DIFFERENTIAL OPERATORS ON GRAPHS WITH A CYCLE Текст научной статьи по специальности «Математика»

Известия Саратовского университета. Новая серия. Серия: Математика. Механика. Информатика. 2021. Т. 21, вып. 3. С. 343-351

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2021, vol. 21, iss. 3, pp. 343-351

https://mmi.sgu.ru https://doi.org/10.18500/1816-9791-2021-21-3-343-351

Article

Differential operators on graphs with a cycle

V. A. Yurko

Saratov State University, 83 Astrakhanskaya St., Saratov 410012, Russia Vjacheslav A. Yurko, [email protected], https://orcid.org/0000-0002-4853-0102

Abstract. An inverse problem of spectral analysis is studied for Sturm - Liouville differential operators on a graph with a cycle. We pay the main attention to the most important nonlinear inverse problem of recovering coefficients of differential equations provided that the structure of the graph is known a priori. We use the standard matching conditions in the interior vertices and Robin boundary conditions in the boundary vertices. For this class of operators properties of spectral characteristics are established, a constructive procedure is obtained for the solution of the inverse problem of recovering coefficients of differential operators from spectra, and the uniqueness of the solution is proved. For solving this inverse problem we use the method of spectral mappings, which allows one to construct the potential on each fixed edge. For transition to the next edge we use a special representation of the characteristic functions. Keywords: Sturm - Liouville operators, geometrical graphs, inverse spectral problems Acknowledgements: This work was supported in part by the Russian Foundation for Basic Research (project No. 19-01-00102).

For citation: Yurko V. A. Differential operators on graphs with a cycle. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2021, vol. 21, iss. 3, pp. 343-351 (in English). https://doi.org/10.18500/1816-9791-2021-21-3-343-351

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0)

Научная статья УДК 539.374

Дифференциальные операторы на графе с циклом

В. А. Юрко

Саратовский национальный исследовательский государственный университет имени Н. Г. Чернышевского, Россия, 410012, г. Саратов, ул. Астраханская, д. 83

Юрко Вячеслав Анатольевич, доктор физико-математических наук, заведующий кафедрой математической физики и вычислительной математики, [email protected], https://orcid.org/0000-0002-4853-0102

Аннотация. Исследуется обратная задача спектрального анализа для дифференциальных операторов Штурма - Лиувилля на графе с циклом. Основное внимание уделяется наиболее важной нелинейной обратной задаче восстановления коэффициентов дифференциальных уравнений при условии, что структура графа известна априори. Используются стандартные

условия склейки во внутренних вершинах и краевые условия Робина в граничных вершинах. Для данного класса операторов установлены свойства спектральных характеристик, получена конструктивная процедура решения обратной задачи восстановления коэффициентов дифференциальных операторов по спектрам и доказана единственность решения. Для решения этой обратной задачи используется метод спектральных отображений, который позволяет строить потенциал на каждом фиксированном ребре. Для перехода к следующему ребру используется специальное представление характеристических функций. Ключевые слова: операторы Штурма - Лиувилля, геометрические графы, обратные спектральные задачи

Благодарности: Исследование выполнено при частичной финансовой поддержке РФФИ (проект № 19-01-00102).

Для цитирования: Yurko V. A. Differential operators on graphs with a cycle [Юрко В. А. Дифференциальные операторы на графе с циклом] // Известия Саратовского университета. Новая серия. Серия: Математика. Механика. Информатика. 2021. Т. 21, вып. 3. С. 343-351. https://doi.org/10.18500/1816-9791-2021-21-3-343-351

Статья опубликована на условиях лицензии Creative Commons Attribution 4.0 International (CC-BY 4.0)

Introduction

We study the inverse spectral problem for Sturm - Liouville differential operators on a graph with a cycle and with standard matching conditions in the internal vertex. Inverse spectral problems consist in recovering operators from their spectral characteristics. The main results on inverse problems for differential operators on an interval are presented in [1,2]. Inverse spectral problems for differential operators on graphs were studied in many works (see the review paper [3] and the references therein). In this paper, we obtain the solution of inverse spectral problems of recovering potentials of Sturm -Liouville operators on a graph with a cycle from the given spectral characteristics and prove the uniqueness of the solution.

Consider a compact graph T in Rm with vertices V = {v0,...,vr} and edges E = {e0,...,er}, where vi,...,vr are the boundary vertices, v0 is the internal ver-

r

tex, ej = [vj,v0], j = 1,r, P| e¿ = {v0}, and e0 is a cycle. Thus, the graph T has one

j=0

cycle e0 and one internal vertex v0. Let Tj, j = 0,r, be the length of the edge ej. Each edge ej £ E is parameterized by the parameter xj £ [0,Tj]. It is convenient for us to choose the following orientation: for j = 1,r, the vertex Vj corresponds to Xj = 0, and the vertex v0 corresponds to Xj = Tj; for j = 0, both ends x0 = +0 and x0 = T0 — 0 correspond to v0. An integrable function Y on T may be represented as Y = {yj}j=0,r, where the function y-(xj), Xj £ [0,Tj], is defined on the edge ej. Let q = {qj}j=0r be an integrable real-valued function on T; q is called the potential. Consider the following differential equation on T:

— yj(xj) + qj(xj)yj(xj) = tyj(xj)5 j = oTr, (i)

where A is the spectral parameter, the functions yj,yj, j = 0,r, are absolutely continuous on [0,Tj] and satisfy the following matching conditions in the internal vertex v0:

r

У0 (0)= yj (Tj), j = 077, y0 (0) = £ yj (Tj), (2)

j=0

Matching conditions (2) are called the standard conditions. In electrical circuits, (2) expresses Kirchhoff's law; in elastic string network, it expresses the balance of tension, and so on.

Let us consider the boundary value problem Bo on T for equation (1) with the matching conditions (2) and with the following boundary conditions at the boundary vertices v1,..., vr:

Uj (Y ) = 0, j = ITT,

where Uj(Y) := yj(0) — bjyj(0), and b = {bjjir is a real vector. Moreover, we also consider the boundary value problems Bk, k = 1,r, on T for equation (1) with the matching conditions (2) and with the boundary conditions

yk(0) = 0, Uj (Y) = 0, j = lT7 \ k.

Denote by Ak := {Akn}n^o the eigenvalues (with multiplicities) of the problem Bk, k = 0,r. In contrast to the case of trees, here the specification of the spectra Ak, k = 0,r does not uniquely determine the potential, and we need additional information. Let Sj(xj, A), Cj(xj, A), j = 0,r be the solutions of equation (1) on the edge ej with the initial conditions Sj(0, A) = Cj(0,A) = 0, Sj(0, A) = Cj(0, A) = l. For each fixed

Xj e [0,Tj], the functions Sjv)(xj, A), Cjv)(xj, A), j = 0,r, v = 0, l are entire in A of order 1/2. Moreover,

(Cj (Xj ,A),Sj (Xj, A)} = 1,

where (y,z} := yz' — y' z is the Wronskian of y and z. Let ^j (xj, A) := Cj (xj, A) + + bj Sj (xj, A), j = 1,7. Denote h(A) := So (To ,A), H (A) := Co (To, A) — SO (To,A). Let {vn}n^1 be zeros of the entire function h(A), and let := signH(vn), Q = {^n}n^1. The inverse problem is formulated as follows.

Inverse problem 1. Given Ak, k = 0,r and Q, construct the potential q on T and the vector b.

Let us formulate the uniqueness theorem for the solution of Inverse Problem 1. For this purpose together with q, b we consider a pair q, b. Everywhere below if a symbol a denotes an object related to q, b, then a will denote the analogous object related to q b.

Theorem 1. If Ak = Ak, k = 0,r, and Q = Q, then q = q and b = b. Thus, the specification of Ak, k = 0, r and Q uniquely determines the potential q on T and the vector b.

This theorem will be proved in section 3. Moreover, we will give a constructive procedure for the solution of Inverse Problem 1. In section 2 we introduce the main notions and prove some auxiliary propositions.

1. Auxiliary propositions

Fix k = l , r. Let = {^kj}j=o~r be the solution of equation (1) satisfying (2) and boundary conditions

Uj ($k) = ¿jk, j =T7r, (3)

where j is the Kronecker symbol. Put Mk(A) := $kk(0, A), k = 1,r. The function Mk(A) is called the Weyl-type function with respect to the boundary vertex vk. Clearly,

$kk(xk, A) = Sk(xk, A) + Mk(A)^k(xk, A), xk e [0, Tk], k = 1,7, (4)

and consequently, (^k(xk, Л), Фкк(xk, Л)) = 1. Using the fundamental system of solutions (xj, Л), Sj (xj, Л), we get

Ф^-(xj, Л) = Mkj(Л)5?(Xj, Л) + Mkj(Л)^-(Xj, Л), Xj G [0, Tj], j = 0,7, k = (5)

In particular, M^(Л) = 1, M°k(Л) = Mk(Л). Substituting (5) into (2^and (3) we obtain a linear algebraic system sk with respect to Mj(Л), v = 0,1, j = 0,r. The determinant A0(Л) of sk does not depend on k and has the form

Ао(Л) = (d(A) - 2) J] ^(Tj, A) + D(A)h(A),

?=i

where

d(A) = Co (To, A) + SO (To, A),

r r

D(A) = ^2 ti(T,A) П ti(Tj,A).

i=1 j = 1,j =«

(6)

(7)

(8)

The function A0(A) is entire in A of order 1/2, and its zeros coincide with the eigenvalues of the boundary value problem B0. Solving the algebraic system sk by Cramer's rule, we get Mj(A) = Ajj(A)/A0(A), s = 0,1, j = 0~r, where the determinant Ajj(A) is obtained from A0(A) by the replacement of the column which corresponds to Mj(A) with the column of free terms. In particular,

Afc (A) Ao(A) '

k = 1, r,

(9)

where

Ak(A) = (d(A) - 2)Sk(Tk,A) Ц ^(Tj, A) + D(A)h(A),

j=1,j=k

(10)

Dk(A) = Sk(Tk,A) П tij(Tj, A) + Sk(Tk,A) £ ^(T, A) f] ti(Tj,A). (11)

j=1, j=k

i=1, i=k

j=1,j=i,k (v ),

We note that Ak(A) is obtained from A0(A) by the replacement of ^k j(Tk, A), v = 0,1, with (Tk, A), v = 0,1. The function Ak(A) is entire in A of order 1/2, and its zeros coincide with the eigenvalues of the boundary value problem Bk. The functions Ak(A) are called the characteristic functions for the boundary value problems Bk.

Let us now study the asymptotic behavior of solutions of equation (1). Let A = p2, Imp ^ 0. Denote A := {p : Im p ^ 0}, A5 := {p : argp £ [5, n - 5]}. For each fixed j = 0,r on the edge ej, there exists a fundamental system of solutions of equation (1) {eji(xj, p),ej2(xj,p)}, Xj £ [0,Tj], p £ A, |p| ^ p* with the properties:

1) the functions e^(xj,p), v = 0,1, are continuous for Xj £ [0,Tj], p £ A, |p| ^ p*;

2) for each fixed Xj £ [0,Tj], the functions ejS)(xj,p), v = 0,1, are analytic for Im p > 0, |p| > p*;

3) uniformly in xj £ [0,Tj], the following asymptotical formulae hold

j (Xj ,p) = (ip)v exp(ipxj )[1], ej 2 (Xj ,p) = (-ip)v exp(-zpxj )[1], p £ Л, |p| ^ oo,(12)

> )

where [1] = 1 + O(p-1 ).

r

Fix k = 1, r. One has

(xj, A) = Aj(p)eji(xj,p) + Akj(p)ej2(xj,p), xj e [0,Tj]. (13)

Substituting (13) into (2) and (3) we obtain a linear algebraic system sk with respect to Akj(p), v = 0,1, j = 0,r. The determinant ¿o(p) of sk does not depend on k and has the form

o

Moreover,

¿0(p) = 2r+1 Aq(Л), p e Л, |p| >p*.

r

¿0(p) = (r + 2)exp( - ip^Tj) [1], p e Л*, |pH<x>. (14)

j

j =0

Solving the algebraic system sk by Cramer's rule and using (12) and (14), we get Akk(P) = [1], Akk(p) = a! exp(2ipTk)[1], p e A', |p| e to,

where ak is a constant. Together with (12) and (13) this yields for each fixed xk e [0, Tk):

^kfc}(xk, A) = (ip)v-i exp(ipxk)[1], p e A', |p| ^ to. (15)

In particular, Mk(A) = (ip)-1 [1], p e A', |p| ^ to. Moreover, uniformly in xj e [0,Tj],

Sjv)(xJ,A) = 2^((ip)vexp(ipxj)[1] — (—ip)vexp(—ipxj)[1]), p e A, |p| ^ to, (16)

j(xj,A) = 1((ip)vexp(ipxj)[1] + (—ip)vexp(—ipxj)[1]), p e A, |p| ^ to. (17)

Let us now we study properties of the spectra and the characteristic functions of Bk. Let Akn = (pnk)2, k = 0,r + 1, be the eigenvalues of the boundary value problems Bk with the zero potential q = 0 and b = 0. Let Ak (A) be the corresponding characteristic functions. According to (6)-(8), (14) and (10)-(11) we have

r r r

ao(a) = 2(cos pTo — 1) JJcos pTj- — sin pTo ^ sin pT JJ cos pTj, (18)

j=1 i=1 j=1,j=i

sin pTk

Ak(Л) = 2(cospTo - J] cospTj +

P j=i,j=k

rp r r Г

cos pTk JJ cos pTj - sin pTk ^ sin pTi JJ cos pTj). (19)

j=1, j=k i=1,i=k j=1, j=i,k Denote t := Im p. It follows from (6)-(8), (10)-(11), (16) and (17) that for |p| ^ to,

Ао(Л) = AQ(Л) + o(|p|-1 exp (|r| £j),

j=o

r

Ak(Л) = Ак(Л) + o(|p|-2 exp (|rTj)), k = 1,7. (20)

=0

Using (18)—(20), by the well-known method (see, for example, [4]), one can obtain the following properties of the characteristic functions Ak(Л) and the eigenvalues Лк of the boundary value problems , k = 0,r. 1. For p G Л, |p| ^ oo,

r

Ao(A) = 0( exp (|тj),

j

j=o

Ak(A) = 0(|p|-1 exp (MEj), k = 1,

j=o

2. There exist h > 0 and Ch > 0 such that

r

|Ao(A)| > Ch exp (|тTj),

j=o

r

|Ak(A)| > Ch|p|-1 exp (|тTj), k = TTr

j

=o

for |t| ^ h. Hence, the eigenvalues Akn = lie in the domain |Imp| < h.

3. The number of zeros of Ak(A) in the rectangle n = {p : | Im p| ^ h, Re p G [£,£ + 1]} is bounded with respect to J.

4. Denote Gs = {p : |p - p0n| ^ S V n ^ 0}, S > 0. Then

r

|Ao(A)| ^ C, exp (|т|E Tj), p £ Gs.

j=o

5. There exist numbers rN ^ to such that for sufficiently small 5 > 0, the circles |p| = rN lie in for all N.

6. For n ^ to,

pnk = p^k + 0(pr) • p

'pnk

Now the reconstruction of the characteristic functions from their zeros is studied. Denote

'AL if Akn = 0,

kn if Akn

vkn ,

1 if At = 0.

Akn = "" (21)

By Hadamard's factorization theorem,

Ak(A) = A П , (22)

where

io _ ( i\sj AO,

UAkn - A

Ao1

n=o kn

Ak = (-1)S^dASk Ak(A)),=o • (23)

and sk ^ 0 is the multiplicity of the zero eigenvalue. Let us show that

Akn - A

ж

Ak (A) = Ak П ^^ • (24)

n=o kn

n=o

r

Indeed, by Hadamard's factorization theorem,

Ak(A)= A П ^

(25)

n=0

where Ak = 0 is a constant, Akn = Akn if Akn = 0, and Akn = 1 if Akn = 0. It follows from (22) and (25) that

\01

Ak(A) = A TT Afcnrr f. +

Ak(A) АоП АкпП=Д +

Akn - Ak

kn

Akn - A

Using properties of the characteristic functions and the eigenvalues of Bk one gets for negative A:

Ak (A) АЧ-» Ak (A)

lim

= 1, lim П 1 +

Л^-оо 11 V

and consequently,

n=0

A1

Ak = Akn Akn

n=0 kn

Akn - Akn

Akn - A

= 1,

Substituting this relation into (25) we arrive at (24).

Thus, the specification of the spectrum Ak = {Aknuniquely determines the characteristic function Ak(A) by (24), where and {A^} are defined by (21), (23), (18), and (19).

2. Solution of the inverse problem

In this section we provide an algorithm for the solution of Inverse Problem 1 and prove the uniqueness of the solution. First we consider auxiliary inverse problems.

Fix k = 1,r, and consider the following auxiliary inverse problem on the edge ek, which is called IP(k).

IP(k). Given Mk(A), construct qk(xk), xk £ [0,Tk] and bk.

In IP(k) we construct the potential only on the edge ek, but the Weyl function Mk(A) brings global information from the whole graph. In other words, IP(k) is not a local inverse problem related to the edge ek. Let us prove the uniqueness theorem for the solution of IP(k).

Theorem 2. If Mk(A) = Mk(A), then qk(xk) = qk(xk) a.e. on [0,Tk] and bk = bk. Thus, the specification of the Weyl function Mk uniquely determines the potential qk on the edge ek and bk.

Proof. Let us define the matrix Pk(xk, A) = [Pk (xk, A)]j,s=i,2 by the formula

Pk (xk, A)

Фkk (xk, A)

Фkk (xk, A)

$kk (xk, A)

i'kk (xk, A) J

^kk (xk, A) Фkk (xk, A) L^kk (xk ,A) Ф^ (xk, A)

(26)

Then (26) yields

^k(xk, A) = Piki(xk, A)^k(xk, A) + Pl2(xk, A)^k(xk, A).

(27)

Since (<pfc(xk, Л), Фкк(xfc, Л)) = 1, one has

PS(xk, Л) = (-1)s-1(^(xk,Л)Фк2к-5)(xk, Л) - ^k2-s)(xk,Л)Фкк(xk,Л)). (28) It follows from (15), (16) and (28) that

Pks(xk,Л)= 6U + O(p-1), p G Л5, |p|^o, xk G (0,Tk]. (29)

According to (4) and (28),

Pk(xk, Л) = (-1)s- ^ (xk,^Sk2-s)(xk, Л) - (¿k2-s)(xk(xk,Л)) + +(Mk(Л) - Mk(Л))^(xk, ^k2-s)(xk, Л)).

Since Mk(Л) = Mk(Л), it follows that for each fixed xk, the functions Piks(xk, Л) are entire in Л of order 1/2. Together with (29) this yields P*(xk, Л) = 1, Pf2(xk, Л) = 0. Substituting these relations into (27) we get (xk, Л) = (xk, Л) for all xk and Л, and consequently, qk(xk) = qk(xk) a.e. on [0, Tk] and bk = bk. Theorem 2 is proved. □

Using the method of spectral mappings [2] for the Sturm - Liouville operator on the edge ek one can get a constructive procedure for the solution of the inverse problem ip(k).

Consider the following auxiliary inverse problem on the edge e0, which is called

IP(0).

IP(0). Given ¿(Л), й(Л), О, construct qo(xo), xo G [0,To].

This inverse problem was studied in many papers (see [5] and the references therein). For convenience of the readers we describe here the solution of IP(0). We remind that ¿(Л) = Co (To ,Л) + SO (To ,Л), H (Л) = Co (To, Л) - SO (То,Л), ^ = sign H (vn), where {vn}n^ 1 are zeros of ^(Л). Clearly,

so (To ,vn) = (d(vn) - H(vn))/2. (30)

Since (Co (To, Л), So (To ,Л)) = 1, it follows that

H 2(Л) - d2 (Л) = -4(1 + Co (To, Л)Ь(Л)),

and, consequently,

H(vn)= d2 (vn) - 4. (31)

pT.о

Denote an := / So(t, vn) dt. Then

o

a„ = h(v„)So(To, v„), Л(Л) := ¿^ЛЛ^• (32)

The data {vn, an}n^ 1 are called the spectral data for the potential qo. It is known (see [1,2]) that the function qo can be uniquely constructed from the given spectral data {vn,an}n^ 1. Thus, IP(0) has been solved, and the following theorem is valid.

Theorem 3. The specification of ¿(Л),^(Л),О uniquely determines the potential qo(xo) on [0, To]. The function qo can be constructed by the following algorithm.

Algorithm 1. Given d(A), h(A), O.

1. Find {vnas the zeros of h(A).

2. Calculate H(vn) by (31).

3. Find S0(To,vn) by (30).

4. Calculate {anusing (32).

5. Construct q0 from the given spectral data {vn, by solving the classical inverse Sturm - Liouville problem.

Let us go on to the solution of Inverse Problem 1. First we give the proof of Theorem 1. Assume that Ak = Afc, k = 0,7, and O = O. Then Ak(A) = Ak(A), k = 0,7. By virtue of (9) this yields Mk(A) = Mk(A), k = 1,r. Applying Theorem 2 for each fixed k = 1,r, we obtain

qk(xk) = qk(xfc) a.e. on [0,Tfc], = , k = 1,7,

and, consequently, Ck(xk, A) = Ck(xk,A), Sk(xk, A) = Sk(xk,A), k = !~r, Xk £ [0,Tk]. Taking (6) and (10) into account we deduce d(A) = d(A), h(A) = h(A). Since O = O, it follows from Theorem 3 that q0(x0) = qo(x0) a.e. on [0,T0], and Theorem 1 is proved.

The solution of Inverse Problem 1 can be constructed by the following algorithm.

Algorithm 2. Given Ak, k = 0,r and O.

1. Construct Ak (A), k = 0,7 by (24), where Ak and {Ak^} are defined by (21), (23), (18), and (19).

2. Find Mk (A), k = 1,_rvia (9).

3. For each fixed k = 1,r, solve the inverse problem IP(k) and find qk(xk), xk £ [0, Tk] on the edge ek and bk.

4. For each fixed k = 1,r, construct Ck(xk, A), Sk(xk, A), xk £ [0,Tk].

5. Calculate d(A) and h(A) using (6) and (10).

6. Construct q0(x0), x0 £ [0,T0] from d(A),h(A), O using Algorithm 1.

References

1. Freiling G., Yurko V. A. Inverse Sturm - Liouville Problems and their Applications. New York, NOVA Science Publishers, 2001. 305 p.

2. Yurko V. A. Method of Spectral Mappings in the Inverse Problem Theory. (Inverse and Ill-posed Problems Series, vol. 31). Utrecht, VSP, 2002. 316 p.

3. Yurko V. A. Inverse spectral problems for differential operators on spatial networks. Russian Mathematical Surveys, 2016, vol. 71, no. 3, pp. 539-584. http://dx.doi.org/10.1070/RM9709

4. Bellmann R., Cooke K. Differential-Difference Equations. New York, Academic Press, 2012. 482 p.

5. Yurko V. A. Inverse problems for differential operators with nonseparated boundary conditions in the central symmetric case. Tamkang Journal of Mathematics, 2017, vol. 48, no. 4, pp. 377-387. https://doi.org/10.5556/j-.tkjm.48.2017.2492

Поступила в редакцию / Received 26.01.2021 Принята к публикации / Accepted 14.03.2021 Опубликована / Published 31.08.2021