# Direct and Inverse Problems for Sturm-Liouville Operator Which Has Discontinuity Conditions and Coulomb Potential

Authors: Yalçın Güldü; Merve Arslantaş
Journal: Chinese Journal of Mathematics
 (2014)

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## Abstract

We give a derivation of the main equation for Sturm-Liouville operator with Coulomb potential and prove its unique solvability.
Using the solution of the main equation, we get an algorithm for the solution of the inverse problem.


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## Body

## 1. Introduction

In this study, we consider the boundary value problemL:
(1)ly=-y′′+(Cx+q(x))y=λy,λ=ρ2,0<x<π,(2)U(y):=y(0)=0,V(y):=y(π)=0,(3)y(d+0)=αy(d-0),y′(d+0)=α-1y′(d-0),
where λ=ρ2 is a spectral parameter, C>0, α∈ℝ+∖{1}, d∈(π/2,π), q(x) is a real valued bounded function, and q(x)∈L2(0,π).In spectral theory, the inverse problem is the usual name for any problem in which it is required to ascertain the spectral data that will determine a differential operator uniquely and a method of construction of this operator from the data. This kind of problem was first formulated and investigated by Ambarzumian in 1929 [1]. Since 1946, various forms of the inverse problem have been considered by numerous authors, Borg [2], Levinson [3], Levitan [4], and so forth, and now there exists extensive literature in [5–9]. Later, the inverse problems having specified singularities were considered by a number of authors [10–12].The method of spectral mappings is an impressive device for investigating a profound class of inverse problems not only for Sturm-Liouville operators, but also for other more complicated classes of operators such as differential operators of arbitrary orders, differential operators with singularities, and pencils of operators. We apply the method of spectral mappings to the solution of the inverse problem for the Sturm-Liouville operator on a finite interval.In the method of spectral mappings we begin from Cauchy’s integral formula for analytic functions. We apply this theorem in the complex plane of the spectral parameter for specially constructed analytic functions having Coulomb singularity connected with the given spectral characteristics. This permits us to reduce the inverse problem to the so-called main equation which is a linear equation in a corresponding Banach space of sequences.In this paper, first, integral representation for solution which satisfies certain initial conditions of differential equation generated by Sturm-Liouville operator with Coulomb potential is handled. Second, properties of spectral characteristics and uniqueness theorems for solution of inverse problem are discussed. After that we give a derivation of the main equation and prove its unique solvability. Finally, we obtain an algorithm for the solution of the inverse problem by using the solution of the main equation.


## 2. Preliminaries

We define(4)(Γy)(x)=y′-u(x)ywhereu(x)=Cln⁡x∈L2(0,π),and let us write the expression of the left-hand side of (1) as follows:
(5)l(y)=-[(Γy)(x)]′-u(x)(Γy)(x)-u2(x)y+q(x)y.Here,(Γy)(x) has a limit as x→0+; that is, lim⁡x→0⁡(Γy)(x)=(Γy)(0).Theorem 1 (see [13]).
Solution of the problemL which satisfies the initial conditions (y1y2)(0)=(1iρ) and the discontinuity conditions (3) has the following form.
Ifx<d,
(6)y1=eiρx+∫-xxK11(x,t)eiρtdt,y2=iρeiρx+b(x)eiρx+∫-xxK21(x,t)eiρtdt+iρ∫-xxK22(x,t)eiρtdt,
if x>d,
(7)y1=α+eiρx+α-eiρ(2d-x)+∫-xxK11(x,t)eiρtdt,y2=iρ(α+eiρx-α-eiρ(2d-x))+b(x)[α+eiρx+α-eiρ(2d-x)]+∫-xxK21(x,t)eiρtdt+iρ∫-xxK22(x,t)eiρtdt,
where
(8)b(x)=-12∫0x[u2(s)-q(s)]e-(1/2)∫sxu(t)dtds,K11(x,x)=α+2u(x),K11(x,-x)=0,K11(x,2d-x+0)-K11(x,2d-x-0)=α-2u(x)K21(x,x)=b′(x)-12∫0x[u2(s)-q(s)]K11(s,s)ds-12∫0xu(s)K21(s,s)ds,K22(x,x)=-α+2[u(x)+2b(x)],∂Kij(x,·)∂x,∂Kij(x,·)∂t∈L2(0,π),i,j=1,2.
In this part, we consider the properties of the spectrum of the problemL. In the case of C=0 and q(x)≡0, the problem L is denoted by L0. Let us verify that a solution φ(x,λ) of (1) with the initial conditions φ(0,λ)=0 and (Γφ)(0,λ)=λ=ρ and discontinuity conditions (3). When C=0 and q(x)≡0, it is easily shown that solution φ0(x,λ) with the same conditions has the form
(9)φ0(x,λ)={1λsinλx,x<d,α+λsinλx+α-λsinλ(2d-x),x>d,(Γφ0)(x,λ)={cos⁡λx,x<d,α+cos⁡λx-α-cos⁡λ(2d-x),x>d.
We denote the characteristic function of the problemL0 by Δ0(λ):
(10)Δ0(λ)=φ0(π,λ)=α+λsinλπ+α-λsinλ(2d-π).
The roots λn0 of equation Δ0(λ)=0 for n∈ℕ are the eigenvalues of the problem L0.
On the other hand, normalizing constantsαn of the problem L are defined as follows:
(11)αn:=∫0πφ2(x,λn)dx.Lemma 2 (see [13]).
The eigenvalues of the problemL satisfy the asymptotic equality
(12)λn=ρn=ρn0+dn(ρn0)2+δn(ρn0)2,
where δn∈ℓ2, ρn0=n+hn, sup⁡|hn|≤M, dn∈ℓ∞, and
(13)dn=α+cos⁡λn0π+α-cos⁡λn0(2d-π)2Δ˙0(λn0)u(π).Lemma 3 (see [13]).
The normalizing constants of the problemL satisfy the asymptotic equality
(14)αn=αn0+κn,
where κn∈ℓ2.On the other hand, eigenfunctionsφ(x,λn) of problem L have the following asymptotic behaviour.Forx<d,
(15)φ(x,λn)=1λnsinλnx+1λn∫0xK~11(x,t)sinλntdt=1λnsinλnx+O(e|Im⁡λn|xλn)=O(e|Im⁡λn|xλn),λn⟶∞,(Γφ)(x,λn)=cos⁡λnx+O(e|Im⁡λn|xλn)=O(e|Im⁡λn|x),λn⟶∞.Forx>d,
(16)φ(x,λn)=α+λnsinλnx+α-λnsinλn(2d-x)+O(e|Im⁡λn|xλn)=O(e|Im⁡λn|xλn),+O(e|Imλn|xλn)=Ohhhhλn⟶∞,(Γφ)(x,λn)=α+cos⁡λnx-α-cos⁡λn(2d-x)+O(e|Im⁡λn|xλn)=O(e|Im⁡λn|x),+O(e|Imλn|xλn)=Ohhλn⟶∞.We consider the inverse problem of the reconstruction of problemL according to Weyl function. For this reason, we can also consider a boundary value problem L~=L(q~(x)) of the same form together with L.Assume thatΦ(x,λ) is a solution of (1) which satisfies the conditions Φ(0,λ)=1 and the conditions (3).LetC(x,λ) and Ψ(x,λ) denote solutions of (1) that satisfy the initial conditions (CC′)(0,λ)=(10) and (ΨΨ′)(π,λ)=(0-1), respectively, and jump conditions (3).We setM(λ):=Φ′(0,λ). The functions Φ(x,λ) and M(λ) are called the Weyl solution and Weyl function of the boundary value problem L, respectively.It is obvious that(17)Φ(x,λ)=Ψ(x,λ)Δ(λ)=C(x,λ)+M(λ)φ(x,λ),(18)M(λ)=Ψ′(0,λ)Δ(λ)(19)〈Φ(x,λ),φ(x,λ)〉≡1.Therefore, the Weyl function is meromorphic with simple poles in the pointsλ=λn,n≥0.Lemma 4.
IfM(λ)=M~(λ), then L=L~. Therefore, the boundary value problem L is uniquely determined by the Weyl function.Proof.
We define the matrixP(x,λ)=[Pjk(x,λ)]j,k=1,2 by the formula
(20)P(x,λ)(φ~Φ~φ~′Φ~′)=(φΦφ′Φ′).
Using (19) and (20), we get
(21)P11(x,λ)=Φ(x,λ)φ~′(x,λ)-φ(x,λ)Φ~′(x,λ)P12(x,λ)=φ(x,λ)Φ~(x,λ)-Φ(x,λ)φ~(x,λ)P21(x,λ)=Φ′(x,λ)φ~′(x,λ)-φ′(x,λ)Φ~′(x,λ)P22(x,λ)=φ′(x,λ)Φ~(x,λ)-Φ′(x,λ)φ~(x,λ),(22)φ(x,λ)=P11(x,λ)φ~(x,λ)+P12(x,λ)φ~′(x,λ)φ′(x,λ)=P21(x,λ)φ~(x,λ)+P22(x,λ)φ~′(x,λ)Φ(x,λ)=P11(x,λ)Φ~(x,λ)+P12(x,λ)Φ~′(x,λ)Φ′(x,λ)=P21(x,λ)Φ~(x,λ)+P22(x,λ)Φ~′(x,λ).
On the other hand, from the hypothesisM(λ)=M~(λ), by using (17) and (21), we get
(23)P11(x,λ)=φ~′(x,λ)C(x,λ)-φ(x,λ)C~′(x,λ)P12(x,λ)=φ(x,λ)C~(x,λ)-C(x,λ)φ~(x,λ)P21(x,λ)=C′(x,λ)φ~′(x,λ)-φ′(x,λ)C~′(x,λ)P22(x,λ)=φ′(x,λ)C~(x,λ)-C′(x,λ)φ~(x,λ).
So, from (23), when M(λ)=M~(λ), Pjk(x,λ) are entire on λ.
Since(24)〈Ψ(x,λ),φ(x,λ)〉=Δ(λ),
relations (17), (21), and (24) yield
(25)P11(x,λ)=φ~′(x,λ)Ψ(x,λ)Δ(λ)-φ(x,λ)Ψ~′(x,λ)Δ~(λ)=1+(φ~(x,λ)-φ(x,λ))Ψ~′(x,λ)Δ~(λ)-(Ψ~(x,λ)Δ~(λ)-Ψ(x,λ)Δ(λ))φ~′(x,λ),P12(x,λ)=φ(x,λ)Ψ~(x,λ)Δ~(λ)-φ~(x,λ)Ψ(x,λ)Δ(λ)=(Ψ~(x,λ)Δ~(λ)-Ψ(x,λ)Δ(λ))φ(x,λ)-(φ~(x,λ)-φ(x,λ))Ψ(x,λ)Δ(λ),P21(x,λ)=φ~′(x,λ)Ψ′(x,λ)Δ(λ)-φ′(x,λ)Ψ~′(x,λ)Δ~(λ)=(φ~′(x,λ)Δ(λ)-φ′(x,λ)Δ~(λ))Ψ~′(x,λ)-(Ψ~′(x,λ)-Ψ′(x,λ))φ~′(x,λ)Δ(λ),P22(x,λ)=φ′(x,λ)Ψ~(x,λ)Δ~(λ)-φ~(x,λ)Ψ′(x,λ)Δ(λ)=1+φ′(x,λ)(Ψ~(x,λ)Δ~(λ)-Ψ(x,λ)Δ(λ))-(φ~(x,λ)-φ(x,λ))Ψ′(x,λ)Δ(λ).
Let Gδ={λ:λ=ρ2,|ρ-ρn0|≥δ,n=0,±1,±2,…} where δ is sufficiently small number. Since |Δ(λ)|>Cδe|Imρ|π in Gδ and from the representations of the solutions
(26)φ(x,λ)=φ0(x,λ)+∫-xxK11(x,t)sinρtdt,Ψ(x,λ)=Ψ0(x,λ)+∫-(π-x)π-xN11(x,t)sinρtdt
we see that Pjk(x,λ) are bounded with respect to λ. Hence it is clear from Liouville’s theorem that Pjk(x,λ) do not depend on λ. Therefore, from the asymptotic formulae of the solutions φ(x,λ), Ψ(x,λ),
(27)lim⁡λ→∞(φ~(x,λ)-φ(x,λ))Ψ~′(x,λ)Δ~(λ)=0,lim⁡λ→∞(Ψ~(x,λ)Δ~(λ)-Ψ(x,λ)Δ(λ))φ~′(x,λ)=0
for allx∈(0,d)∪(d,π) and for λ∈R. Finally,
(28)lim⁡λ→∞(P11(x,λ)-1)=0
uniformly with respect tox. Therefore,
(29)P11(x,λ)≡1.SimilarlyP22(x,λ)≡1,P12(x,λ)≡0≡P21(x,λ).
If we substitute these results in (22), we get φ(i)(x,λ)≡φ~(i)(x,λ), Φ(i)(x,λ)≡Φ~(i)(x,λ)(i=0,1) for all x and λ. Therefore L=L~.


## 3. The Main Equation of the Inverse Problem

In this section, we solve the inverse problem of recoveringL from the given spectral data by using ideas of the contour integral method.We denote that ify(x,λ) and z(x,μ) are solutions of the equations ℓy=λy and ℓz=μz separately, then
(30)ddx〈y,z〉=(λ-μ)yz,〈y,z〉:=yz′-y′z.Note(31)D(x,λ,μ):=〈φ(x,λ),φ(x,μ)〉λ-μ=∫0xφ(t,λ)φ(t,μ)dt.Let us select a model boundary value problemL~=L(q~(x),C~) with real q~(x)∈L2(0,π) such that C=C~. Let {λ~n,α~n}n≥0 be the spectral data of L~.Furthermore, let(32)ξn:=|ρn-ρ~n|+|αn-α~n|It follows from (12) and (14) and analogous formulas for ρ~n and α~n that
(33)Λ:=(∑n=0∞‍[(n+1)ξn]2)1/2<∞,∑n‍ξn<∞.Denote that(34)λn0=λn,λn1=λ~n,αn0=αn,αn1=α~n,φni(x)=φ(x,λni),φ~ni(x)=φ~(x,λni),Pni,j(x)=1αkjD(x,λni,λkj),P~ni,kj(x)=1αkjD~(x,λni,λkj),i,j∈{0,1},n,k≥0.Then, according to (31),
(35)Pni,kj(x)=〈φni(x),φkj(x)〉αkj(λni-λkj)=1αkj∫0xφni(t),φkj(t)dt,P~ni,kj(x)=〈φ~ni(x),φ~kj(x)〉αkj(λni-λkj)=1αkj∫0xφ~ni(t),φ~kj(t)dt.It is clear that(36)Pni,kj′(x)=1αkjφni(x)φkj(x),P~ni,kj′(x)=1αkjφ~ni(x)φ~kj(x).Lemma 5 (Schwarz’s lemma).
Letf(ρ) be an analytic function for |ρ-ρ0|<a such that f(ρ0)=0 and |f(ρ)|≤A. Then |f(ρ)|≤(A/a)|ρ-ρ0| for |ρ-ρ0|<a.Lemma 6.
The following estimates are valid forx∈[0,π]∖{d}, n,k≥0, i,j,v=0,1:
(37)|φni(x)|≤c,|(Γφni)(x)|≤c(n+1),|φn0(x)-φn1(x)|≤cξn,|(Γφn0)(x)-(Γφn1)(x)|≤cξn(n+1),(38)|Pni,kj(x)|≤c|n-k|+1,|Pni,kj(v+1)(x)|≤c(n+k+1)v|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 similar estimates are also current forφ~ni(x),P~ni,kj(x).Proof.
It follows from (12), (14), (15), and (16) that |φni(x)|≤c, |(Γφni)(x)|≤c(n+1) when x<d and x>d.
Besides, for a fixeda>0,
(39)|φ(x,λ)|≤c,|(Γφ)(x,λ)|≤c(n+1),|ρ-ρn1|≤a.
Applying Schwarz’s lemma in theρ-plane to the function f(ρ):=φ(v)(x,λ)-φ(v)(x,λn1) with fixed v,n,x, and a, we obtain
(40)|φ(v)(x,λ)-φ(v)(x,λn1)|≤c(n+1)v|ρ-ρn1|,|ρ-ρn1|≤a.
Here(Γφ)(x,λ) is denoted by φ(v)(x,λ) for v=1.
As a result,(41)|φn0(v)(x)-φn1(v)(x)|≤c(n+1)v|ρn0-ρn1|,
and (37) is proved.Let us show that(42)|D(x,λ,λkj)|≤ce|τ|x|ρ∓k|+1,λ=ρ2,∓Reρ≥0,τ=Im⁡ρ,k≥0.For definiteness, letσ:=Reρ≥0. Get a fixed δ0>0. For |ρ-ρkj|≥δ0, we have by the help of (31), (15), and (12),
(43)|D(x,λ,λkj)|=|〈φ(x,λ),φ(x,λkj)〉||λ-λkj|≤ce|τ|x|ρ|+|ρkj||ρ2-ρkj2|.Since(|ρ|+|ρkj|)/|ρ+ρkj|≤(σ2+τ2+|ρkj|)/σ2+τ2+ρkj2≤2, we obtain
(44)|D(x,λ,λkj)|≤ce|τ|x|ρ-ρkj|,|ρ-k|+1|ρ-ρkj|≤|ρkj-k|+1|ρ-ρkj|≤c,andso1|ρ-ρkj|≤c|ρ-k|+1.Thus, this yields (42) for |ρ-ρkj|≥δ0. For |ρ-ρkj|≤δ0,
(45)|D(x,λ,λkj)|=|∫0xφ(t,λ)φ(t,λkj)dt|≤ce|τ|x.That is, (42) is also valid for |ρ-ρkj|≤δ0. Likewise it can be shown that
(46)|D(x,λ,μ)|≤ce|τ|x|ρ∓θ|+1,μ=θ2,|Im⁡θ|≤c0,∓ReρReθ≥0.Using Schwarz’s lemma, we get(47)|D(x,λ,λk1)-D(x,λ,λk0)|≤ce|τ|xξk|ρ∓k|+1,D(x,λ,λk1)-(x,λ,)∓Reρ≥0,k≥0.Particularly, this yields(48)|D(x,λni,λk1)-D(x,λni,λk0)|≤cξk|n-k|+1.Symmetrically,(49)|D(x,λn1,λkj)-D(x,λn0,λkj)|≤cξn|n-k|+1.If we apply Schwarz’s lemma to the function(50)Qk(x,λ):=D(x,λ,λk1)-D(x,λ,λk0)
for fixed k and x, we obtain
(51)|D(x,λn0,λk0)-D(x,λn1,λk0,)-D(x,λn0,λk1)+D(x,λn1,λk1)|≤cξnξk|n-k|+1.These approximates together with (34), (36), (12), and (14) suggest (38).Lemma 7.
The following relations hold:(52)φ~(x,λ)=φ(x,λ)hhhhhhh+∑k=0∞‍(〈φ~(x,λ),φ~k0(x)〉αk0(λ-λk0)φk0(x)hhhhhhh-〈φ~(x,λ),φ~k1(x)〉αk1(λ-λk1)φk1(x)∑k=0∞‍),(53)〈φ(x,λ),φ(x,μ)〉λ-μ-〈φ~(x,λ),φ~(x,μ)〉λ-μ+∑k=0∞‍{〈φ~(x,λ),φ~k0(x)〉αk0(λ-λk0)+∑k=0∞‍{h×〈φk0(x),φ(x,μ)〉λk0-μ+∑k=0∞‍{h-〈φ~(x,λ),φ~k1(x)〉αk1(λ-λk1)+∑k=0∞‍{h×〈φk1(x),φ(x,μ)〉λk1-μ}=0.Proof.
(1) Denoteλ′=min⁡ni⁡λni and take a fixed δ>0. In the λ-plane we regard closed contours γN=γN+∪γN-∪γ′∪ΓN′, (with counterclockwise circuit) (see Figure 1) where
(54)γN±={λ:±Im⁡λ=δ,Reλ≥λ′,|λ|≤(N+12)2}γ′={λ:λ-λ′=δeiα,α∈(π2,3π2)}ΓN′=ΓN∩{λ:|Im⁡λ|≤δ,Reλ>0},ΓN={λ:|λ|=(N+12)2}.
DenoteγN0=γN+∪γN-∪γ′∪(ΓN∖ΓN′) (with clockwise circuit) (see Figure 1).
LetP(x,λ)=[Pjk(x,λ)]j,k=1,2 be the matrix determined by (20). It follows from (21) and (17) that, for each fixed x, the functions Pjk(x,λ) are meromorphic in λ with simple poles {λn} and {λ~n}. By Cauchy’s integral formula,
(55)P1(x,k)-δ1k=12πi∫γN0P1k(x,ξ)-δ1kλ-ξdξ,k=1,2,…,
where λ∈int⁡(γN0) and δjk is the Kronecker delta. Therefore,
(56)P1(x,k)-δ1k=12πi∫γNP1k(x,ξ)λ-ξdξ-12πi∫ΓNP1k(x,ξ)-δ1kλ-ξdξ,
where ΓN is used with counterclockwise circuit. Substituting into (22), we get
(57)φ(x,λ)=φ~(x,λ)+12πi×∫γNφ~(x,λ)P11(x,ξ)+(Γφ~)(x,λ)P12(x,ξ)λ-ξdξ+ɛN(x,λ),
where
(58)ɛN(x,λ)=-12πi×∫ΓNφ~(x,λ)(P11(x,ξ)-1)+(Γφ~)(x,λ)P12(x,ξ)λ-ξdξ.
By the help of (29),
(59)lim⁡N→∞ɛN(x,λ)=0
uniformly with respect to x∈[0,π] and λ on compact sets.
If we take (25) into account, we calculate
(60)φ(x,λ)=φ~(x,λ)+12πi×∫γN{φ~(x,λ)[φ(x,ξ)(ΓΦ~)(x,ξ)-Φ~(x,ξ)(Γφ~)(x,ξ)]×∫γN{+(Γφ~)(x,λ)[Φ(x,ξ)φ~(x,ξ)-φ(x,ξ)Φ~(x,ξ)]}×dξλ-ξ+ɛN(x,λ).
In view of (17), this supplies
(61)φ~(x,λ)=φ(x,λ)+12πi×∫γN〈φ~(x,λ),φ~(x,ξ)〉λ-ξM^(ξ)φ(x,ξ)dξ+ɛN(x,λ),
where M^(λ)=M(λ)-M~(λ), since the terms with C(x,ξ) vanish by Cauchy’s theorem.
It follows from(62)Resλ=λnM(λ)=-Δ0(λn)Δ˙(λn)=-βnΔ˙(λn)=1αn
that
(63)Resξ=λkj〈φ~(x,λ),φ~(x,ξ)〉λ-ξM^(ξ)φ(x,ξ)=〈φ~(x,λ),φ~kj(x)〉αkj(λ-λkj)φkj(x).
Calculating the integral in (61) by the residue theorem and using (59) we get (52).
Since(1/(λ-μ))((1/(λ-ξ))-(1/(μ-ξ)))=1/((λ-ξ)(ξ-μ)), we have by Cauchy’s integral formula
(64)Pjk(x,λ)-Pjk(x,μ)λ-μ=12πi∫γN0Pjk(x,ξ)(λ-ξ)(ξ-μ)dξ,k,j=1,2,λ,μ∈int⁡(γN0).
Considering in the same way as above and using (29), we get
(65)Pjk(x,λ)-Pjk(x,μ)λ-μ=12πi∫γNPjk(x,ξ)(λ-ξ)(ξ-μ)dξ+ɛNjk(x,λ,μ),
where lim⁡n→∞⁡ɛNjk(x,λ,μ)=0j,k=1,n¯. From (21) and (19), it follows that
(66)P11(x,λ)(Γφ~)(x,λ)-P21(x,λ)φ(x,λ)=(Γφ~)(x,λ),P22(x,λ)φ(x,λ)-P12(x,λ)(Γφ)(x,λ)=φ~(x,λ),(67)P(x,λ)[y(x)y′(x)]=〈y(x),Φ~(x,λ)〉[φ(x,λ)(Γφ)(x,λ)]-〈y(x),φ~(x,λ)〉[Φ(x,λ)(ΓΦ)(x,λ)]
for any y(x)∈C1[0,1].
If we take (65) and (67) into account, we calculate
(68)Pjk(x,λ)-Pjk(x,μ)λ-μ[y(x)y′(x)]=12πi∫γN{〈y(x),Φ~(x,ξ)〉[φ(x,ξ)(Γφ)(x,ξ)]=12πi∫γN‍(hh-〈y(x),φ~(x,ξ)〉[Φ(x,ξ)(ΓΦ)(x,ξ)]}×dξ(λ-ξ)(ξ-μ)+ɛN0(x,λ,μ),lim⁡N→∞ɛN0(x,λ,μ)=0.
According to (20), P(x,λ)[φ~(x,λ)(Γφ~)(x,λ)]=[φ(x,λ)(Γφ)(x,λ)].
Thus,det⁡(P(x,λ)[φ~(x,λ)(Γφ~)(x,λ)],[φ(x,μ)(Γφ)(x,μ)])=〈φ(x,λ),φ(x,μ)〉.
Moreover, using (66) we obtain
(69)det⁡(P(x,μ)[φ~(x,λ)(Γφ~)(x,λ)],[φ(x,μ)(Γφ)(x,μ)])=〈φ~(x,λ),φ~(x,μ)〉.
So,
(70)det⁡((P(x,λ)-P(x,μ))[φ~(x,λ)(Γφ~)(x,λ)],[φ(x,μ)(Γφ)(x,μ)])=〈φ(x,λ),φ(x,μ)〉-〈φ~(x,λ),φ~(x,μ)〉.
As a result, (68) for y(x)=φ~(x,λ) supplies
(71)〈φ(x,λ),φ(x,μ)〉λ-μ-〈φ~(x,λ),φ~(x,μ)〉λ-μ=12πi∫γN{〈φ~(x,λ),Φ~(x,ξ)〉〈φ(x,ξ),φ(x,μ)〉(λ-ξ)(ξ-μ)=12πi∫γN‍(hh-〈φ~(x,λ),φ~(x,ξ)〉〈Φ(x,ξ),φ(x,μ)〉(λ-ξ)(ξ-μ)〈φ~(x,λ),Φ~(x,ξ)〉〈φ(x,ξ),φ(x,μ)〉(λ-ξ)(ξ-μ)}dξ+ɛN′(x,λ,μ),lim⁡N→∞ɛN′(x,λ,μ)=0.
By virtue of (17), (62), and residue theorem, we obtain (53).
Similarly, the following relation can be obtained:(72)Φ~(x,λ)=Φ(x,λ)+∑k=0∞‍(〈φ~(x,λ),φ~k0(x)〉αk0(λ-λk0)φk0(x)+∑k=0∞‍(h-〈φ~(x,λ),φ~k1(x)〉αk1(λ-λk1)φk1(x)).
It follows from the definition ofP~ni,kj(x), Pni,kj(x) and (52), (53) that
(73)φ~ni(x)=φni(x)+∑k=0∞‍(P~ni,k0(x)φk0(x)-P~ni,k1(x)φk1(x)),(74)Pni,lj(x)-P~ni,lj(x)+∑k=0∞‍(P~ni,k0(x)Pk0,lj(x)-P~ni,k1(x)Pk1,lj(x))=0.
Denote
(75)ɛ0(x)=∑k=0∞‍(1αk0φ~k0(x)φk0(x)-1αk1φ~k1(x)φk1(x)),ɛ(x)=-2ɛ0′(x).Figure 1Lemma 8.
The series in (75) converges absolutely and uniformly on [0,π]. The function ɛ0(x) is absolutely continuous and ɛ(x)∈L2(0,π).Proof.
We can writeɛ0(x) in the form
(76)ɛ0(x)=A1(x)+A2(x),
where
(77)A1(x)=∑k=0∞‍(1αk0-1αk1)φ~k0(x)φk0(x),A2(x)=∑k=0∞‍1αk1{(φ~k0(x)-φ~k1(x))φk0(x)∑k=0∞‍1αk1{+φ~k1(x)(φk0(x)-φk1(x))}.
It follows from (12), (14), (33), and (37) that the series in (77) converge absolutely and uniformly on [0,π] and
(78)|Aj(x)|≤c∑k=0∞‍ξk≤cΛ,j=1,2.
Moreover, using the asymptotic formulae (15), (12), (14), and (33) we calculate
(79)A1′(x)=∑k=0∞‍(1αk0-1αk1)ddx(φ~k0(x)φk0(x))=∑k=0∞‍γk(sin2kx+ηk(x)k+1),
where {γk}∈l2 and max⁡0≤x≤π⁡|ηk(x)|≤c for k≥0. Therefore, A1(x)∈W21(0,π). Analogously, we obtain A2(x)∈W21(0,π) and consequently ɛ0(x)∈W21(0,π).Lemma 9.
The following relation holds:(80)q(x)=q~(x)+ɛ(x),
where ɛ(x) is defined by (75).Proof.
If we differentiate (52) twice with respect to x and use (30) and (75), we obtain
(81)(Γφ~)(x,λ)-ɛ0(x)φ~(x,λ)=(Γφ)(x,λ)+∑k=0∞‍(〈φ~(x,λ),φ~k0(x)〉αk0(λ-λk0)(Γφk0)(x)=(Γφ)(x,λ)+∑k=0∞‍(h-〈φ~(x,λ),φ~k1(x)〉αk1(λ-λk1)(Γφk1)(x)),(Γφ~)′(x,λ)=(Γφ)′(x,λ)+∑k=0∞‍(〈φ~(x,λ),φ~k0(x)〉αk0(λ-λk0)(Γφk0)′(x)+∑k=0∞‍(h-〈φ~(x,λ),φ~k1(x)〉αk1(λ-λk1)(Γφk1)′(x))+2φ~(x,λ)∑k=0∞‍(1αk0φ~k0(x)(Γφk0)(x)+2φ~(x,λ)∑k=0∞‍(-1αk1φ~k1(x)(Γφk1)(x))+∑k=0∞‍(1αk0[Γ(φ~(x,λ)φ~k0(x))]φk0(x)+∑k=0∞‍(-1αk1[Γ(φ~(x,λ)φ~k1(x))]φk1(x))+2u(x)φ~(x,λ)×∑k=0∞‍(1αk0φ~k0(x)φk0(x)-1αk1φ~k1(x)φk1(x)).
If we replace(Γφ)′(x,λ) and (Γφ~)′(x,λ) by using (1) and then replace φ(x,λ) using (52), then
(82)q~(x)φ~(x,λ)=q(x)φ~(x,λ)+φ~(x,λ)2ɛ0′(x)q~(x)=q(x)-ɛ(x)q(x)=q~(x)+ɛ(x).Remark 10.
For each fixedx∈[0,π], (73) can be considered as a system of linear equations with respect to φni(x)n≥0, i=0,1. However, the series in (73) converges only “with brackets.” Hence, it is not appropriate to use (73) as the main equation of the inverse problem. Below we will convert (73) to a linear equation in a corresponding Banach space of sequences.
LetT be a set of indices u=(n,i), n≥0, i=0,1. For each fixed x∈[0,π], we define the vector
(83)Ω(x)=[Ωu(x)]u∈T=[Ωn0(x)Ωn1(x)]n≥0=[Ω00,Ω01,Ω10,Ω11,…]T
by the formulae
(84)[Ωn0(x)Ωn1(x)]=[κn-κn01][φn0(x)φn1(x)]=[κn(φn0(x)-φn1(x))φn1(x)]κn={ξn-1,ξn≠0,0,ξn=0.
We also define the block matrix
(85)B(x)=[Bu,v(x)]u,v∈V=[Bn0,k0(x)Bn0,k1(x)Bn1,k0(x)Bn1,k1(x)]n,k≥0,hhhu=(n,i),v=(k,j),
by the formulae(86)[Bn0,k0(x)Bn0,k1(x)Bn1,k0(x)Bn1,k1(x)]=[κn-κn01][Pn0,k0(x)Pn0,k1(x)Pn1,k0(x)Pn1,k1(x)][ξk10-1]=[ξkκn(Pn0,k0(x)-Pn1,k0(x))κn(Pn0,k0(x)-Pn0,k1(x)-Pn1,k0(x)+Pn1,k1(x))ξkPn1,k0(x)Pn1,k0(x)-Pn1,k1(x)].
Analogously we defineΩ~(x),B~(x) by replacing in the former descriptions φni(x) by φ~ni(x) and Pni,kj(x) by P~ni,kj(x). It follows from (37) and (38) that
(87)|Ωni(v)(x)|≤c(n+1)v,|Bni,kj(x)|≤cξk|n-k|+1|Bni,kj(v+1)(x)|≤c(n+k+1)vξk.
Similarly,
(88)|Ω~ni(v)(x)|≤c(n+1)v,|B~ni,kj(x)|≤cξk|n-k|+1|B~ni,kj(v+1)(x)|≤c(n+k+1)vξk
and also
(89)|B~ni,kj(x)-B~ni,kj(x0)|≤c|x-x0|ξk,x,x0∈[0,π],
where c does not depend on x, x0, n, i, j, and k.
Let us consider the Banach spacem of bounded sequences α=[αu]u∈T with the norm ∥α∥m=sup⁡u∈T⁡|αu|. It follows from (87) and (88) that, for each fixed x∈[0,π], the operators E+B~(x) and E-B(x), such that E is the identity operator mapping from m to m, are linear bounded operators and
(90)∥B(x)∥,∥B~(x)∥≤csup⁡n∑k‍ξk|n-k|+1<∞.Theorem 11.
In the Banach spacem, the vector Ω(x)∈m satisfies the equation
(91)Ω~(x)=(E+B~(x))Ω(x)
for each fixed x∈[0,π]. Furthermore, the operator E+B~(x) has a bounded inverse operator; that is, (91) is uniquely solvable.Proof.
We can write (73) in the form
(92)φ~n0(x)-φ~n1(x)=φn0(x)-φn1(x)+∑k=0∞‍{(P~n0,k0(x)-P~n1,k0(x))+∑k=0∞‍(×(φk0(x)-φk1(x))+∑k=0∞‍(+(P~n0,k0(x)-P~n1,k0(x)+∑k=0∞‍(+(-P~n0,k1(x)+P~n1,k1(x))+∑k=0∞‍(×φk1(x)},(93)φ~n1(x)=φn1(x)+∑k=0∞‍{P~n1,k0(x)(φk0(x)-φk1(x))+∑k=0∞‍(+(P~n1,k0(x)-P~n1,k1(x))φk1(x)}.
Considering our notations, we get
(94)Ω~ni(x)=Ωni(x)+∑k,j‍B~ni,kj(x)Ωkj(x),(n,i),(k,j)∈T
which is equivalent to (91). The series in (94) converges absolutely and uniformly for x∈[0,π]. Analogously, (74) takes the form
(95)Bni,kj(x)-B~ni,kj(x)+∑l,s‍B~ni,ls(x)Bls,kj(x)=0,hh(n,i),(k,j),(l,s)∈T
or
(96)(E+B~(x))(E-B(x))=E.
Replacing L and L~, similarly, we get
(97)Ω(x)=(E-B(x))Ω~(x),(E-B(x))(E+B~(x))=E.
Therefore, the operator (E+B~(x))-1 exists and it is a linear bounded operator.Equation (91) is called the main equation of the inverse problem. Solving (91), we find the vector Ω(x) and as a result the functions φni(x). we can construct the function q(x) due to φni(x)=φ(x,λni) are the solutions of (1). So, we get the following algorithm for the solution of inverse problem.Algorithm 12.
Given the numbers{λn,αn}n≥0. First, select L~ such that C=C~ and construct Ω~(x), B~(x). Second, find Ω(x) by solving (91). Finally, calculate q(x) by (75) and (80).



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*Source: 804383-2014-02-18.xml*