A partial inverse problem for the Sturm-Liouville operator on a star-shaped graph
aa r X i v : . [ m a t h . SP ] J a n A partial inverse problem for the Sturm-Liouville operator on astar-shaped graph
Natalia P. BondarenkoAbstract.
The Sturm-Liouville operator on a star-shaped graph is considered. We assumethat the potential is known a priori on all the edges except one, and study the partial inverseproblem, which consists in recovering the potential on the remaining edge from the part of thespectrum. A constructive method is developed for the solution of this problem, based on theRiesz-basicity of some sequence of vector functions. The local solvability of the inverse problemand the stability of its solution are proved.
Keywords: partial inverse problem, differential operator on graph, Sturm-Liouville opera-tor, Weyl function, Riesz basis, local solvability, stability.
AMS Mathematics Subject Classification (2010):
1. Introduction
Differential operators on geometrical graphs (also called quantum graphs) models differentstructures in organic chemistry, mesoscopic physics, nanotechnology, microelectronics, acousticsand other fields of science and engineering (see [1–5] and references therein). In the recentyears, spectral problems on quantum graphs attract much attention of mathematicians. Thereader can find the results on direct problems of studying properties of the spectrum and rootfunctions, for example, in [6–8].
Inverse problems consist in constructing differential operatorsby their spectral characteristics. Inverse problems for quantum graphs were studied in [9–16].We focus our attention on the so-called coefficient inverse problems, which consist in recoveringcoefficients of differential equations (i.e. potentials of Sturm-Liouville equations) on the edgesof the graph, while the structure of the graph and the matching conditions in the vertices areknown a priori. Such problems generalize the classical inverse spectral problems on a finiteinterval (see the monographs [17–20]).In this paper, we consider a partial inverse problem for the Sturm-Liouville operator on agraph. The potential is supposed to be known a priori on some part of the graph, and is beingrecovered on the remaining part. We know only a few results in this direction. V.N. Pivovarchick[21] considered the star-shaped graph with three edges, and assumed that the potential is knownon one or two edges. He proved that two or one spectra, respectively, uniquely determine thepotential on the remaining edges. Later C.-F. Yang [22] showed that some fractional part ofthe spectrum is sufficient to determine the potential on one edge of the star-shaped graph, ifthe potential is known on the other edges. The case, when the potential is unknown only on apart of one edge, was considered in the papers [22–24]. However, the authors of the mentionedpapers proved only uniqueness theorems, and presented neither algorithms for solution norsufficient conditions for the solvability of the partial inverse problems.Another type of partial inverse problems was studied in [25]. We considered the Sturm-Liouville operator on the tree, and showed that if the potential is given on one edge, then weneed one spectrum less to recover the potential on the whole graph, comparing with the fullinverse problem (for example, [10]).We note that partial inverse problems on graphs are related with the Hochstadt-Liebermanproblem [26–29] on a finite interval: the potential is given on the half of the interval, recoverthe potential on the other half by one spectrum. However, the methods developed for theHochstadt-Lieberman problem are not always applicable to partial inverse problems on graphs.1n this paper, we consider the Sturm-Liouville operator on the star-shaped graph, andsuppose the potential is known on all the edges except one. We study the partial inverseproblem, formulated by C.-F. Yang [22]. We develop the constructive algorithm for recoveringthe potential from the part of the spectrum. In addition, the local solvability of the inverseproblem and the stability of the solution are proved. Our method is based on the Riesz-basicityof some system of vector functions. We hope that the ideas of this paper will be useful forinvestigation of more complicated partial inverse problems on graphs, and finally help to findthe minimal data, determining the quantum graph.
2. Problem statement
Consider a star-shaped graph G with edges e j , j = 1 , m , of equal length π . For each edge e j , introduce the parameter x j ∈ [0 , π ]. The value x j = 0 corresponds to the boundary vertex,associated with e j , and x j = π corresponds to the internal vertex.We study the boundary value problem L for the system of the Sturm-Liouville equationson the graph G : − y ′′ j ( x j ) + q j ( x j ) y j ( x j ) = λy j ( x j ) , x j ∈ (0 , π ) , j = 1 , m, (1)with the Dirichlet conditions in the boundary vertices y j (0) = 0 , j = 1 , m, (2)and the standard matching conditions in the internal vertex y ( π ) = y j ( π ) , j = 2 , m, (3) m X j =1 y ′ j ( π ) = 0 . (4)The functions q j in (1) are real-valued and belong to L (0 , π ). We refer to them as the potentials on the edges e j .The eigenvalues of L are described by the following lemma. This result was obtained byV.N. Pivovarchick [21] for m = 3, and can be generalized for an arbitrary m . Lemma 1.
The boundary value problem L has a countable set of real eigenvalues, which can benumbered as { λ nk } n ∈ N , k =1 ,m (counting with their multiplicities) to satisfy the following asymp-totics formulas ρ n = n −
12 + ˆ ωπn + κ n n , (5) ρ nk = n + z k − πn + κ nk n , k = 2 , m, (6) where ρ nk = √ λ nk , { κ nk } n ∈ N ∈ l , k = 1 , m , ˆ ω = m m P j =1 ω j , ω j = π R q j ( x ) dx , and z k , k =1 , m − , are the roots of the characteristic polynomial P ( z ) = ddz m Y k =1 ( z − ω k ) , counting with their multiplicities.
2n this paper, we solve the following partial inverse problem. IP . Given the potentials q j , j = 2 , m , and the sequence { λ nk } n ∈ N , k =1 , of the eigenvalues of L . Find the potential q .In view of the symmetry, one can change the potential q to arbitrary q j , j = 2 , m , and theeigenvalues { λ n } n ∈ N to a sequence { λ nk } n ∈ N with an arbitrary fixed k = 3 , m in the problemstatement.The uniqueness theorem for the IP was proved in [22]. Note that for m = 2 the IP turnsinto the standard Hochstadt-Lieberman problem [26].
3. Solution of the IP
Let S j ( x j , λ ) be solutions of equations (1), satisfying the initial conditions S j (0 , λ ) = 0, S ′ j (0 , λ ) = 1, j = 1 , m . The eigenvalues of L coincide with the zeros of the characteristicfunction , which can be represented in the form∆( λ ) = S ′ ( π, λ ) m Y j =2 S j ( π, λ ) + S ( π, λ ) m X j =2 S ′ j ( π, λ ) m Y k =2 k = j S k ( π, λ ) . (7)Impose the following assumptions :(i) all the eigenvalues { λ nk } n ∈ N , k =1 , are distinct.(ii) λ nk > n ∈ N , k = 1 , S j ( π, λ nk ) = 0, j = 1 , m , n ∈ N , k = 1 , z = ω j , j = 1 , m .(v) S ( π, = 0, S ′ ( π, = 0.One can achieve the conditions (ii) and (v) by a shift q j → q j + C , j = 1 , m . A root z ofthe characteristic polynomial can be chosen to satisfy (iv), if not all ω j are equal to each other.Substituting λ = λ nk , n ∈ N , k = 1 ,
2, into (7) and taking the assumption (iii) into account,one can easily derive the following relation − S ′ ( π, λ nk ) S ( π, λ nk ) = m X j =2 S ′ j ( π, λ nk ) S j ( π, λ nk ) . (8)Note that M j ( λ ) := − S ′ j ( π, λ nk ) S j ( π, λ nk ) is the Weyl function for the Sturm-Liouville problem L j oneach fixed edge e j : − y ′′ j ( x j ) + q j ( x j ) y j ( x j ) = λy j ( x j ) , y j (0) = y j ( π ) = 0 . (9)The Weyl functions M j ( λ ) are meromorphic, their poles are simple and coincide with theeigenvalues of the boundary value problems L j . The potential q j can be uniquely recoveredfrom its Weyl function M j ( λ ) by the classical methods (see [17], [20]).Let the potentials q j , j = 2 , m , and the eigenvalues { λ nk } n ∈ N , k =1 , of the problem L begiven. Rewrite the relation (8) in the following form M ( λ nk ) = − m X j =2 M j ( λ nk ) =: g nk , n ∈ N , k = 1 , . (10)The Weyl functions M j ( λ ), j = 2 , m , can be constructed by the given potentials q j . Thus, thevalues g nk = M ( λ nk ), n ∈ N , k = 1 ,
2, are known, and we have to interpolate the meromorphic3unction M ( λ ) by these values. The subsequences { λ n } n ∈ N and { λ n } n ∈ N are asymptotically“close” to the zeros and the poles of M ( λ ), respectively. However, in view of the assumption(iii), the values { λ nk } n ∈ N , k =1 , do not coincide with the poles of M ( λ ).The solution S ( x, λ ) can be represented in terms of the transformation operator [17, 20]: S ( x, λ ) = sin ρπρ + Z π K ( x, t ) sin ρtρ dt, ρ := √ λ. Using integration by parts and differentiation by x , one can easily derive the relations S ( π, λ ) = sin ρπρ − ω cos ρπρ + 1 ρ Z π K ( t ) cos ρt dt, (11) S ′ ( π, λ ) = cos ρπ + ω sin ρπρ + 1 ρ Z π N ( t ) sin ρt dt, (12)where ω = ω = K ( π, π ) , K ( t ) = ddt K ( π, t ) , N ( t ) = ∂∂x K ( x, t ) | x = π . The functions K ( t ) and N ( t ) are real and belong to L (0 , π ). Note that ω j , j = 2 , m , canbe found by the given q j , ˆ ω can be determined from the subsequence { λ n } (see asymptoticformula (5)), and ω = m ˆ ω − m P j =2 ω j , so the number ω is known.Substituting (11) and (12) into the relation g nk = − S ′ ( π, λ nk ) S ( π, λ nk ) , we get1 ρ nk Z π N ( t ) sin ρ nk t dt + g nk ρ nk Z π K ( t ) cos ρ nk t dt = − cos ρ nk π − ( ω + g nk ) sin ρ nk ρ nk π + ωg nk ρ nk cos ρ nk π. (13)Introduce the notation v n − , ( t ) = (cid:20) sin ρ n t g n ρ n cos ρ n t (cid:21) , v n ( t ) = (cid:20) ρ n g n sin ρ n t cos ρ n t (cid:21) , f ( t ) = (cid:20) N ( t ) K ( t ) (cid:21) , (14) f n − , = − ρ n cos ρ n π − ( ω + g n ) sin ρ n π + ωg n ρ n cos ρ n π, (15) f n = − ρ n g n cos ρ n π − ( ω + g n ) ρ n g n sin ρ n π + ω cos ρ n π. (16)For simplicity, we assume here that g n = 0. In view of Lemma 2, the equality g n = 0 can holdonly for a finite numbers of values n ∈ N , and this case requires minor modifications.The vector functions v nk and f belong to the real Hilbert space H := L (0 , π ) ⊕ L (0 , π ).The scalar product and the norm in H are defined as follows( g, h ) H = Z π ( g ( t ) h ( t ) + g ( t ) h ( t )) dt, k g k H = sZ π ( g ( t ) + g ( t )) dt,g = (cid:20) g g (cid:21) , h = (cid:20) h h (cid:21) , g, h ∈ H . Then (13) can be rewritten in the form( f, v nk ) H = f nk , (17)4here n ∈ N ( N := N ∪ { } ), k = 1 and n ∈ N , k = 2.Note that Z π K ( t ) dt = Z π ddt K ( π, t ) dt = ω, since K ( π,
0) = 0. Put v = (cid:20) (cid:21) , f = ω. (18)Then the relation (17) also holds for n = 0, k = 2.Using the relations (11), (12), (14), (15), (16) and the asymptotics (5), (6), we obtain thefollowing result. Lemma 2.
The following sequences belong to l : { g n − (ˆ ω − ω ) } n ∈ N , (cid:8) g n n − ( ω − z ) (cid:9) n ∈ N , { f nk } n ∈ N , k =1 , , {k v nk − v nk k H } n ∈ N , k =1 , . Here and below v n = (cid:20) sin( n + 1 / t (cid:21) , v n = (cid:20) nt (cid:21) , n ∈ N . Lemma 3.
The vector functions { v nk } n ∈ N , k =1 , form a Riesz basis in H .Proof. By virtue of Lemma 2, the system { v nk } n ∈ N , k =1 , is l -close to the orthogonal basis { v nk } n ∈ N , k =1 , , such that k v nk k H = π for n ∈ N , k = 1 , { v nk } is complete in H . Suppose that, on the contrary, thereexists an element of H , orthogonal to all { v nk } . Then the relations Z π ( h ( t ) ρ nk sin ρ nk t + h ( t ) g nk cos ρ nk t ) dt = 0 , n ∈ N , k = 1 , , Z π h ( t ) dt = 0hold for some functions h , h ∈ L (0 , π ). Recall that g nk = − S ′ ( π, λ nk ) S ( π, λ nk ) and S ( π, λ nk ) = 0.Then the function H ( λ ) := Z π ( h ( t ) S ( π, λ ) ρ sin ρt − h ( t ) S ′ ( π, λ ) cos ρt ) dt has zeros at the points λ = λ nk , n ∈ N , k = 1 ,
2, and λ = 0. Clearly, the function H ( λ ) isentire and satisfies the estimate H ( λ ) = O (exp(2 | Im ρ | π )).Denote D ( λ ) := πλ ∞ Y n =1 λ n − λ ( n − / ∞ Y s =1 λ s − λs . The construction of D ( λ ) resembles representation of characteristic functions of Sturm-Liouvilleoperators in form of infinite products (see, for example, [20, par. 1.1.1]). Note that D ( λ ) ∼ ρ cos ρπ sin ρπ as | ρ | → ∞ . Moreover, it satisfies the estimate | D ( λ ) | ≥ C δ | ρ | exp(2 | τ | π ) for ρ in G δ := { ρ : | ρ | ≥ δ, | ρ − ρ nk | ≥ δ, n ∈ N , k = 1 , } , where δ > H ( λ ) D ( λ ) is entire and H ( λ ) D ( λ ) = O ( ρ − ), | ρ | → ∞ in G δ . Byvirtue of Liouville’s theorem, H ( λ ) ≡
0. Hence Z π ( h ( t ) S ( π, λ ) ρ sin ρt − h ( t ) S ′ ( π, λ ) cos ρt ) dt ≡ . (19)5et { µ n } n ∈ N and { ν n } n ∈ N be the zeros of S ( π, λ ) and S ′ ( π, λ ), respectively, and µ = 0.Substituting λ = µ n and λ = ν n into (19) and using the fact, that S ′ ( π, µ n ) = 0 and S ( π, ν n ) =0, n ∈ N , we obtain Z π h ( t ) cos √ µ n t dt = 0 , Z π h ( t ) sin √ ν n t dt = 0 , n ∈ N . The systems { cos √ µ n t } n ∈ N and { sin √ ν n t } n ∈ N are complete in L (0 , π ) (see [20]), so h ( t ) = 0and h ( t ) = 0 for a.e. x ∈ (0 , π ). Thus, the system { v nk } is complete in H . Hence it is a Rieszbasis.In view of Lemmas 2 and 3, the relation (17) provides the l -sequence of the coordinatesof the unknown vector functon f with respect to the Riesz basis. One can uniquely determinethe function f by these coordinates. Thus, we obtain the following algorithm for the solutionof the IP. Algorithm.
Given the potentials q j , j = 2 , m , and the eigenvalues { λ nk } n ∈ N , k =1 , of theproblem L .1. Find ω j = π R q j ( t ) dt , j = 2 , m , the value of ˆ ω from (5) and ω = m ˆ ω − m P j =2 ω j .2. Construct the Weyl functions M j ( λ ) of the boundary value problems L j , j = 2 , m .3. Calculate g nk , n ∈ N , k = 1 ,
2, via (10), then v nk and f nk , n ∈ N , k = 1 ,
2, via (14), (15),(16), (18).4. Construct the function f ( t ), using its coefficients { f nk } n ∈ N , k =1 , by the Riesz basis { v nk } n ∈ N , k =1 , , find K ( t ) and N ( t ).5. Construct the Weyl function M ( λ ) = − S ′ ( π, λ ) S ( π, λ ) , using (11).6. Solve the classical inverse problem by the Weyl function and find q ( x ), x ∈ (0 , π ).Alternatively to the steps 5 and 6, one can construct the potential q by the Cauchy data K ( π, t ) = R t K ( s ) ds , ∂∂x K ( x, t ) | x = π = N ( t ), using the methods from [30]. Remark . Note that in this section, we have used the assumption (iv) only for j = 1. It willbe required for j = 2 , m in the next section for the proof of local solvability and stability. Infact, we can absolutely omit the assumption (iv) for the algorithm. Indeed, g n stands only inthe denominators in the formulas (14), (16), so the estimate 1 g n = O ( n − ) is sufficient for thepurposes of this section. One can even take g n = ∞ (this corresponds to S ( π, λ n ) = 0) atthe steps 3–6 of the algorithm. The proof of Lemma 3 can be slightly modified to take thiscase into account. Remark . Suppose that the assumption (i) is violated, i.e. the subsequence { λ nk } n ∈ N , k =1 , contains multiple eigenvalues. There can be only a finite number of them because of theasymptotics (5), (6). For instance, let λ ∈ { λ nk } n ∈ N , k =1 , be a double eigenvalue of L . Then∆( λ ) = ddλ ∆( λ ) | λ = λ = 0 . γ S ′ ( π, λ ) + γ S ( π, λ ) = 0 ,γ ddλ S ′ ( π, λ ) | λ = λ + γ ddλ S ( π, λ ) λ = λ + γ S ′ ( π, λ ) + γ S ( π, λ ) = 0 , where γ i , i = 1 ,
4, are the constants, which can be easily found from (7). Take the followingcouple of the vector functions, associated with the eigenvalue λ = ρ : v = (cid:20) γ ρ sin ρ tγ cos ρ t (cid:21) , v = (cid:20) γ ddλ ( ρ sin ρt ) | ρ = ρ + γ ρ sin ρ tγ ddλ (cos ρt ) | ρ = ρ + γ cos ρ t (cid:21) . One can show that the function H ( λ ) from the proof of Lemma 3 has a double zero at λ .Consequently, v and v together with the vector functions, constructed via (14) for simpleeigenvalues, form a Riesz basis. So our method can be applied for multiple eigenvalues withminor modifications. Remark . In view of the previous remarks, the most crucial restriction among (i)-(v) is (iii),while the other assumptions are just technical. Indeed, let (iii) is violated, and S j ( π, λ nk ) = 0for some fixed indices j = 2 , m , n ∈ N , k = 1 ,
2. The relations (7) and ∆( λ nk ) = 0 imply that S p ( π, λ nk ) = 0 also for some p = k . Then the eigenvalue λ nk does not give us any informationabout the potential q .
4. Local solvability and stability
Suppose that the potentials q j , j = 2 , m , and the eigenvalues { λ nk } n ∈ N , k =1 , of the problem L are given, and the assumptions (i)-(v) are satisfied. Let { ˜ λ nk } n ∈ N , k =1 , be arbitrary realnumbers, such that ∞ X n =1 X k =1 , ( n ( ρ nk − ˜ ρ nk )) ! / < ε, ˜ ρ nk := q ˜ λ nk . (20)In this section, we will show, that if ε > { ˜ λ nk } belong tothe spectrum of some boundary value problem ˜ L of the same form as L , but with a differentpotential ˜ q ∈ L (0 , π ) (the other potentials coincide: q j = ˜ q j , j = 2 , m ). Moreover, thepotential ˜ q is sufficiently “close” to q in L -norm, so the solution of the IP is stable. We agreethat if a certain symbol γ denotes an object related to L , then the corresponding symbol ˜ γ with tilde denotes the analogous object related to ˜ L .By virtue of (20), the asymptotic formulas hold˜ ρ n = n −
12 + ˜ˆ ωπn + ˜ κ n n , ˜ ρ n = n + ˜ z πn + ˜ κ n n , where n ∈ N , { ˜ κ nk } ∈ l , k = 1 ,
2, ˜ˆ ω = ˆ ω , ˜ z = z . Moreover, for sufficiently small ε the values { ˜ λ nk } n ∈ N , k =1 , are distinct and positive. Put ˜ ω := ω and˜ g nk := m X j =2 S ′ j ( π, ˜ λ nk ) S j ( π, ˜ λ nk ) , n ∈ N , k = 1 , . (21)7ote that the assumptions (iii) and (iv) hold for { ˜ λ nk } , if j = 2 , m and ε is sufficiently small.Consequently, g nk = ∞ , and we can derive the asymptotic relations˜ g n = m X j =2 ( ω j − ˆ ω + η nj ) = ω − ˆ ω + η n , ˜ g n = m X j =2 n ω j − z (1 + η nj ) = n z − ω (1 + η n ) , where { η njk } ∈ l . Here we have used the fact that z is a root of the characteristic polynomial P ( z ).Construct ˜ v nk and ˜ f nk by formulas, similar to (14), (15), (16), (18). Clearly, these sequencessatisfy the assertion of Lemma 2. Using (20), we obtain the following result. Lemma 4.
There exists ε > , such that for every ε ∈ (0 , ε ] the following estimates hold ∞ X n =1 ( g n − ˜ g n ) ! / < Cε, ∞ X n =1 ( n − ( g n − ˜ g n )) ! / < Cε, ∞ X n =0 X k =1 , ( f nk − ˜ f nk ) ! / < Cε, ∞ X n =0 X k =1 , k v nk − ˜ v nk k H ! / < Cε. where C is some constant depending only on L and ε . Hence for sufficiently small ε >
0, the sequence { ˜ v nk } n ∈ N , k =1 , is a Riesz basis in H , and thesequence { ˜ f nk } n ∈ N , k =1 , belongs to l . Further we need the following abstract result. Lemma 5.
Let { v n } n ∈ N and { ˜ v n } n ∈ N be Riesz bases in a Hilbert space H , f ∈ H , f n = ( f, v n ) H , n ∈ N , and ∞ X n =1 k v n − ˜ v n k H ! / < ε, ∞ X n =1 | f n − ˜ f n | ! / < ε. (22) Let ˜ f be the element of H with coordinates ˜ f n = ( ˜ f , ˜ v n ) H . Then k f − ˜ f k H < Cε , where theconstant C depends only on f , { v n } and ε , if ε ∈ (0 , ε ] .Proof. For the Riesz bases { v n } and { ˜ v n } there exist biorthonormal bases { χ n } , { ˜ χ n } , boundedlinear invertible operators A , ˜ A and an orthonormal basis { e n } , such that Ae n = v n , ( A ∗ ) − e n = χ n , ˜ Ae n = ˜ v n , ( ˜ A ∗ ) − e n = ˜ χ n , n ∈ N . Consequently, v n − ˜ v n = ( A − ˜ A ) e n . By virtue of (22), we have ∞ X n =1 k ( A − ˜ A ) e n k H ! / < ε. Hence k A − A ∗ k H → H ≤ ε . One can also show that k ( A ∗ ) − − ( ˜ A ∗ ) − k H → H < Cε, (23)where the constant C depends only on A and ε , if ε ∈ (0 , ε ]. The elements f and ˜ f can berepresented in the following form f = ∞ X n =1 f n χ n , ˜ f = ∞ X n =1 ˜ f n ˜ χ n . f − ˜ f = ( A ∗ ) − ∞ X n =1 f n e n − ( ˜ A ∗ ) − ∞ X n =1 ˜ f n e n = ( A ∗ ) − ∞ X n =1 ( f n − ˜ f n ) e n + (( A ∗ ) − − ( ˜ A ∗ ) − ) ∞ X n =1 ˜ f n e n . Using the estimates (22) and (23), we arrive at the assertion of the lemma.Let ˜ f ( t ) = (cid:20) ˜ N ( t )˜ K ( t ) (cid:21) be the vector function in H , having the coordinates { ˜ f nk } n ∈ N , k =1 , withrespect to the Riesz basis { ˜ v nk } n ∈ N , k =1 , . (Strictly speaking, they are the coordinates withrespect to the biorthogonal basis). By virtue of Lemma 5, the estimates hold k K ( t ) − ˜ K ( t ) k L < Cε, k N ( t ) − ˜ N ( t ) k L < Cε. (24)Construct the functions˜ S ( π, λ ) = sin ρπρ − ω cos ρπρ + 1 ρ Z π ˜ K ( t ) cos ρt dt, (25)˜ S ′ ( π, λ ) = cos ρπ + ω sin ρπρ + 1 ρ Z π ˜ N ( t ) sin ρt dt. (26)Since Z π ˜ K ( t ) dt = ω, the functions ˜ S ( π, λ ) and ˜ S ′ ( π, λ ) are entire in λ . Denote their zeros by { ˜ µ n } n ∈ N and { ˜ ν n } n ∈ N ,respectively. The following lemma asserts, that the sequences { ˜ µ n } n ∈ N and { ˜ ν n } n ∈ N are suf-ficiently “close” to the zeros { µ n } n ∈ N and { ν n } n ∈ N of the functions S ( π, λ ) and S ′ ( π, λ ),respectively. Lemma 6.
The following estimates take place ∞ X n =1 ( µ n − ˜ µ n ) ! / < Cε, ∞ X n =0 ( ν n − ˜ ν n ) ! / < Cε. (27)The proof of Lemma 6 is based on the estimates (24). One can easily check, that forsufficiently small ε , the numbers { ˜ µ n } n ∈ N and { ˜ ν n } n ∈ N are real. So we can apply the followingresult by G. Borg (see [31], [20, Section 1.8]).Consider the boundary value problem L , defined by (9), and the problem L for the sameequation with the boundary conditions y (0) = y ′ ( π ) = 0. Obviously, the sequences { µ n } n ∈ N and { ν n } n ∈ N are the spectra of L and L , respectively. Theorem 1 (G. Borg) . For the boundary value problems L and L , there exists ε > (whichdepends on L , L ), such that if real numbers { ˜ µ n } n ∈ N and { ˜ ν n } n ∈ N satisfy the condition (27) for Cε ≤ ε , then there exists a unique real function ˜ q ∈ L (0 , π ) , for which the numbers { ˜ µ n } n ∈ N and { ˜ ν n } n ∈ N are the eigenvalues of ˜ L and ˜ L , respectively. Moreover, k q − ˜ q k L < C ε. Let ˜ L be the boundary value problem of the form (1), (2), (3), (4) with the potentials ˜ q and ˜ q j = q j , j = 1 , m . It remains to prove that { ˜ λ nk } n ∈ N , k =1 , are eigenvalues of ˜ L . Indeed,one can easily show that the assumptions (i)-(v) are valid for the given values { ˜ λ nk } n ∈ N , k =1 , and the problem ˜ L for sufficiently small ε . The characteristic functions of the problems ˜ L and˜ L coincide with ˜ S ( π, λ ) and ˜ S ′ ( π, λ ), defined by (25), (26). The functions ˜ N and ˜ K were9onstructed in such a way, that − ˜ S ′ ( π, λ nk )˜ S ( π, λ nk ) = ˜ g nk , n ∈ N , k = 1 ,
2. Together with (21) andthe assumption (iii) this implies that { ˜ λ nk } n ∈ N , k =1 , are zeros of the characteristic function˜∆( λ ) = ˜ S ′ ( π, λ ) m Y j =2 S j ( π, λ ) + ˜ S ( π, λ ) m X j =2 S ′ j ( π, λ ) m Y k =2 k = j S k ( π, λ ) of the problem ˜ L . Thus, we have proved the following theorem. Theorem 2.
For every boundary value problem L , satisfying the assumptions (i)-(v), thereexists ε > , such that for arbitrary real numbers { ˜ λ nk } n ∈ N , k =1 , , satisfying (20) for ε ∈ (0 , ε ] ,there exists a unique real function q ∈ L (0 , π ) , being the solution of the IP for { ˜ λ nk } n ∈ N , k =1 , and q j , j = 2 , m . The following estimate holds k q − ˜ q k L < Cε, where the constant C depends only on L and ε . Theorem 2 gives the local solvability and the stability for the solution of the IP.
Remark . One can obtain a similar result for the more general case, when not only theeigenvalues, but also the potentials q j , j = 2 , m , are perturbed: k q j − ˜ q j k L < ε, j = 2 , m. Acknowledgments . This work was supported by the President grant MK-686.2017.1 andby Grants 15-01-04864 and 16-01-00015 of the Russian Foundation for Basic Research.