for all n≥0, then q(x)=q̃(x) a.e. on (0,π), h=h̃, H=H̃, H1=H̃1 and H2=H̃2. Thus, the specification of the spectral data {λn,γn}n≥0, uniquely determines L.Proof.
Ifλn=λ̃n and γn=γ̃n for all n≥0, then from Lemma 3.2, we get M(λ)=M̃(λ). Hence By virtue of Theorem 3.1, this implies q(x)=q̃(x) a.e. on (0,π), h=h̃, H=H̃, H1=H̃1 and H2=H̃2. Thus, Theorem 3.3 is proved.Remark 3.4.
By virtue of (3.11), the specification of the Weyl function M(λ) is equivalent to the specification of the spectral data {λn,γn}n≥0, that is, the Inverse Problem 1 is equivalent to the Inverse Problem 2.


### 3.3. The Inverse Problem from Two Spectra

Let{λn}n≥0 and {λn0}n≥0 be the eigenvalues of the problems L and L0, respectively. We consider the following inverse problem.Inverse Problem 3
Given two spectra{λn,λn0}n≥0, construct q(x), h, H, H1, and H2.Let us prove a uniqueness theorem for the solution of Inverse Problem3.Theorem 3.5.
Ifλn=λ̃n and λn0=λ̃n0, then q(x)=q̃(x) a.e. on (0,π), h=h̃, H=H̃, H1=H̃1, and H2=H̃2. Thus, the specification of two spectra {λn,λn0}n≥0, uniquely determines L.Proof.
According to Lemma2.1 and Remark 2.9 the sets {λn}n≥0 and {λn0}n≥0 coincide with the set of zeros of the functions Δ(λ) and Δ0(λ), respectively. Using (2.56) and (2.62), we get Δ(λ)=Δ̃(λ) and Δ0(λ)=Δ̃0(λ). Together with (3.2) this yields M(λ)=M̃(λ). By Theorem 3.1 we get q(x)=q̃(x) a.e. on (0,π), h=h̃, H=H̃, H1=H̃1, and H2=H̃2.Remark 3.6.
It follows from Theorems3.1 and 3.5 that the specification of Weyl function M(λ) is equivalent  to the specification of two spectra {λn,λn0}n≥0, that is, the Inverse Problem 1 is equivalent to the Inverse Problem 3.


## 3.1. The Inverse Problem from the Weyl Function

LetΦ(x,λ) be the solution of (1.1) under the conditions U(Φ)=1 and V(Φ)=0. We set M(λ):=Φ(0,λ). The functions Φ(x,λ) and M(λ) 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]). Clearly(3.1)Φ(x,λ)=-ψ(x,λ)Δ(λ)=S(x,λ)+M(λ)φ(x,λ),(3.2)M(λ)=-Δ0(λ)Δ(λ),(3.3)〈φ(x,λ),Φ(x,λ)〉≡1.
Since Δ(λ) and Δ0(λ) have no common zeros, it follows from (3.2) that M(λ) is a meromorphic function with poles {λn}n≥0 and zeros {λn0}n≥0. We consider the following inverse problem.Inverse Problem 1
Given the Weyl functionM(λ), construct q(x), h, H, H1, and H2.Let us prove the uniqueness theorem for the solution of the Inverse Problem1. For this purpose we agree that together with L we consider a boundary value problem L̃ of the same form but with different coefficients q̃(x), h̃, H̃, H̃1, and H̃2. Everywhere below if a certain symbol α denotes an object related to L, then the corresponding symbol α̃ with tilde denotes the analogous object related to L̃.Theorem 3.1.
IfM(λ)=M̃(λ), then q(x)=q̃(x) a.e. on (0,π), h=h̃, H=H̃, H1=H̃1 and H2=H̃2. Thus, the specification of the Weyl function M(λ) uniquely determines L.Proof.
Let us define the matrixP(x,λ)=[Pjk(x,λ)]j,k=1,2 by the formula
(3.4)P(x,λ)[φ̃(x,λ)Φ̃(x,λ)φ̃′(x,λ)Φ̃′(x,λ)]=[φ(x,λ)Φ(x,λ)φ′(x,λ)Φ′(x,λ)].
Using (3.3) and (3.4) we calculate for j=1,2:
(3.5)Pj1(x,λ)=φ(j-1)(x,λ)Φ̃′(x,λ)-Φ(j-1)(x,λ)φ̃′(x,λ),Pj2(x,λ)=Φ(j-1)(x,λ)φ̃(x,λ)-φ(j-1)(x,λ)Φ̃(x,λ),(3.6)φ(x,λ)=P11(x,λ)φ̃(x,λ)+P12(x,λ)φ̃′(x,λ),Φ(x,λ)=P11(x,λ)Φ̃(x,λ)+P12(x,λ)Φ̃′(x,λ).
It follows from (3.1), (3.3), and (3.5) that
(3.7)P11(x,λ)=1+1Δ(λ)(ψ(x,λ)(φ̃′(x,λ)-φ′(x,λ))-φ(x,λ)(ψ̃′(x,λ)-ψ′(x,λ))),P12(x,λ)=1Δ(λ)(φ(x,λ)ψ̃(x,λ)-ψ(x,λ)φ̃(x,λ)).
By virtue of (2.27), (2.28), and (2.47), this yields
(3.8)|P11(x,λ)-1|≤Cδ|ρ|,|P12(x,λ)|≤Cδ|ρ|,ρ∈Gδ,|ρ|≥ρ*.
Uniformly with respect to x∈[0,π]. Similarly, we have
(3.9)|P22(x,λ)-1|≤Cδ|ρ|,|P21(x,λ)|≤Cδ,ρ∈Gδ,|ρ|≥ρ*,
Uniformly with respect to x∈[0,π]. On the other hand according to (3.1) and (3.5),
(3.10)P11(x,λ)=φ(x,λ)S̃′(x,λ)-S(x,λ)φ̃′(x,λ)+(M̃(λ)-M(λ))φ(x,λ)φ̃′(x,λ),P12(x,λ)=S(x,λ)φ̃(x,λ)-φ(x,λ)S̃(x,λ)+(M(λ)-M̃(λ))φ(x,λ)φ̃(x,λ).
Since M(λ)≡M̃(λ), it follows that for each fixed x, the functions P11(x,λ) and P12(x,λ) are entire in λ. With the help of (3.8), this yields P11(x,λ)≡1, P12(x,λ)≡0. Substituting into (3.6), we get φ(x,λ)≡φ̃(x,λ),  Φ(x,λ)≡Φ̃(x,λ) for all x and λ. Hence from (1.1) and (2.1) we get q(x)=q̃(x) a.e. on (0,π), h=h̃, H=H̃, H1=H̃1, and H2=H̃2. Consequently, L=L̃.


## 3.2. The Inverse Problem from the Spectral Data

Let{λn}n≥0 and {γn}n≥0 be the eigenvalues and norming constants of L, respectively. We consider the following inverse problem.Inverse Problem 2
Given the spectral data{λn,γn}n≥0, construct q(x),h,H,H1, and H2.Lemma 3.2.
The following representation holds:(3.11)M(λ)=∑n=0∞1γn(λ-λn).Proof.
Consider the contour integral(3.12)JN(λ)=12πi∫ΓNM(μ)λ-μdμ,λ∈intΓN,
where the contour ΓN is assumed to have the counterclockwise circuit. Since Δ0(λ)=ψ(0,λ), it follows from (2.28) that |Δ0(λ)|≤C|ρ|2e|τ|π. Then, using (3.1) and (2.47), we get for sufficiently large ρ*>0,
(3.13)|M(λ)|≤Cδ|ρ|,ρ∈Gδ,|ρ|≥ρ*.
Moreover, using (3.2) and (2.23), we calculate
(3.14)Resλ=λnM(λ)=-Δ0(λn)Δ̇(λn)=-knΔ̇(λn)=1γn.
In view of (3.13), limN→∞JN(λ)=0. By virtue of (3.14) and residue theorem [21, page 112], we have
(3.15)JN(λ)=-M(λ)+∑n=0N1γn(λ-λn),
and consequently (3.11) is proved.Let us prove a uniqueness theorem for the solution of Inverse Problem2.Theorem 3.3.
Ifλn=λ̃n and γn=γ̃n for all n≥0, then q(x)=q̃(x) a.e. on (0,π), h=h̃, H=H̃, H1=H̃1 and H2=H̃2. Thus, the specification of the spectral data {λn,γn}n≥0, uniquely determines L.Proof.
Ifλn=λ̃n and γn=γ̃n for all n≥0, then from Lemma 3.2, we get M(λ)=M̃(λ). Hence By virtue of Theorem 3.1, this implies q(x)=q̃(x) a.e. on (0,π), h=h̃, H=H̃, H1=H̃1 and H2=H̃2. Thus, Theorem 3.3 is proved.Remark 3.4.
By virtue of (3.11), the specification of the Weyl function M(λ) is equivalent to the specification of the spectral data {λn,γn}n≥0, that is, the Inverse Problem 1 is equivalent to the Inverse Problem 2.


## 3.3. The Inverse Problem from Two Spectra

Let{λn}n≥0 and {λn0}n≥0 be the eigenvalues of the problems L and L0, respectively. We consider the following inverse problem.Inverse Problem 3
Given two spectra{λn,λn0}n≥0, construct q(x), h, H, H1, and H2.Let us prove a uniqueness theorem for the solution of Inverse Problem3.Theorem 3.5.
Ifλn=λ̃n and λn0=λ̃n0, then q(x)=q̃(x) a.e. on (0,π), h=h̃, H=H̃, H1=H̃1, and H2=H̃2. Thus, the specification of two spectra {λn,λn0}n≥0, uniquely determines L.Proof.
According to Lemma2.1 and Remark 2.9 the sets {λn}n≥0 and {λn0}n≥0 coincide with the set of zeros of the functions Δ(λ) and Δ0(λ), respectively. Using (2.56) and (2.62), we get Δ(λ)=Δ̃(λ) and Δ0(λ)=Δ̃0(λ). Together with (3.2) this yields M(λ)=M̃(λ). By Theorem 3.1 we get q(x)=q̃(x) a.e. on (0,π), h=h̃, H=H̃, H1=H̃1, and H2=H̃2.Remark 3.6.
It follows from Theorems3.1 and 3.5 that the specification of Weyl function M(λ) is equivalent  to the specification of two spectra {λn,λn0}n≥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 recoveringL from the given spectral data {λn,γn}n≥0 by the method of spectral mappings and state necessary and sufficient solvability conditions.Let{λn,γn}n≥0 be the spectral data of L. Denote(4.1)D(x,λ,μ):=〈φ(x,λ),φ(x,μ)〉λ-μ=∫0xφ(t,λ)φ(t,μ)dt.Let us choose a model boundary value problemL̃=L(q̃(x),h̃,H̃,H̃1,H̃2) with real q̃(x)∈L2(0,π), h̃, H̃, H̃1, H̃2, and r̃:=H̃H̃1-H̃2>0 such that ω̃=ω (take, e.g., q̃(x)≡0, h̃=0, H̃=ω). Let {λ̃n,γ̃n}n≥0 be the spectral data of L̃. Denote(4.2)ξn:=|ρn-ρ̃n|+|γn-γ̃n|.
Since ω=ω̃, it follows from (2.37), (2.39), and analogous formulae for ρ̃n and γ̃n that(4.3)∑n=0∞ξn<∞.
Denote(4.4)λn0=λn,λn1=λ̃n,γn0=γn,γn1=γ̃n,φni(x)=φ(x,λni),φ̃ni(x)=φ̃(x,λni),Pni,kj(x)=1γkjD(x,λni,λkj),P̃ni,kj(x)=1γkjD̃(x,λni,λkj),
where i,j=0,1, n,k≥0. Let x∈[0,π], n≥0, i,ν=0,1. It follows from (2.27) and (2.37) that(4.5)|φni(ν)(x)|≤C(n+1)ν.
Moreover, for a fixed a>0(4.6)|φ(ν)(x,λ)|≤C(n+1)ν,|ρ-ρn1|≤a.
Applying Schwarz’s lemma [21, page 130] in the ρ-plane to the circle |ρ-ρn1|≤a and to the function f(ρ):=φ(ν)(x,λ)-φ(ν)(x,λn1) with fixed ν, n, x, and a, we get(4.7)|φ(ν)(x,λ)-φ(ν)(x,λn1)|≤C(n+1)ν|ρ-ρn1|,|ρ-ρn1|≤a.
In particular, this yields(4.8)|φn0(ν)(x)-φn1(ν)(x)|≤Cξn(n+1)ν.
By similar arguments (see [20, page 48]) one gets that the following estimates are valid for x∈[0,π],  n,k≥0,  i,j,ν=0,1:(4.9)|Pni,kj(x)|≤C|n-k|+1,|Pni,k0(x)-Pni,k1(x)|≤Cξk|n-k|+1,|Pn0,kj(x)-Pn1,kj(x)|≤Cξn|n-k|+1,|Pn0,k0(x)-Pn1,k0(x)-Pn0,k1(x)+Pn1,k1(x)|≤Cξnξk|n-k|+1.
The analogous estimates are also valid for φ̃ni(x) and P̃ni,kj(x).Lemma 4.1.
The following relations hold:(4.10)φ̃(x,λ)=φ(x,λ)+∑k=0∞(〈φ̃(x,λ),φ̃k0(x)〉γk0(λ-λk0)φk0(x)-〈φ̃(x,λ),φ̃k1(x)〉γk1(λ-λk1)φk1(x)),(4.11)〈φ(x,λ),φ(x,μ)〉λ-μ-〈φ̃(x,λ),φ̃(x,μ)〉λ-μ+∑k=0∞(〈φ̃(x,λ),φ̃k0(x)〉γk0(λ-λk0)〈φk0(x),φ(x,μ)〉(λk0-μ)-〈φ̃(x,λ),φ̃k1(x)〉γk1(λ-λk1)〈φk1(x),φ(x,μ)〉(λk1-μ))=0.
Both series converge absolutely and uniformly with respect to x∈[0,π] and λ, μ on compact sets.Proof.
(1) Denoteλ′=minniλni and take a fixed δ>0.
LetI:={λ:|Imλ|≤δ,Reλ≥λ′-δ}, and let θ:=∂I be the boundary of I. Denote ΓN′=ΓN∩I, ΓN′′=ΓN∖ΓN′, θN′=θ∩intΓN. In the λ-plane we consider closed contours θN=θN′∪ΓN′ (with counterclockwise circuit), θN0=θN′∪ΓN′′ (with clockwise circuit). Let P(x,λ)=[pjk(x,λ)]j,k=1,2 be the matrix defined by (3.4). It follows from (3.5) and (3.1) that for each fixed x, the functions Pjk(x,λ) are meromorphic in λ with simple poles {λn}n≥0 and {λ̃n}n≥0. By Cauchy’s integral formula [21, page 84],(4.12)P1k(x,λ)-δ1k=12πi∫θN0P1k(x,ξ)-δ1kλ-ξdξ,k=1,2,λ∈int⁡θN0,
where δjk is the Kronecker delta. Hence
(4.13)P1k(x,λ)-δ1k=12πi∫θNP1k(x,ξ)λ-ξdξ-12πi∫ΓNP1k(x,ξ)-δ1kλ-ξdξ,
where ΓN is used with counterclockwise circuit. Substituting into (3.6) we obtain
(4.14)φ(x,λ)=φ̃(x,λ)+12πi∫θNφ̃(x,λ)P11(x,ξ)+φ̃′(x,λ)P12(x,ξ)λ-ξdξ+ɛN(x,λ),
where
(4.15)ɛN(x,λ)=-12πi∫ΓNφ̃(x,λ)(P11(x,ξ)-1)+φ̃′(x,λ)P12(x,ξ)λ-ξdξ.
By virtue of (3.8) we have that
(4.16)lim⁡N→∞ɛN(x,λ)=0
uniformly with respect to x∈[0,π] and λ on compact sets. Using (3.5) we calculate
(4.17)φ(x,λ)=φ̃(x,λ)+12πi∫θN(φ̃(x,λ)(φ(x,ξ)Φ̃′(x,ξ)-Φ(x,ξ)φ̃′(x,ξ))+φ̃′(x,λ)(Φ(x,ξ)φ̃(x,ξ)-φ(x,ξ)Φ̃(x,ξ)))dξλ-ξ+ɛN(x,λ).
This, in combination with (3.1), implies that
(4.18)φ̃(x,λ)=φ(x,λ)+12πi∫θN〈φ̃(x,λ),φ̃(x,ξ)〉λ-ξM̂(ξ)φ(x,ξ)dξ-ɛN(x,λ),
where M̂(λ)=M(λ)-M̃(λ), since the terms with S(x,ξ) vanish by Cauchy’s theorem [21, page 85]. It follows from (3.14) that
(4.19)Resξ=λkj〈φ̃(x,λ),φ̃(x,ξ)〉λ-ξM̂(ξ)φ(x,ξ)=(-1)j〈φ̃(x,λ),φ̃kj(x)〉γkj(λ-λkj)φkj(x).
Applying residue theorem to the integral in (4.18) and using (4.16) we obtain (4.10).
(2) Since(4.20)1λ-μ(1λ-ξ-1μ-ξ)=1(λ-ξ)(ξ-μ),
we have by Cauchy’s integral formula
(4.21)Pjk(x,λ)-Pjk(x,μ)λ-μ=12πi∫θN0Pjk(x,ξ)(λ-ξ)(ξ-μ)dξ,j,k=1,2,λ,μ∈int⁡θN0.
Acting in the same way as above and using (3.8) and (3.9), we obtain
(4.22)Pjk(x,λ)-Pjk(x,μ)λ-μ=12πi∫θNPjk(x,ξ)(λ-ξ)(ξ-μ)dξ+ɛNjk(x,λ,μ),
where limN→∞ɛNjk(x,λ,μ)=0, j,k=1,2.
It follows from (3.5) and (3.3) that(4.23)P11(x,λ)φ′(x,λ)-P21(x,λ)φ(x,λ)=φ̃′(x,λ),P22(x,λ)φ(x,λ)-P12(x,λ)φ′(x,λ)=φ̃(x,λ),(4.24)P(x,λ)[y(x)y′(x)]=〈y(x),Φ̃(x,λ)〉[φ(x,λ)φ′(x,λ)]-〈y(x),φ̃(x,λ)〉[Φ(x,λ)Φ′(x,λ)],
for any y(x)∈C1[0,π]. Using (4.22) and (4.24), we get
(4.25)P(x,λ)-P(x,μ)λ-μ[y(x)y′(x)]=12πi∫θN(〈y(x),Φ̃(x,ξ)〉[φ(x,ξ)φ′(x,ξ)]-〈y(x),φ̃(x,ξ)〉[Φ(x,ξ)Φ′(x,ξ)])×dξ(λ-ξ)(ξ-μ)+ɛN0(x,λ,μ),
where limN→∞ɛN0(x,λ,μ)=0. From (3.4) and (4.23), we get
(4.26)det⁡((P(x,λ)-P(x,μ))[φ̃(x,λ)φ̃′(x,λ)],[φ(x,μ)φ′(x,μ)])=〈φ(x,λ),φ(x,μ)〉-〈φ̃(x,λ),φ̃(x,μ)〉.
Hence, for y(x)=φ̃(x,λ), (4.25) gives
(4.27)〈φ(x,λ),φ(x,μ)〉λ-μ-〈φ̃(x,λ),φ̃(x,μ)〉λ-μ=12πi∫θN(〈φ̃(x,λ),Φ̃(x,ξ)〉〈φ(x,ξ),φ(x,μ)〉(λ-ξ)(ξ-μ)-〈φ̃(x,λ),φ̃(x,ξ)〉〈Φ(x,ξ),φ(x,μ)〉(λ-ξ)(ξ-μ))dξ+ɛN1(x,λ,μ),
where limN→∞ɛN1(x,λ,μ)=0. By virtue of (3.1), (3.14), and the residue theorem, we get (4.11).It follows from the definition ofP̃ni,kj(x), Pni,kj(x), and from (4.10) and (4.11) that(4.28)φ̃ni(x)=φni(x)+∑k=0∞(P̃ni,k0(x)φk0(x)-P̃ni,k1(x)φk1(x)),(4.29)Pni,lj(x)-P̃ni,lj(x)+∑k=0∞(P̃ni,k0(x)Pk0,lj(x)-P̃ni,k1(x)Pk1,lj(x))=0.
For each x∈[0,π], the relation (4.28) can be considered as a system of linear equations with respect to φni(x), n≥0, i=0,1. But the series in (4.28) converges only “with 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)).LetV be a set of indices u=(n,i), n≥0, i=0,1. For each fixed x∈[0,π], we define the vector(4.30)ψ(x)=[ψu(x)]u∈V=[ψn0(x)ψn1(x)]n≥0=[ψ00,ψ01,ψ10,ψ11,…]T
by the formulae(4.31)ψn0(x)=χn(φn0(x)-φn1(x)),ψn1(x)=φn1(x),
where(4.32)χn={ξn-1,ξn≠0,0,ξn=0.
We also define the block matrix(4.33)H(x)=[Hu,v(x)]u,v∈V=[Hn0,k0(x)Hn0,k1(x)Hn1,k0(x)Hn1,k1(x)]n,k≥0,u=(n,i),v=(k,j),
by the formulae(4.34)Hn0,k0(x)=ξkχn(Pn0,k0(x)-Pn1,k0(x)),Hn0,k1(x)=χn(Pn0,k0(x)-Pn0,k1(x)-Pn1,k0(x)+Pn1,k1(x)),Hn1,k0(x)=ξkPn1,k0(x),Hn1,k1(x)=Pn1,k0(x)-Pn1,k1(x).
Analogously we define ψ̃(x), H̃(x) by replacing in the previous definitions, φni(x) by φ̃ni(x) and Pni,kj(x) by P̃ni,kj(x). It follows from (4.5)–(4.9) that(4.35)|ψni(ν)(x)|≤C(n+1)ν,|Hni,kj(x)|≤Cξk|n-k|+1.
Similarly(4.36)|ψ̃ni(ν)(x)|≤C(n+1)ν,|H̃ni,kj(x)|≤Cξk|n-k|+1.Let us consider the Banach spacel∞(V) of bounded sequences α=[αu]u∈V with the norm ∥α∥∞=supu∈V|αu|. It follows from (4.35) and (4.36) that for each fixed x∈[0,π], the operators E+H̃(x) and E-H(x) (here E is the identity operator), acting from l∞(V) to l∞(V), are linear bounded operators, and(4.37)‖H(x)‖,‖H̃(x)‖≤Csup⁡n∑kξk|n-k|+1<∞.Theorem 4.2.
For each fixedx∈[0,π], the vector ψ(x)∈l∞(V) satisfies the equation
(4.38)ψ̃(x)=(E+H̃(x))ψ(x)
in the Banach space l∞(V). Moreover, the operator E+H̃(x) has a bounded inverse operator, that is, (4.38) is uniquely solvable.Proof.
From (4.28), we have
(4.39)φ̃n0(x)-φ̃n1(x)=φn0(x)-φn1(x)+∑k=0∞((P̃n0,k0(x)-P̃n1,k0(x))(φk0(x)-φk1(x))+(P̃n0,k0(x)-P̃n1,k0(x)-P̃n0,k1(x)+P̃n1,k1(x))φk1(x)(P̃n0,k0(x)-P̃n1,k0(x))),φ̃n1(x)=φn1(x)+∑k=0∞(P̃n1,k0(x)(φk0(x)-φk1(x))+(P̃n1,k0(x)-P̃n1,k1(x))φk1(x)).
For our notations this gives
(4.40)ψ̃ni(x)=ψni(x)+∑k,jH̃ni,kj(x)ψkj(x),(n,i),(k,j)∈V,
and (4.38) is proved. The series in (4.40) converges absolutely and uniformly for x∈[0,π]. Similarly, From (4.29), we get
(4.41)Hni,kj(x)-H̃ni,kj(x)+∑l,sH̃ni,ls(x)Hls,kj(x)=0,(n,i),(k,j),(l,s)∈V,
which is  equivalent to H-H̃+H̃H=0. Thus,
(4.42)(E+H̃(x))(E-H(x))=E.
Interchanging places of L and L̃, we get analogously
(4.43)ψ(x)=(E-H(x))ψ̃(x),(E-H(x))(E+H̃(x))=E.
Therefore, the operator (E+H̃(x))-1 exists, and it is a bounded linear operator.Equation (4.38) is called the main equation of the inverse problem. Solving (4.38) we find the vector ψ(x) and consequently, the functions φni(x), n≥0, i=0,1. Since φni(x)=φ(x,λni) are the solutions of (1.1), we can construct the function q(x) by the formula(4.44)q(x)=λni+φni′′(x)φni(x).
We get the coefficient h by(4.45)h=φ′(0,λni).
We obtain the coefficients H, H1, and H2 from the linear system of equations(4.46)(λn0-H1)φn0′(π)+(λn0H-H2)φn0(π)=0,n≥0.
Thus, we get the following algorithm for the solution of the inverse problem of recovering L from the given spectral data {λn,γn}n≥0.Algorithm 4.3.
Let the numbers{λn,γn}n≥0 be given.(1)
ChooseL̃ such that ω̃=ω, and construct ψ̃(x) and H̃(x).(2)
Findψ(x) by solving (4.38).(3)
Calculateq(x), h, H, H1, and H2 by (4.44), (4.45), and (4.46).Remark 4.4.
Using the method of spectral mappings [15], one can show that relations (2.37) and (2.39) are not only necessary but also sufficient for the solvability of the inverse problem. In other words, for real numbers {λn,γn}n≥0 to be the spectral data for certain problem L=L(q(x),h,H,H1,H2) with q(x)∈L2(0,π), it is necessary and sufficient that λm≠λn for m≠n, γn>0 and the relations (2.37) and (2.39) hold.



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*Source: 754718-2011-12-25.xml*