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Results. Math. Online Firstc© 2013 Springer BaselDOI 10.1007/s00025-013-0352-4 Results in Mathematics
On the Reconstruction of the Sturm-LiouvilleProblems with Spectral Parameterin the Discontinuity Conditions
Yongxia Guo and Guangsheng Wei
Abstract. In this paper, we are concerned with the problem of recoveringthe Sturm–Liouville problem under the circumstance of the discontinu-ity conditions involved spectral parameter at finite interior points of afinite interval. We provide procedures for constructing their potentialsand boundary conditions either from the Weyl function, or from spectraldata, or from two spectra in terms of the method of spectral mappings.
Mathematics Subject Classification (1991). Primary 34A55, 34L05;Secondary 34L40.
Keywords. Inverse spectral problem, interior point condition,method of spectral mapping.
1. Introduction
Let 0 := x0 < x1 < · · · < xm < xm+1 =: 1 be given and X := {xs}ms=1.
Consider the following Sturm–Liouville problem L := L(q(x), h,H):
ly := −y′′ + q(x)y = λy (1.1)
on the direct-sum interval [0, 1]\X =: I̊ with the Robin boundary conditions{U(y) := y′(0) − hy(0) = 0,
V (y) := y′(1) +Hy(1) = 0,(1.2)
and with the interior point conditions{y(x+
s ) = asy(x−s ),
y′(x+s ) = a−1
s y′(x−s ) + (bs − λms)y(x−
s ),(1.3)
Y. Guo and G. Wei Results. Math.
where s = 1, ...,m, λ is the spectral parameter; q(x) ∈ L2[0, 1], h,H and bs areall real; as > 0,ms > 0 for s = 1, ...,m. It is known [10] that the problem hasa discrete spectrum consisting of simple real and bounded below eigenvalues,denoted by σ(L) = {λn}∞
n=0.Here we will confine ourself to the problem of recovering L from the given
sets of spectral characteristics. This paper can be viewed as a continuation of[10] by the authors, in which we gave a self-adjiont operator-theoretic formula-tion in an appropriate produce space L2[0, 1] × C
m for this problem, obtainedsome properties of the spectrum and proved some uniqueness theorems for theinverse problems of L.
Boundary value problems with discontinuities inside the interval oftenappear in mathematics, mechanics, physics, geophysics and other branches ofnatural sciences. As a rule, such problems are connected with discontinuousmaterial properties. For example, the discontinuous Sturm–Liouville problem(1.1)–(1.3) appears in the one-dimensional wave equation corresponding to astring with finitely many embedded point masses (see [3]). Moreover, boundaryvalue problems with discontinuities in an interior point also appear in geophys-ical models for oscillations of the Earth (see [11]). Here the main discontinuityis caused by reflection of the shear waves at the base of the crust. The inverseproblem of reconstructing the material properties of a medium from data col-lected outside of the medium is of central importance, where the data is generalconnected with the spectral information for the Sturm–Liouville problems.
Inverse problems for Sturm–Liouville operators either with the spectralparameter λ in the boundary conditions, or with interior jump point condi-tions (not containing spectral parameter), have been studied fairly completely(see [1,6,11–13] and the references therein). Inverse Sturm–Liouville problemswith interior point conditions depending on the spectral parameter are less toinvestigate, and nowadays there are only a rather limited number of papers inthis direction (see e.g. [3,10]). In particular, Carlson [3] considered the spectraland inverse spectral problems for string equations with finitely many embed-ded masses, proved this problem is isospectral to a regular Sturm–Liouvilleproblem with interior point conditions depending on the spectral parameter,and gave the algorithms for recovering the locations and masses of the string.However, the problem for recovering the potential remains open.
In the present paper, we provide a constructive solution of the inversespectral problems for L from three appropriately given sets of the spectralcharacteristics. In other words, we solve three inverse problems of recoveringL either from the Weyl function, or from spectral data or from two spectra. Themain method of the present paper is by virtue of the spectral mapping theoremdeveloped by Yurko [14]. This approach has been employed skillfully by Freilingand Yurko et al. [5–9] to deal with some variant of inverse Sturm–Liouvilleproblems in a series of papers [2,14]. The obtained results here are naturalgeneralizations of the well-known results on the classical inverse problem sincethe interior point conditions (1.3) involve the spectral parameter.
Inverse Discontinuous SL Problems
The paper is organized as follows. In Sect. 2, we formulate the inverseproblems of the reconstruction of the problem L from the given spectral char-acteristics. Developing the ideas of the method of spectral mappings, we solvethese inverse problems of recovering the L in Sect. 3.
2. Formulation of the Inverse Problem
In this section, we provide the spectral characteristics of the boundary valueproblem L and present the relationship among these spectral characteristics.Moreover, we formulate the inverse problem of the reconstruction of the prob-lem L: from the Weyl function, from the spectral data, and from two spectra.
Let ϕ(x, λ), ψ(x, λ) be the solutions of Eq. (1.1) under the initial condi-tions
ϕ(0, λ) = 1, ϕ′(0, λ) = h; ψ(1, λ) = 1, ψ′(1, λ) = −H, (2.1)
and the interior point conditions (1.3) respectively. For each fixed x ∈ I̊ thefunctions ϕ(x, λ), ψ(x, λ) together with their derivatives with respect to x areentire in λ. For convenience, we denote ρ =
√λ = σ + iτ, Is = (xs, xs+1), ls =
xs+1 − xs,Ms = m1 · · ·ms, M′s = mm · · ·ms+1. It is known [10] that the
following asymptotic estimates hold uniformly with respect to x ∈ Is, 0 ≤ s ≤m, as |ρ| → ∞:⎧⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎩
ϕ(x, λ) = (−1)sρsMs cos(ρl0) sin(ρl1) · · · sin(ρls−1)
× sin(ρ(x− xs)) +O(|ρ|s−1e|τ |x),
ϕ′(x, λ) = (−1)sρs+1Ms cos(ρl0) sin(ρl1) · · · sin(ρls−1)
× cos(ρ(x− xs)) +O(|ρ|se|τ |x),
(2.2)
⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩
ψ(x, λ) = (−1)m−sρm−sM ′s cos(ρlm) sin(ρlm−1) · · · sin(ρls+1)
× sin(ρ(xs+1 − x)) +O(|ρ|m−s−1e|τ |(1−x)),
ψ′(x, λ) = (−1)m−s+1ρm−s+1M ′s cos(ρlm) sin(ρlm−1) · · · sin(ρls+1)
× cos(ρ(xs+1 − x)) +O(|ρ|m−se|τ |(1−x)).
(2.3)
Define the Wronskians determinant 〈y, z〉(x) := (yz′ − y′z)(x) for thefunctions y(x) and z(x) which are all continuously differentiable on [0, 1]\X.It is easy to verify from (1.3) and (2.1) that 〈ϕ,ψ〉(x−
s , λ) = 〈ϕ,ψ〉(x+s , λ) for
s = 1, · · · ,m, which implies that the Wronskians 〈ϕ,ψ〉(x, λ) is continuouson [0, 1]. From this fact and by virtue of Liouville′s formula, we infer that〈ϕ(x, λ), ψ(x, λ)〉 does not depend on x. Denote
Δ(λ) = 〈ϕ(x, λ), ψ(x, λ)〉. (2.4)
Substituting x = 0 and x = 1 into (2.4), one has Δ(λ) = −V (ϕ) = U(ψ).Let {λn}∞
n=0 be the zeros of the function Δ(λ). Then the numbers {λn}∞n=0
coincide with the eigenvalues of the problem L defined by (1.1)–(1.3) and
Y. Guo and G. Wei Results. Math.
therefore the function Δ(λ) is called the characteristic function of L. Thefunctions ϕ(x, λn), φ(x, λn) are eigenfunctions, and there exists a sequence{βn} such that
ψ(x, λn) = βnϕ(x, λn), βn = 0, (2.5)
where βn = (ϕ(1, λn))−1. Throughout this paper we use the notation
αn =
1∫0
ϕ2(x, λn)dx+m∑
s=1
asmsϕ2(x−
s , λn) (2.6)
as the norming constant corresponding to eigenvalue λn. The data Ω ={λn, αn}∞
n=0 are called the spectral data associated with problem (1.1)–(1.3).We note that the spectral data Ω is a generalization of the spectral data forthe classical Sturm–Liouville operators [12].
Lemma 2.1 (see [10]). The following statements hold.(i) For each eigenvalue λn, we have
Δ̇(λn) = αnβn, (2.7)
where Δ̇(λ) = dΔ(λ)/dλ.(ii) Let λn be the nth eigenvalue of the problem (1.1)–(1.3). Let {rk}∞
k=0 bethe zeros (counting with multiplicities if any) of the entire function
Δ00(λ) = ρm+1 cos(ρl0) sin(ρl1) · · · sin(ρlm−1) cos(ρlm).
If rn′ is the closest point to λn, then as n → ∞ρn =
√rn′ +O(n− 1
m+1 ). (2.8)
In addition, if there exists a positive number ν0 such that {rk}∞k=0 satisfy
|√rk − √rk′ | > ν0 as rk = rk′ , (2.9)
then (2.8) holds for n = n′.(iii) Fix δ > 0. Then there exists a constant Cδ such that
|Δ(λ)| > Cδ|ρ|m+1e|τ |, ρ ∈ Gδ, |ρ| ≥ ρ∗ (2.10)
for sufficiently large ρ∗, where Gδ = {ρ : |ρ− √rn| ≥ δ, n ≥ 0}.
It should be noted that {rn}∞n=0 are independent of the potential q and
the boundary conditions (1.2) and some of them may turn out to be repeated,but of multiplicity not exceeding m. Furthermore, it is easy to see that
limn→∞
√rn′
n= A, lim
n→∞ρn√rn′
= A1,
where both A and A1 are positive constants.We define the Weyl function by
M(λ) =ψ(0, λ)Δ(λ)
. (2.11)
Inverse Discontinuous SL Problems
Here the function ψ(0, λ) is the characteristic function of the boundary valueproblem for Eq. (1.1) subject to the boundary conditions y(0) = V (y) = 0and the interior point conditions (1.3). Let {μn}∞
n=0 be the zeros of the entirefunction ψ(0, λ). Obviously, ψ(0, λ) and Δ(λ) have no common zeros. Thus,the Weyl function M(λ) is meromophic with poles in {λn}∞
n=0 and zeros in{μn}∞
n=0.The following lemma provides the relationship among the spectral char-
acteristics of L: the Weyl function M(λ), the spectral data Ω and two spectra{λn, μn}∞
n=0.
Lemma 2.2 (see [10]). Let M(λ), Ω and Δ(λ) be defined as above. Then thefollowing representation holds
M(λ) =∞∑
n=0
1αn(λ− λn)
. (2.12)
Moreover, if condition (2.9) holds, then
Δ(λ) = −C0Mm
m−1∏n=0
(λn − λ)∞∏
n=m
λn − λ
rn, (2.13)
where Mm = m1 · · ·mm and C0 = l1 · · · lm−1.
We shall consider the following inverse problems of recovering L:• Inverse problem 1: Given the spectral data Ω = {λn, αn}∞
n=0, construct q(x)and h,H.
• Inverse problem 2: Given the Weyl function M(λ), construct q(x) and h,H.• Inverse problem 3: Suppose (2.9) holds, given two spectra {λn, μn}∞
n=0, con-struct q(x) and h,H.
It was proved in [10] that the specification of the given spectral charac-teristics uniquely determine the problem (1.1)–(1.3). This gives us the desireduniqueness result for the solutions of the above inverse problems. Furthermore,according to (2.11), (2.12) and (2.13), the inverse problems of recovering theL from the spectral data and from two spectra are particular inverse problemof recovering the L from the Weyl function. Consequently, Inverse problems1-3 are equivalent.
The inverse problems considered here are generalizations of the inverseproblems for the classical Sturm–Liouville operators (see [5, ch. 1] ). In nextsection we will give a constructive procedure for the solution of these inverseproblems. For this purpose we use the ideas of the method of spectral mappingsdeveloped by Yurko [5,14].
3. Solution of the Inverse Problem
In this section, we shall solve the inverse problems of recovering the L(q(x),h,H) by the method of spectral mappings with the help of Cauchy′s integral
Y. Guo and G. Wei Results. Math.
formula and Residue theorem. We reduce the inverse problem to the so-calledmain equation which is a linear equation in a corresponding Banach space ofsequences. Finally, we provide the algorithms for the solution of the inverseproblems by using the solution of the main equation.
For this purpose, relative to L we consider problem L̃ of the same form butwith different coefficients q̃, h̃, H̃. We agree that if a certain symbol γ denotesan object related to L , then γ̃ will denote the analogous object related to L̃.For the convenience of following discussion, we introduce the notations:⎧⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎩
λn0 = λn, λn1 = λ̃n, αn0 = αn, αn1 = α̃n,
ϕni(x) = ϕ(x, λni), ϕ̃ni(x) = ϕ̃(x, λni),
Qkj(x, λ) =〈ϕ(x, λ), ϕkj(x)〉αkj(λ− λkj)
=1αkj
∫ x
0
ϕ(t, λ)ϕkj(t)dt,
Qni,kj(x) = Qkj(x, λni)
for i, j = 0, 1 and n, k ≥ 0, where ϕ̃(x, λ) is the solution of Eq. (1.1) with thepotential q̃ under the initial conditions ϕ̃(0, λ) = 1, ϕ̃′(0, λ) = h̃ and the interiorpoint conditions (1.3). Analogously, we can define Q̃kj(x, λ) by replacing ϕwith ϕ̃ in the above definition. Using Schwarz’s lemma [4, p.130] and (2.2),(2.8) we get the following asymptotic estimates.
Lemma 3.1. Let ϕni(x), Qni,kj(x) be defined as above. Then the following es-timates are valid for x ∈ Is, 0 ≤ s ≤ m:
|ϕni(x)| ≤ C(n+ 1)s, |ϕn0(x) − ϕn1(x)| ≤ C(n+ 1)s− 1m+1 , (3.1)⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎩
|Qni,kj(x)| ≤ C(n+ 1)s
(|n− k| + 1)(k + 1)2m−s,
|Qni,k0(x) −Qni,k1(x)| ≤ C(n+ 1)s
(|n− k| + 1)(k + 1)2m−s+ 1m+1
,
|Qn0,kj(x) −Qn1,kj(x)| ≤ C(n+ 1)s− 1m+1
(|n− k| + 1)(k + 1)2m−s,
(3.2)
where n, k ≥ 0, i, j = 0, 1 and C is a positive constant. The analogous estimatesare also valid for ϕ̃ni(x), Q̃ni,kj(x).
Proof. It follows from (2.2), (2.8) and Schwarz′s lemma that (3.1) is valid.Following, we will only check the first inequality in (3.2), others are similar.Let us show that for x ∈ Is
|〈ϕ(x, λ), ϕkj(x)〉||λ− λkj | ≤ C
|ρ|s(k + 1)se|τ |x
|ρ− k| + 1, (3.3)
where λ = ρ2, ρ = σ + iτ and k ≥ 0. Indeed, if take a fixed δ0 > 0, then for|ρ− ρkj | ≥ δ0 we have by virtue of (2.2) that
|〈ϕ(x, λ), ϕkj(x)〉||λ− λkj | ≤ C
|ρ|s|ρkj |s(|ρ| + |ρkj |)e|τ |x
|ρ2 − ρ2kj |
.
Inverse Discontinuous SL Problems
Moreover |ρ|+ |ρkj | ≤ √2(|ρ+ρkj |) and |ρ−√
rk′ |+1
|ρ−ρkj | ≤ 1+ |ρkj−√rk′ |+1
|ρ−ρkj | ≤ 1+ 2δ0
yield that
|〈ϕ(x, λ), ϕkj(x)〉||λ− λkj | ≤ C
|ρ|s|ρkj |se|τ |x
|ρ− √rk′ | + 1
≤ C|ρ|s(k + 1)se|τ |x
|ρ− k| + 1.
On the other hands, if ρ satisfies |ρ − ρkj | ≤ δ0, then by (2.2) and (2.8) wehave
|〈ϕ(x, λ), ϕkj(x)〉||λ− λkj | =
∣∣∣∣∣∣x∫
0
ϕ(t, λ)ϕkj(t)dt
∣∣∣∣∣∣ ≤ C|ρ|s(k + 1)se|τ |x.
Thus, it is clear that the inequality (3.3) holds for all k ∈ N0 and λ ∈ C.Furthermore, according to (2.2) and (2.6), we deduce αn = O(n2m). Hence,by letting λ = λni in (3.3), we get the estimate for Qni,kj(x). The proofcompletes. �
Let us define the matrix P (x, λ) = [Pjk(x, λ)]j,k=1,2 by the formula
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)
), (3.4)
where
Φ(x, λ) =ψ(x, λ)�(λ)
= S(x, λ) +M(λ)ϕ(x, λ) (3.5)
and S(x, λ) is the solution of Eq. (1.1) with the initial conditions S(0, λ) =0, S′(0, λ) = 1 and the interior point conditions (1.3). The function Φ(x, λ) iscalled the Weyl solution for L. By virtue of (3.5), we calculate{
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)
and {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).(3.7)
Taking (2.2), (2.3) and (2.10) into account, we infer
|P11(x, λ) − 1| ≤ Cδ
|ρ| , |P12(x, λ)| ≤ Cδ
|ρ| , ρ ∈ Gδ, |ρ| ≥ ρ∗, (3.8)
where Gδ is defined in (2.10) and Cδ is a constant.
Y. Guo and G. Wei Results. Math.
Lemma 3.2. Let ϕni(x) and Qni,lj(x) be defined as above. Then the followingrepresentations hold for i, j = 0, 1 and n, l ≥ 0:
ϕ̃ni(x) = ϕni(x) +∞∑
k=0
(Q̃ni,k0(x)ϕk0(x) − Q̃ni,k1(x)ϕk1(x)
), (3.9)
Qni,lj(x) − Q̃ni,lj(x) +∞∑
k=0
(Q̃ni,k0(x)Qk0,lj(x) − Q̃ni,k1(x)Qk1,lj(x)) = 0.
(3.10)
Both series converge absolutely and uniformly with respect to x ∈ [0, 1]\X.Proof. Denote λ′ = minni λni and take a fixed δ > 0. In the λ-plane, weconsider closed contour γn (with counterclockwise circuit) of the form: γn =γ+
n ∪ γ−n ∪ γ′
n ∪ Γ′n. Here
γ±n =
{λ : ±Im(λ) = δ,Re(λ) ≥ λ′, |λ| ≤ a2
n
},
γ′n = {λ : λ− λ′ = δ exp(iθ), θ ∈ (π/2, 3π/2)} ,
Γ′n = Γn ∩ {λ : |Im(λ)| ≤ δ,Re(λ) > 0}, Γn = {λ : |λ| = a2
n},where an ∈ [nπ, (n + 1)π) depends on Δ00(λ). Denote γ0
n = γ+n ∪ γ−
n ∪ γ′n ∪
(Γn\Γ′n) (with clockwise circuit). Let P (x, λ) = [Pjk(x, λ)]j,k=1,2 be the matrix
defined by (3.4). It follows from (3.6) that for each fixed x, the functionsPjk(x, λ) are meromorphic in λ with simple poles λn and λ̃n. We will provethis lemma through the following two steps.
(1) We first prove Eq. (3.9). By Cauchy’s integral formula [4, p.84], we get
P1k(x, λ) − δ1k =1
2πi
∫γ0
n
P1k(x, ξ) − δ1k
λ− ξdξ, k = 1, 2,
where λ ∈ intγ0n and δ1k is the Kronecker delta. Hence
P1k(x, λ) − δ1k =1
2πi
∫γn
P1k(x, ξ)λ− ξ
dξ − 12πi
∫Γn
P1k(x, ξ) − δ1k
λ− ξdξ,
where Γn is used with counterclockwise circuit. Substituting this into (3.7)we obtain
ϕ(x, λ) = ϕ̃(x, λ) +1
2πi
∫γn
ϕ̃(x, λ)P11(x, ξ) + ϕ̃′(x, λ)P12(x, ξ)λ− ξ
dξ + εn(x, λ),
where
εn(x, λ) = − 12πi
∫Γn
ϕ̃(x, λ)(P11(x, ξ) − 1) + ϕ̃′(x, λ)P12(x, ξ)λ− ξ
dξ.
Inverse Discontinuous SL Problems
By virtue of (3.8), limn→∞ εn(x, λ) = 0 uniformly with respect to x ∈[0, 1]\X and λ on compact sets. Taking (3.6) into account we calculate
ϕ(x, λ) = ϕ̃(x, λ) +1
2πi
∫γn
[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
+ εn(x, λ).
Using (3.5) we obtain
ϕ̃(x, λ) = ϕ(x, λ) +1
2πi
∫γn
〈ϕ̃(x, λ), ϕ̃(x, ξ)〉λ− ξ
M̂(ξ)ϕ(x, ξ)dξ − εn(x, λ),
(3.11)
where M̂(λ) = M(λ) − M̃(λ), since the terms with S(x, ξ) and S̃(x, ξ)vanish by Cauchy’s theorem. Moreover, it follows from (2.12) that
Resξ=λkj
〈ϕ̃(x, λ), ϕ̃(x, ξ)〉λ− ξ
M̂(ξ)ϕ(x, ξ) = (−1)jQ̃kj(x, λ)ϕkj(x).
Consequently, calculating the integral in (3.11) by the residue theorem[4, p.112] and then taking λ = λni, we arrive at (3.9) as n → ∞. Fur-thermore, according to asymptotic formulaes (3.1) and (3.2), we derivefor x ∈ Is, 0 ≤ s ≤ m that∣∣∣∣∣
∞∑k=0
(Q̃ni,k0(x)ϕk0(x) − Q̃ni,k1(x)ϕk1(x))
∣∣∣∣∣≤
∞∑k=0
(|Q̃ni,k0(x)(ϕk0(x) − ϕk1(x))|+ |ϕk1(x)(Q̃ni,k0(x)− Q̃ni,k1(x))|
)
≤ C∞∑
k=0
(n+ 1)s
(|n− k| + 1)(k + 1)2m−2s+ 1m+1
.
Thus the series converges absolutely and uniformly on x ∈ [0, 1]\X.(2) Second, we will prove Eq. (3.10). In virtue of the identity
1λ− μ
(1
λ− ξ− 1μ− ξ
)=
1(λ− ξ)(ξ − μ)
,
we have by Cauchy’s integral formula that
Pjk(x, λ) − Pjk(x, μ)λ− μ
=1
2πi
∫γ0
n
Pjk(x, ξ) − δjk
(λ− ξ)(ξ − μ)dξ,
Y. Guo and G. Wei Results. Math.
where j, k = 1, 2, λ, μ ∈ intγ0n. Acting in the same way as above and using
(3.8) we obtain
Pjk(x, λ) − Pjk(x, μ)λ− μ
=1
2πi
∫γn
Pjk(x, ξ)(λ− ξ)(ξ − μ)
dξ + εnjk(x, λ, μ), (3.12)
where limn→∞ εnjk(x, λ, μ) = 0. From the definition of P (x, λ), which isdefined by (3.4), it follows that
P (x, ξ)[ϕ̃(x, λ)ϕ̃′(x, λ)
]= 〈ϕ̃(x, λ), Φ̃(x, ξ)〉
[ϕ(x, ξ)ϕ′(x, ξ)
]−〈ϕ̃(x, λ), ϕ̃(x, ξ)〉
×[
Φ(x, ξ)Φ′(x, ξ)
].
Thus taking (3.12) into account, we calculate
P (x, λ) − P (x, μ)λ− μ
[ϕ̃(x, λ)ϕ̃′(x, λ)
]=
12πi
∫γn
(〈ϕ̃(x, λ), Φ̃(x, ξ)〉[ϕ(x, ξ)ϕ′(x, ξ)
]
−〈ϕ̃(x, λ), ϕ̃(x, ξ)〉[
Φ(x, ξ)Φ′(x, ξ)
])
dξ
(λ− ξ)(ξ − μ)+ ε0n(x, λ, μ) (3.13)
where limn→∞ ε0n(x, λ, μ) = 0. On the other hand, according to (3.4)
P (x, λ)[ϕ̃(x, λ)ϕ̃′(x, λ)
]=
[ϕ(x, λ)ϕ′(x, λ)
].
Therefore,
det(P (x, λ)
[ϕ̃(x, λ)ϕ̃′(x, λ)
],
[ϕ(x, μ)ϕ′(x, μ)
])= 〈ϕ(x, λ), ϕ(x, μ)〉,
and
det(P (x, μ)
[ϕ̃(x, λ)ϕ̃′(x, λ)
],
[ϕ(x, μ)ϕ′(x, μ)
])= 〈ϕ̃(x, λ), ϕ̃(x, μ)〉.
Thus
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)〉. (3.14)
Consequently, (3.13) yields that
〈ϕ(x, λ), ϕ(x, μ)〉λ − μ
− 〈ϕ̃(x, λ), ϕ̃(x, μ)〉λ − μ
=1
2πi
∫γn
〈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 + ε1n(x, λ, μ),
where limn→∞ ε1n(x, λ, μ) = 0. By virtue of (3.5) and the residue theorem,we arrive at (3.10) for λ = λni, μ = λlj . Similar to (3.9), according to
Inverse Discontinuous SL Problems
(3.1) and (3.2), the series in (3.10) converges absolutely and uniformly forx ∈ [0, 1]\X. This proof is completed.
�
As the above arguments, it is seen that, for each fixed x ∈ [0, 1]\X, therelation (3.9) can be considered as a system of linear equations with respectto ϕni(x) for n ≥ 0 and i = 0, 1. But the series in (3.9) converges only “withbrackets”. Therefore, it is not convenient to use (3.9) as a main equation ofthe inverse problem. Below we will transfer (3.9) to a linear equation in acorresponding Banach space of sequences (see (3.19) or (3.20) below).
Let V be a set of indices u = (n, i), n ≥ 0, i = 0, 1. For each fixedx ∈ [0, 1]\X, we define the vector
φ(x) = [φu(x)]u∈V =
[φn0(x)φn1(x)
]n≥0
(3.15)
by the formulae[φn0(x)φn1(x)
]=
[n+1
(ρn)m − n+1(ρn)m
0 1(ρn)m
][ϕn0(x)ϕn1(x)
]=
⎡⎣ (n+1)(ϕn0(x)−ϕn1(x))
(ρn)m
ϕn1(x)(ρn)m
⎤⎦ .
Note that if φn0, φn1 are given, then ϕn0, ϕn1 can be found by the formula[ϕn0(x)ϕn1(x)
]=
[(ρn)m
n+1 (ρn)m
0 (ρn)m
] [φn0(x)φn1(x)
]. (3.16)
Further, we define the block matrix
H(x) = [Hu,v(x)]u,v∈V =[Hn0,k0(x) Hn0,k1(x)Hn1,k0(x) Hn1,k1(x)
]n,k≥0
, (3.17)
where u = (n, i), v = (k, j) and[Hn0,k0(x) Hn0,k1(x)Hn1,k0(x) Hn1,k1(x)
]=
[n+1
(ρn)m − n+1(ρn)m
0 1(ρn)m
]
×[Qn0,k0(x) Qn0,k1(x)Qn1,k0(x) Qn1,k1(x)
] [(ρk)m
k+1 (ρk)m
0 −(ρk)m
].
Analogously we define φ̃(x), H̃(x) by replacing in the previous definitionsϕni(x), Qni,kj(x) by ϕ̃ni(x) and Q̃ni,kj(x), respectively. In view of (2.8), (3.1)and (3.2), we infer for x ∈ Is, 0 ≤ s ≤ m that
|φni(x)|, |φ̃ni(x)| ≤ C
(n+ 1)m−s,
and
|Hni,kj(x)|, |H̃ni,kj(x)| ≤ C
(|n− k| + 1)(n+ 1)m−s(k + 1)m−s+1. (3.18)
Y. Guo and G. Wei Results. Math.
Definition 3.3. The normed space B := {α = [αu]u∈V , ||α||B < ∞} is aBanach space with the norm ||α||B = supu∈V |αu|.
It follows from (3.18) that for each fixed x ∈ Is, the operator E + H̃(x)and E−H(x) (here E is the identity operator), acting from B to B, are linearbounded operators, and
||H(x)||, ||H̃(x)|| ≤ C supn
∞∑k=0
1(|n− k| + 1)(n+ 1)m−s(k + 1)m−s+1
< ∞.
Here, we give the main result of the section.
Theorem 3.4. For each fixed x ∈ [0, 1]/X, the vector φ(x) ∈ B satisfies theequation
φ̃(x) = (E + H̃(x))φ(x) (3.19)
in the Banach space B. Moreover, the operator E+H̃(x) has a bounded inverseoperator, i.e. equation (3.19) is uniquely solvable.
Proof. We rewrite (3.9) in the form[ϕ̃n0(x)ϕ̃n1(x)
]=
[ϕn0(x)ϕn1(x)
]+
∞∑k=0
[Q̃n0,k0(x) −Q̃n0,k1(x)Q̃n1,k0(x) −Q̃n1,k1(x)
] [ϕk0(x)ϕk1(x)
].
Taking into account our notations, we obtain[φ̃n0(x)φ̃n1(x)
]=
[φn0(x)φn1(x)
]+
∞∑k=0
[H̃n0,k0(x) H̃n0,k1(x)H̃n1,k0(x) H̃n1,k1(x)
] [φk0(x)φk1(x)
].
or
φ̃ni(x) = φni(x) +∑k,j
H̃ni,kj(x)φkj(x), (n, i) ∈ V, (k, j) ∈ V, (3.20)
which is equivalent to (3.19). The series in (3.20) converges absolutely anduniformly for x ∈ Is, 0 ≤ s ≤ m. Similarly, using the notations of H(x) werewrite (3.10) as follows
Hni,lj(x) − H̃ni,lj(x) +∑k,t
(H̃ni,kt(x)Hkt,lj(x)) = 0
for (n, i), (l, j), (k, t) ∈ V or in the other form
(E + H̃(x))(E −H(x)) = E. (3.21)
Interchanging places for L and L̃, we obtain analogously
φ(x) = (E −H(x))φ̃(x), (E −H(x))(E + H̃(x)) = E.
Hence the operator (E + H̃(x))−1 exists, and it is a linear bounded operator.This proof is completed. �
Inverse Discontinuous SL Problems
Eq. (3.19) is called the main equation of the inverse problem. Solving(3.19) we find the vector φ(x), and consequently, the functions ϕni(x). Sinceϕni(x) = ϕ(x, λni) are the solutions of (1.1), we can construct the functionq(x) and the coefficients h and H. Thus, we obtain the following algorithm forthe solution of our inverse problems.Algorithm 1. Given the spectral data Ω = {λn, αn}∞
n=0, construct q(x) andh,H.
(1) Choose L̃ and calculate φ̃(x) and H̃(x);(2) Find φ(x) by solving Eq. (3.19) and calculate ϕn0(x) via (3.16);(3) Choose some n (e.g., n = 0) and construct q(x) and h,H by formulae
q(x) =ϕ′′
n0(x)ϕn0(x)
+ λn, h = ϕ′n0(0), H = −ϕ′
n0(1)ϕn0(1)
. (3.22)
Algorithm 2. Given M(λ), construct q(x) and h,H.(1) According to (2.12) construct the spectral data Ω;(2) Choose L̃ and calculate φ̃(x) and H̃(x);(3) Find φ(x) by solving Eq. (3.19) and calculate ϕn0(x) via (3.16);(4) Construct q(x) and h,H by (3.22) for some fixed n (e.g., n = 0).
Algorithm 3. Suppose (2.9) holds, given two spectra {λn, μn}∞n=0, construct
q(x) and h,H.(1) By virtue of (2.11) calculate M(λ);(2) According to (2.12) construct the spectral data Ω;(3) Choose L̃ and calculate φ̃(x) and H̃(x);(4) Find φ(x) by solving Eq. (3.19) and calculate ϕn0(x) via (3.16);(5) Construct q(x) and h,H by (3.22) for some fixed n (e.g., n = 0).
Remark 3.5. Using the main equation (3.19) and the method of spectralmappings [14] one can also obtain necessary and sufficient conditions for thesolvability of the inverse problem similarly to the classical Sturm–Liouvilleproblem [5].
Acknowledgements
The research was supported in part by the NNSF (No. 11171198) and Fun-damental Research Funds for the Central Universities (No. GK 201304001) ofChina.
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Yongxia Guo and Guangsheng WeiCollege of Mathematics and Information ScienceShaanxi Normal UniversityXi’an 710062, People’s Republic of Chinae-mail: [email protected];
Received: November 6, 2013.
Accepted: November 8, 2013.
