doi:10.5556/j.tkjm.50.2019.3356
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--- -This paper is available online at http://journals.math.tku.edu.tw/index.php/TKJM/pages/view/onlinefirst
AN INVERSE SPECTRAL PROBLEM FOR
STURM-LIOUVILLE OPERATORS WITH SINGULAR POTENTIALS
ON ARBITRARY COMPACT GRAPHS
S. V. VASILIEV
Abstract. Sturm-Liouville differential operators with singular potentials on arbitrary com-pact graphs are studied. The uniqueness of recovering operators from Weyl functions is proved and a constructive procedure for the solution of this class of inverse problems is provided.
1. Introduction
The paper is devoted to the theory of inverse spectral problems for differential opera-tors on geometrical graphs. The inverse problem consists in recovering the potential from the given spectral characteristics. Differential operators on graphs are intensively studied by mathematicians in recent years and have applications in different branches of science and en-gineering. The inverse problem for the classical Sturm-Liouville operator on the interval has been studied comprehensively in the papers [3] - [6]. The case of inverse problem for Sturm-Liouville operators with potentials from the classW2−1, which we call the singular potentials, on an interval was extensively studied in [7]-[9]. The inverse problems for the classical Strum-Liouville operator on the graphs were investigated in many papers [15]-[21]. The main result for such operators was obtain in [21], where the arbitrary graph has been considered. The case of inverse problem for Sturm-Liouville operators with singular potentials on graphs is more difficult for investigating, and nowadays there are only a few papers in this area. The inverse problem on star-type graph with such type of potentials has been studied in [33]. Also, some specific types of graph has been considered in papers [30]-[32]. The inverse spectral problem for Sturm-Liouville operators with singular potentials on arbitrary graph has not been studied yet. In this paper we consider the solution of the inverse spectral problem for Sturm-Liouville differential operators with singular potentials on compact arbitrary graphs. As the spectral characteristic we consider the Weyl functions, as itî ˝Uÿ done in [33]. A constructive procedure
2010Mathematics Subject Classification. 34A55, 34K29, 34B45.
Key words and phrases. Sturm–Liouville operator, singular potentials, spectral problems, method of spectral mappings.
for the solution of the inverse problem from the given spectrums are provided. We develop the ideas of the method of spectral mappings [6] for studying this inverse problem.
LetGbe a metric graph with a set of verticesV(G) and a set of edgesE(G). We assume that all edges are the smooth curves which can intersect only in the vertices. Let le be the
length of the edgee∈E(G). We consider each edgeeas a segment [0,le] and parameterize it by the parameterxe ∈[0,le]. Letµbe the map, which assigns to each edge an order pair of verticese±∈V(G): µ(e)=[e−, e+], wheree− ande+areinitialandterminalvertices ofe
respectively. It is convenient for us to choose the orientation such thatxe=lecorresponds to the vertexe−andx
e=0 corresponds to the vertexe+. For every vertexv, we denote byI(v,G)
the set of all the edges incidental tov. The number of elements inI(v,G) is called the valency of vertexvand is denoted byv al(v). The vertexvis called aboundary vertex, ifv al(v)=1. All other vertices are calledinternal. LetVB(G) be a set of boundary vertices andVI(G) be a set of internal vertices. The edgeeis called aboundary edge, ife+∈VB(G) ore−∈VB(G). All other edges are calledinternal. LetEB(G) be a set of boundary edges andEI(G) be a set of internal edges.
A chain of edges {e1, . . . ,en} is called acycleif it forms a closed curve. The edge is called simple, if it is not a part of any cycle. The set of simple edges is denoted byES(G). LetEC(G) be the set of edges, which form the set of cycles. For definiteness, we suppose, that setVB(G) is not empty and contain at least two vertices. Fix some boundary vertexv0 and call it the root. The corresponding edger0∈I(v0,G) is called therooted edge. We agree, that ife∈ES(G), thane−is nearer to the root, thane+. For each edgee∈EB(G) we define
µB(e,G) :=
(
e+, ife+∈VB(G),
e−, ife−∈VB(G).
If we contract each cycle to a point, then we obtain a new graphG∗, such thatE(G∗)=
ES(G). Clearly,G∗ is a tree. Fix somee∈G∗. The minimal numberχe of edges on theG∗
between rooted edge and edgee, includinge, we called theorderof edgee. The order of the rooted edger0is equal to zero. The number
χ:= max
e∈E(G∗)χe
is called theorderof theG∗. The set of edges of orderνis denoted byE(ν),ν=0,χ.
A functionyon graphGis considered asy=[ye(xe)]e∈E(G),xe∈[0,le]. Letq=[qe(xe)]e∈E(G) be a real-valued function onGsuch thatqe∈W2−1[0,le], i.e.qe(xe)=σ′e(xe),σe(xe)∈L2[0,le],
where the derivative is considered in the sense of distributions. Functionσ=[σe(xe)]e∈E(G) we call thepotential. The Sturm-Liouville differential operator on the edges e∈E(G) is de-fined by the following expression:
ℓeye:= −
¡
whereye[1]:=ye′−σe(xe)ye- is a quasi-derivative, and
d om(ℓe)={ye|ye∈W21[0,le], ye[1]∈W11[0,le],ℓyee∈L2[0,le]}.
We consider the Sturm-Liouville equation one∈E(G):
(ℓeye)(xe)=λye(xe), xe∈(0,le), ye∈d om(ℓe). (1)
At internal vertexvwe consider the following matching conditionsMC(v,G):
ye|v=yr|v,e,r∈I(v,G),
X
e∈I(v,G)
∂eye|v=0, (2)
where
ye|v:=
(
ye(0), v=e+
ye(le), v=e− , ∂eye|v:=
(
y[1]e (0), v=e+
−ye[1](le), v=e−
We denote byMC(G) the matching conditionsMC(v,G),v∈VI(G).
Fix somek∈EC(G). In the casek+6=k−at internal vertexu=k+we consider the
follow-ing matchfollow-ing conditionsMCk(u,G):
ye|u=yr|u, e,r∈I(u,G)\{k}, X
e∈I(u,G)\{k}
∂eye|u=0. (3)
In the case|I(u,G)\{k}| =1 this matching condition has the form∂eye|u=0,e∈I(u,G)\{k}.
In the casek+=k−at internal vertexu=k+we consider the following matching condi-tionsMCk(u,G):
yk|k−=ye|u, ye|u=yr|u, e,r∈I(u,G)\{k}, X
e∈I(u,G)\{k}
∂eye|u+∂kyk|k−=0. (4)
We denote byMCk(G) the matching conditionsMC(v,G),v∈VI(G)\{k+} and the matching conditionMCk(k+,G).
Fix somer ∈EB(G)\{r0}. Let us consider the solution ϕer of the equation (1) on edge e∈E(G), satisfyingMC(G) and the boundary conditions:
∂eϕer|µB(e,G)=δer, e∈EB(G), (5)
whereδer is the Kronecker delta. The functionMr(λ,G) :=ϕr r|µB(r,G),r∈EB(G)\{r0} we call theWeyl functionfor (1) with respect to the edger∈EB(G)\{r0}.
Fix somek∈EC(G). Let us consider the solutionϕekof the equation (1) on edgee∈E(G),
satisfyingMCk(G) and the boundary conditions:
The functionMk(λ,G) :=ϕkk|k+,k∈EC(G) we call theWeyl functionfor (1) with respect to the edgek∈EC(G). DenoteM(λ,G)=[Me(λ,G)]e∈EB(G)\{r
0}∪EC(G). The vectorM(λ,G) is called the
Weyl vector. The inverse problem is formulated as follows.
Inverse problem 1.Given the Weyl vectorM(λ,G), construct the potentialσ.
Everywhere below if a symbolαdenotes an object, related toσ, thenαewill denote the analogous object, related toσeand ˆα=α−αe. Let us formulate the uniqueness theorem for the solution of Inverse Problem 1.
Theorem 1. If M(λ,G)=Mf(λ,G), thenσ=σe. Thus, the specification of the Weyl vector M(λ,G)
uniquely determines the potentialσon G.
The paper is structured as follows. Section 2 contains some auxiliary propositions. Sec-tion 3 is devoted to the soluSec-tion the so-called auxiliary inverse problems. In the secSec-tion 4 we prove Theorem 1, provide the descent procedure and the solution of the global inverse problem on the graph.
2. Auxiliary propositions
Let us consider the boundary value problemLΩ(G),Ω⊂EB(G), for equation (1) with the matching conditionsMC(G) and boundary conditions
∂eye|µB(e,G)=0, e∈EB(G)\{Ω}, yr|µB(r,G)=0, r∈Ω. (7)
We defineL(G) :=L∅(G). Also we consider the boundary value problemLν
k(G),ν=0, 1,k∈ EC(G) for equation (1) with the matching conditionsMCk(G) and boundary conditions
∂eye|µB(e,G)=0, e∈E
B(G), ∂ν
kyk|k+=0, (8)
where∂0kyk|k+=yk|k+and∂1
kyk|k+=∂kyk|k+.
LetCe(xe,λ),Se(xe,λ),ψe(xe,λ) andζe(xe,λ) be the solutions of equation (1) on the edge e∈E(G) under initial conditions
Ce(0,λ)=S[1]e (0,λ)=1, Ce[1](0,λ)=Se(0,λ)=0,
ζe(le,λ)= −ψ[1]e (le,λ)=1, ψe(le,λ)=ζ[1]e (le,λ)=0.
(9)
For each fixedxe∈[0,le] the functionsCe(xe,λ),Se(xe,λ),Ce[1](xe,λ),S[1]e (xe,λ) andζe(xe,λ),
ψe(xe,λ),ζ[1]e (xe,λ),ψ[1]e (xe,λ),e∈E(G) are entire inλ. Moreover,
where〈y,z〉:=y z[1]−y[1]zis the Wronskian ofyandz. LetY ={ye(xe)}e∈E(G)be a solution of equation (1) on graphG. Then
ye(xe,λ)=Me0(λ)Se(xe,λ)+Me1(λ)Ce(xe,λ). (10)
Substituting this representation into matching and boundary conditions of the boundary
value problemLin fixed order (analogous to [21]), we obtain a linear algebraic system with respect toae(λ),be(λ),e∈E(G). The determinant∆(λ,L) of this system is an entire function.
The zeros of∆(λ,L) coincide with the eigenvalues ofL. The function∆(λ,L) is called char-acteristic functionfor boundary value problemsL. We denote byΛΩ:={λΩn}n≥0the zeros of
∆(λ,LΩ(G)) and defineλΩn=ρ2Ωn,n≥0.
As in the classical case [21] one can show that the functionsMe(λ),e∈EB(G)\{r0}∪EC(G)
are meromorphic inλ, namely:
Me(λ)= −
∆(λ,Le(G))
∆(λ,L(G)) ,e∈E B(G)\{r
0}, Mr(λ)= −
∆(λ,L0r(G))
∆(λ,L1r(G)),r∈E
C(G) (11)
LetHe,e∈E(G), be the classes of functions, which are entire inρfor allx∈[0,le] and fixed
potentialσe, such that forηe(xe,ρ,σe)∈Hefollowing conditions are valid:
1) ηe(xe,ρ,σe)=o(exp(xe|I mρ|)) forρ→ ∞and any fixedxe∈[0,le] andσe∈L2[0,le].
2) ηe(xe,·,σe)∈L2(γ) for allxe∈[0,le], realτand fixedσe∈L2[0,le], where
γ=γ(τ) :=(−∞ +iτ,+∞ +iτ).
3) ηe(·,·,σe)∈L2[0,le]×γand bounded uniformly on [0,le]×γfor any fixed realτandσe∈ L2[0,le].
4) ηe(xe,ρ,σe) depends continuously on the potential in the following sense: ifσen(xe)→
σe(xe) inL2[0,le] asn→ ∞, then the correspondingηe(xe,ρ,σen)∈Heconverges toηe(xe,ρ,σe)∈ Heuniformly asn→ ∞on [0,le]×γfor allτ>τ0and
max
xe∈[0,le]
||ηe(xe,·,σen)−ηe(xe,·,σe)||L2(γ)→0.
Obviously, ifηe(xe,ρ,σe), η∗e(xe,ρ,σe)∈He, thanηe(xe,ρ,σe)+η∗e(xe,ρ,σe)∈He.
DefineAε(τ0) :={ρ:I mρ≥0,d i st(ρ,Z)>ε}, whereZ⊂{ρ: 0≤I mρ≤τ0} is countable set with constrained number of points inReρ∈[t,t+1],I mρ∈[0,τ0]. LetK be the class of meromorphic functions, such that forκ(ρ,σ)∈Kfollowing conditions are valid:
1) κ(ρ,σ)=o(1) forρ→ ∞and fixedσ∈L2(G),ρ∈Aε(τ0), whereτ0depends onκ.
3) κ(ρ,σ) depends continuously onσ, in the following sense: ifσn(x)→σ(x) inL2(G) as n→ ∞, thenκ(ρ,σn)∈Kconverges toκ(ρ,σ)∈K uniformly asn→ ∞onγfor allτ>τ0 and
lim
n→∞||κ(·,σn)−κ(·,σ)||L2(γ)→0.
Obviously, ifκ(ρ,σ),κ∗(ρ,σ)∈K, thanκ(ρ,σ)+κ∗(ρ,σ)∈K andκ(ρ,σ)κ∗(ρ,σ)∈K. De-fine [1] :=1+κ(ρ),κ(ρ)∈K. We consider the solutions of equation (1):
ξe(xe,λ) :=Ce(xe,λ)−iρSe(xe,λ), Ee(xe,λ) :=ζe(x,λ)−iρψe(x,λ), e∈E(G)
Letλ=ρ2,I mρ≥0. Analogous to [33], we obtain
ξe(le,λ)=e−iρle[1], Ee(0,λ)=e−iρle[1],
ξ[1]e (le,λ)= −iρe−iρle[1], Ee[1](0,λ)=iρe−iρle[1],
(12)
Using Liouville’s formula and (12), we obtain〈ξe,Ee〉 =2iρe−iρle[1]. Clearly, that〈ξe,Ee〉 6≡0
and consequently {ξe(xe,λ),Ee(xe,λ)} is a fundamental system of solutions. We consider the
case of some fixedr∈EB(G). Then
ϕer(xe,λ)=Aer(λ)ξe(xe,λ)+Ber(λ)Ee(xe,λ), e∈E(G). (13)
Substituting (13) into (2) and (5) in fixed order (analogous to [21]), we obtain the system of linear equations with variables Aer(λ) andBer(λ). Determinant of this system we define as
∆E(λ,G).
Lemma 1. DefineΘ:={ P
e∈E(G)
θele,θe∈{0, 1, 2}}. The following representation is valid:
∆E(λ,G)=(iρ)nX
l∈Θ
Al(G)e−iρl[1], A|G|6=0, (14)
where n:= |VI(G)| + |VB(G)|,|G|:=2 P
e∈E(G) le.
Proof.In the each internal vertexu∈VI(G) we define a variableαuand we define the
indica-tor function
Js±(u) :=
(
1, s±=u,
0, s±6=u. (15)
Substitute (13) into (2) and (5), we obtain
ξe(le,λ)Aer(λ)+Ber(λ)−
X
v∈VI(G)
Je−(v)αv=0,
Aer(λ)+Ee(0,λ)Ber(λ)− X
v∈VI(G)
Je+(v)αv=0,
X
e∈E(G) n
Je+(u)
h
Ee[1](0,λ)Ber(λ)−iρAer(λ)
i
−Je−(u)
h
ξ[1]e (le,λ)Aer(λ)+iρBer(λ)
io
fore∈E(G) andu∈VI(G) and boundary conditions fork∈EB(G)
−iρAkr(λ)+Ek[1](0,λ)Bkr(λ)=δkr, ifk+∈VB(G),
ξ[1]k (lk,λ)Akr(λ)+iρBkr(λ)=δkr, ifk−∈VB(G),
(17)
Using (16), (17) and (12), we obtain
∆E(λ,G)=(iρ)nX
l∈Θ
Al(G)e−iρl[1],
whereA|G|(G) is the determinant of system
Aer(λ)− X
v∈VI(G)
Je−(v)αv=0, e∈E(G), (18)
Ber(λ)− X
v∈VI(G)
Je+(v)αv=0, e∈E(G), (19)
X
e∈E(G) ³
Aer(λ)Je−(u)+Ber(λ)Je+(u)´=0, u∈VI(G), (20)
We denote byP the matrix of this system (18)-(20) and byPe1, Pe2 andPu the rows of the
matrixP, which correspond to the equation (18), (19) and (20) respectively fore∈E(G) and
u∈VI(G). Transform the matrixPby following equation
Pu=Pu−
X
e∈E(G)
Je−(u)Pe1− X
e∈E(G)
Je+(u)Pe2, u∈VI(G),
we obtain, thatA|G|(G) is the determinant of system
Aer(λ)− X
v∈VI(G)
Je−(v)αv=0,
Ber(λ)− X
v∈VI(G)
Je+(v)αv=0,
X
e∈E(G) ³
Je−(u)+Je+(u)´αu=0,u∈VI(G)
(21)
In each vertexu∈VI(G) we denote
N±(u) := X
e∈E(G) Je±(u).
GraphGis connected, thusN+(u)+N−(u)6=0 for eachu∈VI(G). Consequently, using (21), we obtain
A|G|= Y
u∈VI(G)
(N+(u)+N−(u))6=0,
Lemma 2. For sufficiently large |ρ|, such thatρ∈ Aε(τ0), τ0 is some fix number, following inequality is valid
C1|ρ|ne|G|Imρ< |∆E(λ,G)| <C2|ρ|ne|G|Imρ. (22)
Consequently, we obtain for fixedr∈EB(G):
Lemma 3. For each fixed xe∈[0,le]and forρ∈Aε(τ0),τ0is some fix number,ρ→ ∞, following representations are valid
ϕer(xe,λ)=O
³1 ρe
−xeImρ
´
, ϕ[1]er(xe,λ)=O
³
e−xeImρ
´ ,
ˆ
ϕer(xe,λ)=
1 ρe
iρxeκˆ(ρ), κ(ρ),eκ(ρ)∈K.
(23)
Proof.Analogous to [21], using (22) and Cramer’s rule, we obtain
Aer=O³1
ρe
−2leImρ´, Ber=O³1 ρe
−leImρ´.,
ˆ
Ae(λ)=1 ρe
2iρleκˆ(ρ), ˆBe(λ)=1 ρe
iρleκˆ(ρ), κ(ρ),eκ(ρ)∈K.
(24)
Together with (12) and (13), this yields for each fixedxe∈[0,le] to (23).
Using the same arguments, one can prove, that Lemma 3 is valid also for some fixed
r∈EC(G).
3. Auxiliary inverse problem
Fix some edgee∈EB(G)\{r0} and consider the following auxiliary inverse problem on the edgee, which is calledI P(e,G):
Auxiliary inverse problem I P(e,G). GivenMe(λ,G), constructσe(xe),xe∈[0,le].
Using properties of functions from classK, analogous to [33] one can prove following theorem:
Theorem 2. If Me(λ,G)≡Mef(λ,G), thenσe(xe)≡σee(xe)almost everywhere on[0,le].
In theρ-plane consider the contourγ=γ(τ), whereτ>0 is such that inf{Λe∪Λee}> −τ2. LetΓbe the contour in theλ-plane which is the image ofγunder the mappingλ=ρ2. Denote byD+the image of the half-plane {I mρ>τ} andD−:=C\D+. We define the sequence of real numbersδn, n ∈N, such that∀n,k ∈Nδn 6= |ρek|,δn 6= |ρeek|,δk <δk+1, andδn → ∞ as n→ ∞. LetCN :={|λ| =δ2N},γN =γ∩{ρ:|ρ|2=δ2N}, whereCN− :=CN∩D−be the contours
with clockwise orientation. DenoteΓN=Γ∩i ntCN, Γ−
N=ΓN∪C −
N. Denoteθ2=µ. Define
the functions
De(xe,λ,µ) :=
〈Ce(xe,λ),Ce(xe,µ)〉
λ−µ =
Zxe
0
e
De(xe,λ,µ) :=
〈Cee(xe,λ),Cee(xe,µ)〉
λ−µ =
Zxe
0 e
Ce(t,λ)Cee (t,µ)d t,
re(xe,ρ,θ) :=De(xe,λ,µ)θMˆe(µ), ree(xe,ρ,θ) :=Dee(xe,λ,µ)θMˆe(µ).
Everywhere below we chose contourγ(τ) such thatθMeˆ (µ)∈L2(γ). It is always possible to choose such contour because of the properties of functions from classKand (23). Analogous to [33], one can obtain the main equation
Ψe(xe,ρ)=Uee (xe)Ψe(xe,ρ)+Fee(xe,ρ), (25)
whereΨe(xe,ρ) :=Ce(xe,λ)−Cee (xe,λ),
e
Fe(xe,ρ) := 1
2πiN→∞lim
Z
ΓN e
De(xe,λ,µ) ˆMe(µ)Cee (xe,µ)dµ, λ=ρ2, (26)
and for all fixedxe∈[0,le]
e
Ue(xe)f(ρ) := 1 πi
Z
γe
re(xe,ρ,θ)f(θ)dθ, Ue(xe)f(ρ) := 1 πi
Z
γ
re(xe,ρ,θ)f(θ)dθ
OperatorUee (xe) is a Hilbert-Schmidt operator inL2(γ). Also from [33] we obtain the validity of the following theorem:
Theorem 3. For each fixed xe∈[0,le]equation(25)is uniquely solvable in L2(γ).
Using the solutionΨe(xe,ρ) of the main equation (25), one can calculate the function Ce(xe,λ) and then constructσe(xe) according to the next theorem [33].
Theorem 4. The solutionσe(xe)of the I P(e,G)can be found by the formula
σe(x)= −
1 πi
Z
Γ e
Ce(xe,µ) ˆCe(xe,µ) ˆMe(µ)dµ+ 1
πil.i.m.N→∞
Z
γN
ρcos 2ρxeMeˆ (ρ2)dρ (27)
Thus, the solution of the auxiliary inverse problemI P(e,G) can be constructed by the following algorithm.
Algorithm 1.GivenMe(λ).
1) Takeσe=0 and calculateCee(xe,λ),Mfe(λ),Dee(xe,λ,µ) andree(xe,ρ,θ).
2) ConstructFee(xe,ρ)) by (26).
3) FindΨe(xe,ρ) by solving the main equation (25) for eachx∈[0,le].
4) Constructσe(xe) using (27), where ˆCe(xe,λ)=Ψe(xe,ρ).
Auxiliary inverse problem I P(k,G). GivenMk(λ,G), constructσk(xk),xk∈[0,lk].
Fix somek∈EC(G). If we unlink the vertexk+=v∈VI(G) and move it slightly to new vertexv1∉V(G) without moving other edges and without changing the length ofk, we tear apart the cycle, which contains the edgek, and obtain a new graphGk with edgek1instead of kand with the new vertexv1, such thatv16=v andk1+=v1. Moreover,k1∉I(v,Gk) and
the edgek1 is a boundary edge forGk and v1∈VB(Gk). For example, ifE(G)={k,r0,r1}, EC(G)={k,r1},k−6=k+,k+=v∈VI(G) andI(v,G)={k,r1}, then after this procedure we obtain graphGk with edgek1instead ofk and with the new vertexv1, such thatk1+=v1, v16=v,I(v1,Gk)={k1},I(v,Gk)={r1}. Clearly, that in this caser1∈EI(G) andr1∈EB(Gk).
Clearly, inverse problemI P(k,G),k∈EC(G), is equivalent to the inverse problemI P(k1,Gk).
Therefore,I P(k,G),k∈EC(G), is solved by the same arguments asI P(r,G),r∈EB(G)\{r0}.
4. Descent procedure. Solution of the inverse problem 1
Consider the solution of the equation (1) on the edgese∈E(G), represented by (10). Let us construct graphsT andQ. Fix the edger ∈ES(G)∩EI(G)∪{r0}. Denotev:=r+,v∈V(G). The vertexvdivide graphGon two parts:G=Q∪T, whereV(Q)∩V(T)=v,E(T)∩I(v,G)=r,
r∈EB(T) andr∉E(Q)∩I(v,G). Moreover, the rooted edger0∈E(T).
Consider the boundary value problemLΩ(Q,v) for equation (1) with the matching con-ditionsMC(u,Q),u∈VI(Q)\{v} and boundary conditions
∂eye|µB(e,G)=0, e∈EB(G)\{Ω}, yr|µB(r,G)=0, r∈Ω, ∂ryr|v=0, r∈I(v,Q).
We defineL∅(Q,v)=L(Q,v). Using the Laplace expansion by the columns, corresponding to
Me0(λ),Me1(λ),e∈E(T), we obtain two possible cases:
1. The vertexv∈VB(Q),I(v,Q)=:e. Clearly, fork∈EB(Q)∩EB(G) Ã
∆(λ,Le(Q))∆(λ,L(Q))
∆(λ,Lek(Q))∆(λ,Lk(Q))
!Ã
∆(λ,L(T))
∆(λ,Lr(T))
!
=
Ã
∆(λ,L(G))
∆(λ,Lk(G))
!
Using Cramer’s rule and (11), we obtain
Mr(λ,T)=Mk(λ,G)∆(λ,Le(Q))+∆(λ,Lek(Q))
∆(λ,Lk(Q))+∆(λ,Q)Mk(λ,G) . (28)
2. The vertexv∈VI(Q). Clearly, fork∈EB(Q)∩EB(G) Ã
∆(λ,L(Q))∆(λ,L(Q,v))
∆(λ,Lk(Q))∆(λ,Lk(Q,v))
!Ã
∆(λ,L(T))
∆(λ,Lr(T))
!
=
Ã
∆(λ,L(G))
∆(λ,Lk(G))
Analogous to the first case, using Cramer’s rule and (11), we obtain
Mr(λ,T)=
Mk(λ,G)∆(λ,L(Q))+∆(λ,Lk(Q))
∆(λ,Lk(Q,v))+∆(λ,L(Q,v))Mk(λ,G). (29)
Descent procedure. Fix edger ∈ES(G)∩EI(G)∪{r0} and suppose, thatr ∈E(ν) be a fixed simple edge of orderν. Denotev=r−. The vertexvdivide graphGon two partsG=Q∪T, whereV(Q)∩V(T)=v,E(T)∩I(v,G)=r,r∈EB(T) andr ∉E(Q)∩I(v,G). We consider the potentialσ on the graphQ are known. Fixe ∈EB(G)∩E(Q). We consider theMe(λ,G) is known.
1. Weyl functionsMr(λ,T) find from (28) or (29).
2. Solving the inverse problemI P(r,T), we constructσronr.
We suppose, thatMe(λ,G),e∈EC(G)∪EB(G)\{r0} are known. The solution of the inverse problem can be found by the following algorithm
Algorithm 2.
1. For each fixed edgek ∈EC(G) we solve I P(k,G) by the algorithm 1 and findσk on the
edgek.
2. For each fixed edger∈EB(G)\{r0} we solveI P(r,G) by the algorithm 1 and findσr on the
edger.
3. Forν=χ−1, . . . , 0 we perform the following operations: for each fixed edger ∈E(ν)(G)∩
³
ES(G)∩EI(G)∪{r0} ´
using the descent procedure, we findσr.
The considerations above show that the solution of Inverse Problem 1 is uniquely deter-mined, and Theorem 1 is proved.
Acknowledgement
This work was supported by Grant 17-11-01193 of the Russian Science Foundation.
References
[1] V. A. Marchenko,Sturm-Liouville operators and their applications, “Naukova Dumka”, Kiev, 1977; English transl., Birkhauser, 1986.
[2] B. M. Levitan, Inverse Sturm-Liouville problems, Nauka, Moscow, 1984; English transl., VNU Sci.Press, Utrecht, 1987.
[3] G. Freiling and V. A. Yurko, Inverse Sturm-Liouville Problems and their Applications, NOVA Science Publish-ers, New York, 2001. 305 p.
[5] V. A. Yurko, Inverse Spectral Problems for Differential Operators and their Applications, Gordon and Breach, Amsterdam, 2000.
[6] V. A. Yurko, Method of Spectral Mappings in the Inverse Problem Theory, Inverse and Illposed Problems Series, Utrecht, 2002. 303 p.
[7] R. O. Hryniv and Ya. V. Mykytyuk,Inverse spectral problems for Sturm-Liouville operators with singular
po-tentials, Inverse Problems,19(2003), 665–684.
[8] R. O. Hryniv and Ya. V. Mykytyuk,Transformation operators for Sturm-Liouville operators with singular
po-tentials, Mathematical Physics, Analysis and Geometry,7(2004), 119–149.
[9] A. A. Shkalikov and A. M. Savchuk, Sturm-Liouville operators with singular potentials, Math. Notes,66(2003), 741–753.
[10] Yu. V. Pokornyi and A. V. Borovskikh,Differential equations on networks(geometric graphs), J. Math. Sci. (N.Y.),119(2004), 691–718.
[11] Yu. Pokornyi and V. Pryadiev,The qualitative Sturm-Liouville theory on spatial networks, J. Math. Sci. (N.Y.),
119(2004), 788–835.
[12] M. I. Belishev,Boundary spectral inverse problem on a class of graphs(trees)by the BC method, Inverse Prob-lems,20(2004), 647–672.
[13] V. A. Yurko,Inverse spectral problems for Sturm-Liouville operators on graphs, Inverse Problems,21(2005), 1075–1086.
[14] R. Bellmann and K. L. Coike,Differential-difference Equations, Santa Monica, CA: RAND Corporation, 1963, 548p.
[15] G. Freiling and V. A. Yurko,Inverse problems for differential operators on trees with general matching
condi-tions, Applicable Analysis,86(2007), no.6, 653–667.
[16] V. A. Yurko,Inverse problems for Sturm-Liouville operators on graphs with a cycle, Operators and Matrices,2
(2008), 543=-553.
[17] V. A. Yurko,Inverse problem for Sturmî˘a ˇEiouville operators on hedgehog-type graphs. Math. Notes,89(2011), no.3, 438–449.
[18] V. A. Yurko,Inverse problems for Sturm-Liouville operators on bush-type graphs, Inverse Problems,25(2009), 125–127.
[19] V. A. Yurko,Uniqueness of recovering Sturm-Liouville operators on A-graphs from spectra, Results in Mathe-matics,55(2009), 199–207.
[20] V. A. Yurko,On recovering Sturmî˘a ˇEiouville operators on graphs, Mat. Zametki,79(2006), 619–630.
[21] V. A. Yurko,Inverse spectral problems for differential operators on arbitrary compact graphs, Journal of Inverse and Ill-Posed Problems,18(2010), 245–261.
[22] V. A. Yurko,Inverse problems for differential of any order on trees, Matemat. Zametki,83(2008), 139–152; English transl. in Math. Notes,83(2008), 125–137.
[23] I. V. Stankevich,An inverse problem of spectral analysis for Hill’s equation, Doklady Akad. Nauk SSSR,192
(1970), 34–37. (in Russian); English transl. in Soviet Math. Dokl.,11(1970), 582–586.
[24] M. Y. Ignatiev,Inverse spectral problem for Sturm-Liouville operator on non-compact A-graph : Uniqueness
result, Tamkang Journal of Mathematics,44(2013), 25 p.
[25] M. Y. Ignatiev,Inverse scattering problem for Sturm-Liouville operator on one-vertex noncompact graph with
a cycle, Tamkan J. of Mathematics,42s(2011), 154–166.
[26] J. B. Conway, Functions of One Complex Variable, vol.I, 2nd edn., Springer–Verlag, New York, 1995, 412p. [27] M. A. Naimark, Linear Differential Operators. Harrap, London; Toronto, 1968.
[28] A. Yu. Evnin, Polynomial as a sum of periodic functions, Vestn. Yuzhno-Ural. Gos. Un-ta. Ser. Matem. Mekh. Fiz., 2013.
[29] P. Vellucci,A simple pointview for kadec-1/4 theorem in the complex case, Ricerche di Matematica, 2014, 87– 92.
[30] N. P. Bondarenko,A 2-edge partial inverse problem for the Sturm-Liouville operators with singular potentials
[31] N. P. Bondarenko and C.-F. Yang,A partial inverse problem for the Sturm-Liouville operator on the
lasso-graph, Inverse Problems and Imaging,13(2019), 69.
[32] N. P. Bondarenko,An inverse problem for Sturm-Liouville operators on trees with partial information given
on the potentials, Mathematical Methods in the Applied Sciences,42(2019), 1512–1528.
[33] G. Freiling, M. Y. Ignatiev and V. A. Yurko,An inverse spectral problem for Sturm-Liouville operators with
singular potentials on star-type graph, Proc. Symp. Pure Math.,77(2008), 397–408.
Department of Mathematics, Saratov State University, Astrakhanskaya 83, Saratov 410012, Russia.
