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НАУЧНЫЙ ОТДЕЛ
МАТЕМАТИКА
An Inverse Spectral Problem for Sturm - Liouville Operators with Singular Potentials on Graphs with a Cycle
S. V. Vasilev
Sergei V. Vasilev, https://orcid.org/0000-0001-5870-8966, Saratov State University, 83 Astrakhanskaya St., Saratov 410012, Russia, VasilievSV@info.sgu.ru
This paper is devoted to the solution of inverse spectral problems for Sturm - Liouville operators with singular potentials from class W—1 on graphs with a cycle. We consider the lengths of the edges of investigated graphs as commensurable quantities. For the spectral characteristics, we take the spectra of specific boundary value problems and special signs, how it is done in the case of classical Sturm - Liouville operators on graphs with a cycle. From the spectra, we recover the characteristic functions using Hadamard's theorem. Using characteristic functions and specific signs from the spectral characteristics, we construct Weyl functions (m-function) on the edges of the investigated graph. We show that the specification of Weyl functions uniquely determines the coefficients of differential equation on a graph and we obtain a constructive procedure for the solution of an inverse problem from the given spectral characteristics. In order to study this inverse problem, the ideas of spectral mappings method are applied. The obtained results are natural generalizations of the well-known results of on solving inverse problems for classical differential operators.
Keywords: Sturm - Liouville operator, singular potential, graph with a cycle.
Received: 26.02.2019 / Accepted: 05.05.2019/ Published: 02.12.2019
This is an open access article distributed under the terms of
Creative Commons Attribution License (CC-BY 4.0)
DOI: https://doi.org/10.18500/1816-9791-2019-19-4-366-376
INTRODUCTION
The paper concerns the theory of inverse spectral problems for differential operators 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 engineering. The inverse problem for the classical Sturm - Liouville operator on an interval has been studied comprehensively in the papers [1-4]. The case of inverse problem for Sturm - Liouville operators with potentials from class W2-1, which we call the singular potentials, on an interval was extensively studied in [5-7]. The inverse problems for a classical Sturm - Liouville operator on graphs was investigated in many papers [8-14]. The main result for such operators was obtained in [14], 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 investigation, and nowadays there is only a number of papers in this area. The inverse problem on startype graph with such type of potentials has been studied in [15]. Also, some specific types of graphes have been considered in papers [16,17]. The inverse spectral problem for Sturm - Liouville operators with singular potentials on a graph with a cycle has not been studied yet because of the procedure of recovering characteristic functions from the spectra. In this paper we consider the solution of an inverse spectral problem for Sturm - Liouville differential operators with singular potentials on compact graphs with a cycle. As the spectral characteristics we consider the eigenvalues of specific boundary value problems and specific signs, as it is done in [9]. The lengths of the edges of a graph we consider as commensurable quantities. We provide a constructive procedure for the solution of inverse problem from given spectrums.
Let G be a graph with a set of vertices {vj}p=0 and a set of edges {ej}p=0, ej = [v0,vj], where edge e0 generates a cycle. We suppose that the length of edge ej is equal to |ej|. We consider each edge ej as a segment [0, |ej|] and parameterize it by the parameter x e [0, |ej|]. It is convenient for us to choose the orientation such that Xj = |ej| corresponds to the vertex v0. We consider lengths of edges ej, j = 0,p, as commensurable quantities.
Function y on graph G is considered as y = [yj(xj)]p=0, Xj e [0, |ej|]. Let q = [qj(xj)]p=0 be a real-valued function on G such that qj e W2-1[0, |ej|], i.e. qj(xj) = aj(xj), where the derivative is considered in the sense of distributions. We call function a = [aj(xj)]p=0 the potential. The Sturm - Liouville differential operator on the edges ej, j = 0,p is defined by the following expression:
j := -(yj1)' - aj (xj )yj1]- aj (xj ,
where yj11 := yj — aj(x)y3- is a quasi-derivative, and
dom (I3) = {yj | yj e W[0, |ej|], yj11 € W[0, |ej|], I3yj € L[0, ^|]}. We consider the Sturm - Liouville equation on G:
(ijyj)(xj) = Ay,(xj), Xj € (0, |ej j), yj € dom (ij), j = (1)
At the internal vertex v0 we consider the following matching conditions:
yo(0)= yk(jekj), k = 07p, ^yj1](|ejj) = y01](0). (2)
j=o
Let us consider the boundary value problem L(G) for equation (1) with matching conditions (2) and boundary conditions
yi11 (0) = 0, j =T7p. (3)
We define the eigenvalues of L(G) by Л. We also consider the boundary value problem Lk(G), k = 1,p, for equation (1) with matching conditions (2) and boundary conditions
yj1](0) = 0, j = 1^ \ k, yk(0) = 0. (4)
The symbol A(A, L) denotes the characteristic function of some boundary value problem L. The eigenvalues of Lk(G) we define by Лк. Let = ]p=1, k = 1,p, be the solutions of equation (1), satisfying (2) and boundary conditions:
^ц(0)= Skj, (5)
where £kj is the Kronecker delta. Denote Mk(A) := ^kk(0). The function Mk(A) is called the Weyl function for (1) with respect to the vertex vk.
Graph G is partitioned into two parts by the vertex v0: G = G0UT, where G0 is a startype graph and T is a cycle, generated by edge e0. Let Q(A) := C0(|e0|, A) — S01](|e01, A). Taking into account (2) и (3), one can show, that
A(A,L(T)) = C0 (|e0 |,A) + S01] (|eQ |,A) — 2. (6)
Let l(ej), j = 0,p, be the boundary value problem for equation (1) on ej with Dirichlet boundary condition in the end points of edge ej, and let lj(ej) be boundary value problem for equation (1) on ej with boundary conditions yj1](0) = 0, yj(|ej |) = 0. Zeros of function A(A,l(e0)) we denote as {zn}n>1. Also we define шп := signQ(zn), Q = {^n}n>1. The inverse problem is formulated as follows.
Inverse Problem 1. Given the Л, Лк, k = 1,p, Q, construct the potential q on G. The paper is structured as follows. Section 1 contains some auxiliary propositions. Section 2 is devoted to the solution of so-called local inverse problems and a solution of the global inverse problem on a graph.
1. AUXILIARY PROPOSITIONS
Let Cj(xj, A), Sj(xj, A) be the solutions of equation (1) on edge ej, j = 0,p under initial conditions
Cj (0, A) = Sj1](0, A) = 1, Cj1](0, A) = Sj(0, A) = 0. (7)
As in the classical case [13] one can show that functions Mj(A), j = 1,p are meromorphic in A, namely:
M m A(A,Lj(G)) M (A) A(A,l0(eg)) (8)
M(A) = — A(A,L(G)) ' M0(A) = — A(A,l(e0)) ' (8)
Denote by Br the class of Paley-Wiener functions of exponential type not greater than r e R, belonging to L2(R). It follows from [5-7] that
Cj (|ej |, A) = cos p|ej | + j (p), Sj (|e, |, A) = Sinpj + pj (p), (9)
where j(p) e B\e.\ are even functions and j(p) e B\e.\ are odd functions. Clearly,
|ej1 |ej1
j(p) = / Ko(t) sinptdt, Zj,e(p) = / Ke(t) cospt dt, K0, Ke e L(0, |ej|).
Let Hj, j = 0,p, be the classes of functions, which are entire in p for all x e [0, |ej|] and fixed potential aj, such that for nj(x,p, aj) e Hj following conditions are valid:
1) nj(x,p, ) = o(exp(x| Imp|)) for p ^ to and any fixed x e [0, ej] and
aj e L2[0, |ej|];
2) nj(x, •,aj) e l2(y) for all x e [0, ej], real t and fixed aj e L2[0, |ej|], where
Y = y(t) := (-to + zt, +to + zt);
3) nj(•, •,aj) e L2[0, |ej|] x y and bounded uniformly on [0, |ej|] x y for any fixed real t and aj e L2 [0, |ej |];
4) nj(x,p, aj) depends continuously on the potential in the following sense: if (x) ^ aj(x) in L2[0, |ej|], then the corresponding nj(x, p,ajn) e Hj converges to
nj(x,p, aj) e Hj uniformly on [0, |ej|] x y for all t > t0 and
max Hn-(x, •, j - nj(x •,aj)||l2(7) ^ 0-
xG[0,|ei
Obviously, if nj(x,p, aj), nj(x,p, aj) e Hj, then nj(x,p, aj) + nj/(x, p, aj) e Hj. Define Ae(t0) := {p : Im p > 0,dzst(p, K) > e}, where Z C {p : 0 < Im p < t0} is a countable set with a constrained number of points in Rep e [m + 1], Im p e [0,t0]. Let K be the class of meromorphic functions, such that for k(p, a) e K following conditions are valid:
1) k(p, a) = o(1) for p ^ to and fixed a e L2(G), p e Ae(t0), where t0 depends on k;
2) ft(-,a) e L2(y) for all t > t0 and fixed a e L2(G);
3) k(p, a) depends continuously on a, in the following sense: if ajn(x) ^ aj(x) in L2(G), then k(p, an) e K converges to k(p, a) e K uniformly on y for all t > t0 and
lim ||ft(-,an) - ft(-,a)|U2(7) ^ 0.
n—^^o
Obviously, if k(p, a), K*(p, a) e K, then k(p, a) + k*(p, a) e K and k(p, a)k*(p, a) e e K. Define [1] := 1 + k(p), k(p) e K. It follows from [5-7] that the following lemma holds.
Lemma 1. Following representations are valid
sin px 1
Cj (x,A)=cos px + nj (x, p, aj), Sj (x, A) = —---b ^nj (x,p,aj). (10)
Using (10), we obtain
Cj(|ej|, A) = cosp|ej|[1], Sj(|e,|,A) = sinpj[1],
Cj^bA) = —psinp|ej|[1], S'11 (jej|,A) = cosp|e,|[1].
P (11)
j ^' '— ^ 0111 1 l^j > ^j vII ' 'V — cos pjej We consider the solutions of equation (1):
Cj(x, A) := Cj(x, A) - ipSj(x, A), Ej(x, A) := Cj(x, A) + ipSj(x, A), j = 0~p. Clearly, that (Cj ,Ej> = 2ip and
Cj(jejj, A) = e-ipM[1], Ej(0, A) = e-ip|ej|[1], Cj11 (jejj, A) = —ipe-ip|ej1 [1], Ej11 (0, A) = zpe-^1 [1]. ( )
_Изв. С арат, ун-та. Нов. сер. Сер. Математика. Механика. Информатика. 2019. Т. 19, вып. 4 We consider the representation
x, Л) = j(A)£,(x, A) + Bjk(A)Ej(x, A), j = 0^. (13)
Substituting (13) into (2) and (5), we obtain the system of linear equations with variables A,k(A) and Bjk(A). The determinant of this system we define as AE(A, G).
Lemma 2. Following representation is valid
Ae(A,G) = (-i;p)p+^£ (e]1'(|ej|, A) + (|ej|,A)) n (E(^U) + Ci(N|,A)) +
]=0 i=0, i=]
p
;^Eo(|eoA) - Co(|eoA)) - 2ip] JJ (^(h|, A) + Ci(h|, A))]. (14)
Proof. We define variable a in the internal vertex v0. Then AE(A, G) is the determinant of this system
-ipAjk + ipBjk = j, j = (|ejA)Ajk + Ej(|ejA)Bjk = a, j =
( j j jk j j
^Tj(|ejA)Ajk + Ej1](|ejA)B,k) + ipAok - zpBok = 0. j=o
Using Laplace expansion for rows, which defines the matching condition for edge e0, we obtain
Ae(A, G) = (E0(|e0|,G) + C0(|e01, G))Ae(A, G0)+
+(-ip)p+1 n (Ej(|ej|,A)+ C](|ej|,A))[(^01](|e0|,A)+ ^'(^0|,A)) + ]=1
- (C0(|e01, A)E01](|e01, A) - ^'(^0A)E0(K1, A) + ;pC0(M, A) - ;pE0(K1, A))], (15)
where G0 is a star-type graph. Let us define the determinant of the system of linear equations by dm(A)
(Ej(|ej|,A) + Cj(|ej|, A})a, = a, j = mp, £ je]|,A) + jde,|,A))a, = 0
]=m
with variables a,, j = 1,p, a. We add each j-th column to (j + p)-th column, j = 1,p, and get AE(A, G0) = d1(A). Using Laplace expansion for dm(A), m < p, for the first row, we obtain
dm (A) = (Em(|em A) + Cm (|em |,A)) dm+i(A) + + (Em'(|em|,A)+ em'(|em|,A)) n (Ei(h|,A)+ Ci(h|,A)).
i=m+1
It is obvious, that dP(A) = ^E|m](|em|,A)+ ^(|em|,A)^. Using mathematical induction, one can show, that for 1 < m < p following representations are valid:
Ae(A, Go) = (ip)P+1 ^ (j(|ej|, A) + jfe-1, A)) fl (|e<|, A) + &(|e<|, A)).
j=1 i=1, i=j
Substitute this representation into (15), we obtain (14). □
Taking into account (12) and properties of the function k(p) e K, one can show
p p
Corollary 1. Define 0 := {J^ kj|ej|, kj e {0,1}}, a |G| := |ejThen following
j=o j=o
representation is valid:
Ae(A, G) = (-ip)P+1 Y A(G)e-ipi [1], = 0, (16)
From [18] one can obtain the following lemma.
Lemma 3. For sufficiently large |p|, such that p e A(t0), t0 is fixed, following estimate is valid
C1 |p|P+1e|G|Imp < |Ae(A,G)| < G2|p|p+1 e|G|Imp. (17)
The following lemma describes the asymptotic representations of the solutions .
Lemma 4. For fixed x e [0, |ej|] and for p e A(t0), where t0 is fixed and p ^ to, following estimates are valid:
j(x, A) = o(pe-xImp), = o(e-xImp), j(x, A) = pe"pxK(p). (18)
Proof. Using (13) by Cramer's rule, we obtain
j (A) = jfe, (A) = Aeag) , (1Q)
where j(A) and djk(A) are determinants of the matrices, formed by replacing the corresponding column by the column of free terms. Analogous to proof of Lemma 2, we obtain
j(A) = -(V)P+1 Y B(G)e-ipi[1], B|g|-2|ek| = 0,
lG©j
djk (A) = -(V)p+1 Y G; (G)e-'pi [1], G|g| = 0,
(20)
lG©
P
j j j
B:={E0j|ej| 0j e{0,1}, j = 1,p},
j=1
p
Bk :={E 0j|ej| 0j e{0,1}, j = \ k, 0fc = -1}.
j=1
Thus, we obtain
Ci|p|pe(|G|-2|ej|)Imp < fe(A)| < C2|p|pe(|G|-2|ej|)Imp, Ci|p|pe|G|Imp < |djk(A)| < Ci|p|pe|G|Imp.
(21)
Using (16) and (20), we obtain
j(A) - [1], j (A) - - ^__ [1] (22)
Ae(A, G) £ A,(G)e-,Pl L J' Ae(A, G) £ Ai(G)e-ip
lee lee
for p G A(to).
Consequently, we get
djk(A) = Ao ril djk(A) = Bo Г11 (23)
AE^ATG) = Ajkri]' AE(ATG) = j[1]' (23)
where A°k and Bjk are coefficients of an expansion in a fundamental system of solutions (C,(x, A), Ej(x, A)} in case a = 0. Substitute (17) and (21) into (22), for sufficiently large p g Ae we obtain following estimates:
Kk| < Cpe-2|e'|ImP• IBokI < 7
Thus, using (23), we obtain analogously, that for p ^ то and p g A(to) following estimates are valid:
Ajk = 10(e-2|ej1 Im p), Bjk = o(1). (24)
Substitute this estimates into (13), we obtain (18). □
Analogous to Lemma 2 and Corollary 1, one can prove
Lemma 5. Following formulas are valid
A(A, L(G)) = A(A, l(eo))A(A, L(Q)) + (A(A, L(T)) + 1) ^ A(A,
k=1
Aj (A,L(G)) = A(A,l(eo ))Aj (A,L(Q))+ (25)
+(A(A,L(T)) + 1)A(A,l(ej)) Д A(A,lk (ek)),
k=1^\{j }
where A(A,L(Q)) = £ Cj1](|ej|,A) П C(|e,|,A).
j=1 i=1, i=j
Corollary 2. Following representation is valid
A(A,L(G)) = (-1)p £ Di (G)e-ipl [1], D|G| = 0. (26)
lee
The eigenvalues A can be numbered as {Ank}k=0^-i,neN u {Ank}k=;^;m,nez, where m e N, p0 is a multiplicity of a zero eigenvalue of the boundary value problem L(G) with zero potential. Analogously, the eigenvalues Ak can be numbered as
{Ankj}j=0X-I,neN u {Ankjj^.nez, where mk e N MS is a multiPlicity of zero eigenvalue of the boundary value problem Lk(G) with zero potential. From [19] it follows, that characteristic functions A(A, L(G)) and A(A, Lk(G)), k = 1,p, can be constructed from spectra by Hadamard's theorem:
Theorem 1. The specification of spectrums A and Aj uniquely determines the characteristic functions respectively by the formula
A(A,L(G)) = (-I)«^A°(A,L(G)) II T Il II
d "0
■to "o — 1 A .to m . .
Ank — A ^ TT Ank — A
A(A,L,(G)) = ( — 1)"°dA-Aq(A, Lj(G))x=^^n jj^ II II
a=o , Aq1 ^ , Aq1
n=0 k=0 n=—TO k="°
to "0—1
Ajnk — A tto IT Ajnk — A
(27)
\=011 H A 01 11 11 A01
=0 k=0 n=—TO k="j
2. SOLUTION OF THE INVERSE PROBLEM
Fix ek, k e 1,p and consider the following auxiliary inverse problem. Local inverse problems IP(k,G): Given Mk(A), construct ak(x), x e [0, |ek|]. Everywhere below if a symbol a denotes an object, related to a, then a will denote the analogous object, related to a and a = a — a. Using the properties of functions from class K, analogous to [15] one can prove the following theorem:
Theorem 2. If Mk(A) = Mk(A), then ak(x) = Ck(x) almost everywhere on [0, |ek|].
In p-plane consider the contour 7 = 7(r) := (—to + ir, +to + ir), where r > 0 is such that
inf{As U Ak } > —r2.
Let r be the contour in A-plane which is an image of 7 under the mapping A = p2. Denote by D+ the image of the half-plane {Im p > r} and D" := C \ D+. Let CN := {|A| = (N+1/4)2} and C" := CNnD" be the contours with clockwise orientation. Denote rN = r n intCN, r^ = rN U C". Denote 02 = p. Define the functions
x
Dk(x, A, p) := (Ck(x,A),C7k(x,P)) = / Ck(t, A)Ck(t, p) dt,
A — p J
A — p
Dk(x, A, p) := (Ck(x,A),Ck(x,p)) = i Ck(t, A)Ck(t, p) dt,
A — p J
o
rk(x, p, 0) := Dk(x, A, p)0Mk(p), Ck(x, p, 0) := Dk(x, A, p)0Mk(p).
Everywhere below we chose contour 7(r) such that 0Me(p) e L2(7). Analogous to [15], one can obtain the main equation
^k (x) = HHk (x)*k (x) + F (x), (28)
0
x
_Пзв. С арат, ун-та. Нов. сер. Сер. Математика. Механика. Информатика. 2019. Т. 19, вып. 4
where (x, р) := Ck(x, Л) - Ck(x, Л),
F(x):=-^lim/ Dk(xAp)Mk(p)Ck(x,p)dp, (29)
and for all fixed x e [0, |ek|]
Hk(x)f (р) := - i h(x, р, 0)/(0)d0, Hk(x)f (р) := - i n(x, р, 0)/(0)d0, nu Y Y
and Hk(x) is a Hilbert - Schmidt operator in L2(7). Also from [15] we obtain the validity of the following theorems:
Theorem 3. For each fixed x e [0, |ek|] equation (28) is uniquely solvable in L2(7).
Using the solution (x,p) of the main equation (28), one can calculate the function Ck(x, A) and then construct ok(x) according to the next theorem.
Theorem 4. The solution ok(x) of the Problem IP(k) can be found by the formula
ok(x) = —1 [ Ck(x,p)Ck(x,p)Mk(p)dp + — lim i pcos2pxMk(p2)dp, (30) ni Jr ni N^ J7N
where = 7 n {p : |p|2 = (N + 1/4)2}.
Thus, the solution of the local inverse problem IP(k) can be constructed by the following algorithm.
Algorithm 1. Given Mk(A)
1. Take a = 0 and calculate Ck(x, A), Mk(A), Dk(x, A,p) and rk(x,p, 0).
2. Construct F(x,p)) by (29).
3. Find (x,p) by solving the main equation (28) for each x e [0, |ek|].
4. Construct ok(x) using (30), where Ck(x, A) = (x,p).
Using A(A,l0(eo)) = —C0(|e0|,A), A(A,l(e0)) = S0(|e0|,A) and Lemma 1, analogous to [16], we obtain
M0(A)= V ^, M„ = — A(z"'l0(e0)}, (31)
n=0 A A(Zn,1(60))' '
where A(A,l(e0)) = fxA(A,l(e0)) and Mn is Weyl sequence. The solution of the inverse problem 1 can be constructed by the following algorithm. Algorithm 2. Given {An}n>o, {AnkWo, k = T~p.
1. Construct A(A,G), Ak (A,G), k = 1,p by (27). Find Mk (A) using (8). For edges ej, j = 1,p, we find oj by solving local inverse problems by Algorithm 1.
2. Calculate A(A,l(eo)) and A(A, L(T)) via (25).
3. Find zeros {zn}n>i of the function A(A,l(e0)).
4. Define D(A) = A(A, L(T)) + 2. Calculate Q(zn) = D2(zn) - 4.
5. Calculate A(zn,Iq(eo)) = 2(D(zn) + Q(zn)).
6. Find Mn by (31).
7. Calculate M0(A) via (31) and find o0 by solving local inverse problem on edge e0 by Algorithm 1.
Acknowledgements: This work was supported in part by the Russian Ministry of Education and Science (project No. 1.1660.2017/4.6) and by the Russian Foundation for Basic Research (project No. 19-01-00102).
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Cite this article as:
Vasilev S. V. An Inverse Spectral Problem for Sturm - Liouville Operators with Singular Potentials on Graphs with a Cycle. Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform., 2019, vol. 19, iss. 4, pp. 366-376. DOI: https://doi.org/10.18500/1816-9791-2019-19-4-366-376
УДК 517.984
Обратная задача для операторов Штурма - Лиувилля с сингулярными потенциалами на графах с циклами
С. В. Васильев
Васильев Сергей Владимирович, аспирант кафедры математической физики и вычислительной математики, Саратовский национальный исследовательский государственный университет имени Н. Г. Чернышевского, Россия, 410012, г. Саратов, ул. Астраханская, д. 83, VasilievSV@info.sgu.ru
В данной статье исследуются обратные спектральные задачи для дифференциальных операторов Штурма - Лиувилля с сингулярными потенциалами из класса W—1 на графе с циклом. Длины рёбер рассматриваемого графа мы будем считать соизмеримыми величинами. В качестве спектральных характеристик мы рассмотрим спектры некоторых краевых задач, а также специальные знаки, аналогично тому, как это сделано в случае классических операторов Штурма - Лиувилля, заданных на графе с циклом. Используя теорему Адамара, мы восстановим характеристические функции по заданным спектрам краевых задач. Применяя восстановленные характеристические функции, мы построим функции Вейля (так называемые m-функции) на рёбрах рассматриваемого графа. Мы покажем, что задание функций Вейля однозначно определяет коэффициенты дифференциального уравнения на исследуемом графе. Также мы получим конструктивную процедуру решения обратной задачи по заданным спектральным характеристикам. Для решения поставленной задачи в работе используются идеи метода спектральных отображений, применённого для решения обратной задачи для классических операторов Штурма - Лиувилля. Полученный результат является обобщением хорошо известных результатов для обратных задач для классических дифференциальных операторов.
Ключевые слова: оператор Штурма - Лиувилля, сингулярный потенциал, граф с циклом.
Поступила в редакцию: 26.02.2019 / Принята: 05.05.2019 / Опубликована: 02.12.2019
Статья опубликована на условиях лицензии Creative Commons Attribution License (CC-BY4.0)
Благодарности. Работа выполнена при финансовой поддержке Минобрнауки РФ (проект № 1.1660.2017/4.6) и РФФИ (проект № 19-01-00102).
Образец для цитирования:
Vasilev S. V. An Inverse Spectral Problem for Sturm - Liouville Operators with Singular Potentials on Graphs with a Cycle [Васильев С. В. Обратная задача для операторов Штурма -Лиувилля с сингулярными потенциалами на графах с циклами] // Изв. Сарат. ун-та. Нов. сер. Сер. Математика. Механика. Информатика. 2019. Т. 19, вып. 4. С. 366-376. DOI: https://doi.org/10.18500/1816-9791-2019-19-4-366-376
