On recovering the Sturm--Liouville differential operators on time scales
aa r X i v : . [ m a t h . SP ] S e p On recovering the Sturm–Liouville differential operators on timescales
M.A. Kuznetsova Abstract.
We study Sturm–Liouville differential operators on the time scales consistingof a finite number of isolated points and segments. In a previous paper it was establishedthat such operators are uniquely determined by their spectral characteristics. In the presentpaper, an algorithm for their recovery based on the method of spectral mappings is obtained.We also prove that the eigenvalues of two Sturm–Liouville boundary value problems with onecommon boundary condition alternate.
Keywords: differential operators; Sturm–Liouville equation; time scales; closed sets; inversespectral problems.
AMS Mathematics Subject Classification (2010):
We study an inverse spectral problem for the Sturm–Liouville operator on the time scales con-sisting of a finite number of isolated points and segments. Inverse spectral problems consistin recovering operators from their spectral characteristics. Such problems have many appli-cations in natural sciences and engineering. For the classical Sturm–Liouville operator on aninterval, inverse problems have been studied fairly completely; the basic results can be foundin [1–3].Differential operators on time scales, i.e. closed subsets of the real line, unify classical dif-ferential operators and difference operators. Namely, they involve the so-called ∆ -derivativewhich may generalize both the classical derivative and the divided difference, depending ontime scales structure, see [4, 5]. Differential operators on time scales frequently appear inapplications, see [5–7].Posing and studying inverse spectral problems essentially depend on time scale structure,which causes difficulties for time scales of the general form. The single work devoted to aninverse problem on an arbitrary time scale is [8], where Ambarzumian-type theorem is obtainedbeing an analogue to the simplest result in the inverse spectral theory for the Sturm–Liouvilleoperator [9]. Any further studies always require putting for definiteness some restrictions onthe time scale under consideration, see [10–13]. For example, in [10] a partial inverse problemwas studied on the time scale of the form T = [0 , a ] ∪ T , where T is a time scale onwhich the potential was assumed to be given a priori. In [11] recovery of the potential fromthe spectral characteristics was ivestigated on the time scales consisting of a finite numberof segments. In [12] the case of two segments and a finite number of isolated points betweenthem was studied. The time scales considered in [13] generalize those ones in [11] and [12].In the present paper, as in [13], we consider the Sturm–Liouville operator ℓ given byformula (2) below on quite general class of time scales T consisting of N < ∞ segments and M < ∞ isolated points: T = N + M [ l =1 [ a l , b l ] , a l − ≤ b l − < a l ≤ b l , l = 2 , N + M , a l < b l iff l ∈ { l k } Nk =1 , (1) Department of Mathematics, Saratov State University, Astrakhanskaya 83, Saratov, 410012, Russia, email:[email protected], ORCID: https://orcid.org/0000-0003-1083-0799 where l k equals to the indice corresponding to the k -th segment. In the case when N = 1and M = 0 the operator ℓ becomes the classical Sturm–Liouville operator. If T consistsonly of isolated points, i.e. N = 0 , it is a difference operator. Inverse spectral problems inthe latter case consist in recovering coefficients in the recurrent relations. Such problems werestudied in [14–20] and other works. If N > , then the Sturm–Liouville equation on T isequivalent to N classical Sturm–Liouville equations (3) subject to matching conditions (4),see the next section. This partially resembles differential equations with discontinuity condi-tions, matrix equations, and equations on geometrical graphs, see [21–25]. However, unlikethe matching conditions usually imposed along with the mentioned equations, conditions (4)possess special dependence on the spectral parameter, which essentially complicates the study.In [13] we proved that the spectral data uniquely determine the Sturm–Liouville operatoron the time scales of the form (1) having assumed that q ∈ W [ a l k , b l k ] , k = 1 , N . In [11, 12]the uniqueness theorems were proved under the weaker assumption q ( x ) ∈ C ( T ) , but theauthors were forced to restrict the study to a more particular structure of T than (1) andto prespecify q ( x ) in all isolated points. Note that our assumption on q allows to recover italso in the isolated points.In this paper, we obtain an algorithm for solving the inverse problem based on the methodof spectral mappings. For this purpose, we need asymptotic formulae proved in [13], which weprovide in Section 4 for convenience of the reader. We also prove that the eigenvalues of twoboundary value problems with one common boundary condition alternate, see Theorem 1. Ourresults generalize the well-known ones for the classical continuous and discrete Sturm–Liouvilleoperators. For convenience of the reader, let us first provide some basic notions of time scale theory. Let T be so far an arbitrary closed subset of R , which we refer to as a time scale . Define theso-called jump functions σ and σ − on T in the following way: σ ( x ) := ( inf { s ∈ T : s > x } , x = max T, max T, x = max
T, σ − ( x ) := ( sup { s ∈ T : s < x } , x = min T, min T, x = min T. A point x ∈ T is called left-dense , left-isolated , right-dense , and right-isolated , if σ − ( x ) = x,σ − ( x ) < x, σ ( x ) = x, and σ ( x ) > x, respectively. If σ − ( x ) < x < σ ( x ) , then x is called isolated ; if σ − ( x ) = x = σ ( x ) , then x is called dense .Denote T := T \ { max T } , if max T exists and is left-isolated, and T := T, otherwise.We also denote by C ( B ) the class of functions continuous on a subset B ⊆ T. A function f on T is called ∆ - differentiable at t ∈ T , if for any ε > δ > | f ( σ ( t )) − f ( s ) − f ∆ ( t )( σ ( t ) − s ) | ≤ ε | σ ( t ) − s | for all s ∈ ( t − δ, t + δ ) ∩ T. The value f ∆ ( t ) is called the ∆ - derivative of the function f at the point t. We also introduce derivatives of the higher order n ≥ . Let the ( n −
1) -th ∆ -derivative f ∆ n − of f be defined on T n − , where a n = a . . . a | {z } n for any symbol a. If f ∆ n − , in turn, is∆ -differentiable on T n := ( T n − ) , then f ∆ n := ( f ∆ n − ) ∆ is called the n -th ∆ -derivative of f on T n . For n ≥ , we also denote by C n ( T ) the class of functions f for whichthere exists the n -th ∆ -derivative f ∆ n and f ∆ n ∈ C ( T n ) . From now on, f ∆ ν ( x , . . . , x n ) , ν ∈ N , denotes the ν -th partial ∆ -derivative of the function f ( x , . . . , x n ) with respect tothe first argument; we agree that f ∆ ( x , . . . , x n ) := f ( x , . . . , x n ) . Further notions of timescale theory can be found in [5].Consider the following Sturm–Liouville equation on T : ℓy := − y ∆∆ ( x ) + q ( x ) y ( σ ( x )) = λy ( σ ( x )) , x ∈ T , (2)where T has form (1). There λ is a spectral parameter, and q ( x ) ∈ C ( T ) is a real-valuedfunction. Let N > M ≥ , otherwise Eq. (2) degenerates.A function y is called a solution of Eq. (2), if y ∈ C ( T ) and equality (2) is fulfilled. Forsuch functions, (2) is equivalent to the system of equations ℓ k y := − y ′′ ( x k ) + q ( x k ) y ( x k ) = λy ( x k ) , x k ∈ ( a l k , b l k ) , k = 1 , N , (3)equipped with the jump conditions y ( a l +1 ) = α l ( λ ) y ( b l ) + α l ( λ ) y ∆ ( b l ) , l = 1 , N + M − ,y ∆ ( a l +1 ) = α l ( λ ) y ( b l ) + α l ( λ ) y ∆ ( b l ) , l = 1 , N + M − − µ , ) (4)where α l ( λ ) = 1 , α l ( λ ) = a l +1 − b l ,α l ( λ ) = ( a l +1 − b l )( q ( b l ) − λ ) , α l ( λ ) = 1 + ( a l +1 − b l ) ( q ( b l ) − λ ) ,µ := δ ( a N + M , b N + M ) , δ ( k, n ) := (cid:26) , k = n, , k = n. We arrange the coefficients of the jump conditions into the matrices α l ( λ ) := (cid:18) α l ( λ ) α l ( λ ) α l ( λ ) α l ( λ ) (cid:19) , l ∈ , N + M − − µ ; α N + M − ( λ ) := (cid:0) α N + M − ( λ ) , α N + M − ( λ ) (cid:1) if µ = 1 . Without loss of generality, we assume that l k < l k +1 , k = 1 , N − . Denote also l := 1 ,l N +1 := N + M, µ := δ ( a , b ) , and β l k − s ( λ ) := α l k − ( λ ) . . . α l k − s ( λ ) , k = 1 , N + µ , s = 1 , l k − l k − ,β l N ( λ ) := (1 , , l N = N + M. By the definition of β l ( λ ) and the jump conditions, we have (cid:0) y ( a l k ) , y ∆ ( a l k ) (cid:1) T = β l k − ( λ ) (cid:0) y ( b l k − ) , y ∆ ( b l k − ) (cid:1) T , k = 2 − µ , N ,y ( a l N +1 ) = β l N ( λ ) (cid:0) y ( b l N ) , y ∆ ( b l N ) (cid:1) T if N + M > l N , ) (5)where the superscript T is the transposition sign.Consider the elements of the matrices β l ( λ ) : β l ( λ ) =: (cid:18) β l ( λ ) β l ( λ ) β l ( λ ) β l ( λ ) (cid:19) , l = 1 , l N − , (cid:0) β l ( λ ) , β l ( λ ) (cid:1) , l = l N , l N +1 − µ , It is easy to see that the coefficients β l k ij ( λ ) are polynomials of the form β l k ij ( λ ) = ( a l k − +1 − b l k − ) j ( a l k − b l k − ) i l k − Y l = l k − +1 ( a l +1 − b l ) ( − λ ) l k − l k − − i + O ( λ l k − l k − − i ) , (6)where k = 2 − µ , N + µ , i = 1 , − δ ( k, N + 1) and j = 1 , . The more precise formulaethan (6) can be found in [13, Lemma 1].Thus, the study of Eq. (2) is reduced to the study of system (3) subject to conditions (5)with the coefficients polynomially dependent on λ. This allows us to use some auxiliary resultsfrom classical Sturm–Liouville theory.
Denote by L j ( B ) the boundary value problem for Eq. (2) on a closed subset B ⊆ T withthe boundary conditions y ∆ j (min B ) = y (max B ) = 0 , j = 0 , . (7)A value λ is called an eigenvalue of the boundary value problem L j ( B ) if there exists anon-zero solution y of Eq. (2) satisfying boundary conditions (7).Let S ( x, λ ) and C ( x, λ ) be solutions of Eq. (2) on T satisfying the initial conditions S ∆ ( a , λ ) = C ( a , λ ) = 1 , S ( a , λ ) = C ∆ ( a , λ ) = 0 . For each fixed x, the functions S ( x, λ ) and C ( x, λ ) are entire in λ. We introduce thefunctions Θ ( λ ) := S ( b N + M , λ ) , Θ ( λ ) := C ( b N + M , λ ) . For j = 0 , , the eigenvalues { λ nj } n ≥ of the boundary value problem L j ( T ) coincide withthe zeros of the entire function Θ j ( λ ) , which is called the characteristic function of L j ( T ) . For k = 1 , N , denote d k := b l k − a l k , δ k := δ ( l k , N + M ) ,f k ( x ) := (cid:26) sin d k x, δ k = 1 , cos d k x, δ k = 0 , f k ( x ) := (cid:26) cos d k x, δ k = 1 , sin d k x, δ k = 0 . In what follows, let us agree that √ z is the principal square root of a complex number z. In [13] we proved the following asymptotic formulae for the characteristic functions:Θ j ( λ ) = ( F j ( λ ) + O (cid:0) exp( γ τ ) λ N + M − j (1 − µ ) / − µ / (cid:1) , N > ,β , − j ( λ ) , N = 0 , j = 0 , , (8)where γ := N X k =1 d k , τ := Im ρ ≥ , ρ := √ λ,F j ( λ ) := ( − j (1 − δ )(1 − µ ) ρ µ + j (1 − µ ) − N − Y k =1 − µ β l k , − jδ (0 ,k ) ( λ ) β l N , µ ( λ ) N Y k =2 f k ( ρ ) f , (1 − µ ) j ( ρ ) . From asymptotics (8) it follows that Θ j ( λ ) are entire functions of order 1 / N > M − . Then Hadamard’s factorization theoremgives Θ j ( λ ) = C j p j ( λ ) , p j ( λ ) = λ s j Y λ nj =0 (cid:16) − λλ nj (cid:17) , j = 0 , , where C j are non-zero constants, while s j is the multiplicity of the zero eigenvalue in thespectrum { λ nj } n ≥ . By virtue of (8), we have C j = lim λ →−∞ F j ( λ ) p j ( λ ) . Hence, the characteristic functions Θ j ( λ ) are uniquely determined by their zeros { λ nj } n ≥ . We also introduce the
Weyl solution Φ( x, λ ) as the solution of Eq. (2) under the boundaryconditions Φ ∆ ( a , λ ) = 1 , Φ( b N + M , λ ) = 0 . (9)We call M ( λ ) := Φ( a , λ ) the Weyl function , which generalizes the classical Weyl function.It is obvious that Φ( x, λ ) = S ( x, λ ) + M ( λ ) C ( x, λ ) , M ( λ ) = − Θ ( λ )Θ ( λ ) . (10)In accordance with the second assertion in Proposition 1 below, we put α n := Res λ = λ n M ( λ ) = − Θ ( λ n )Θ ′ ( λ n ) , n ≥ . We call α n a weight number . The numbers { /α n } n ≥ generalize the classical normingconstants for the Sturm–Liouville operator (see, e.g. [3]). Spectral characteristics include the Weyl function, two spectra { λ nj } n ≥ , and the weightnumbers { α n } n ≥ . The following properties of the eigenvalues and the weight numbers areestablished in [13].
Proposition 1.
1. The spectra { λ n } n ≥ and { λ n } n ≥ have no common elements.2. All zeros of Θ j ( λ ) , j = 0 , , are real and simple.3. All weight numbers are positive.4. For N > , the spectra { λ n } n ≥ and { λ n } n ≥ are infinite. Otherwise each of themconsists of M − elements. For j = 0 , , assume that eigenvalues λ nj are numbered in the increasing order: λ nj < λ n +1 ,j , n = 1 , Θ NM , Θ NM := (cid:26) + ∞ , N > ,M − , N = 0 . The following theorem holds.
Theorem 1.
The eigenvalues alternate in the following way: λ n < λ n < λ n +1 , , n = 1 , Θ NM , where we put λ M − , := + ∞ if Θ NM = M − . Proof.
Let us consider another time scale X = T S { b N + M + 1 } and continue q on X \ T arbitrarily. Then the functions S and C can be extended to X to satisfy Eq. (2)on X . We also can determine the values S ∆ ( b N + M , λ ) and C ∆ ( b N + M , λ ) . Let ϕ ( x, λ ) bea solution of Eq. (2) on X satisfying the initial conditions ϕ ( b N + M , λ ) = 0 , ϕ ∆ ( b N + M , λ ) = − . The Wronskian-type determinant W ( y, z ) := y ( t ) z ∆ ( t ) − y ∆ ( t ) z ( t ) is constant on X if y, z obey Eq. (2) on X . Then ϕ ( a , λ ) = Θ ( λ ) and ϕ ∆ ( a , λ ) = − Θ ( λ ) . From relation (2) on X with y = ϕ we get( λ − µ ) Z b N + M a ϕ ( σ ( t ) , λ ) ϕ ( σ ( t ) , µ )∆ t = − Θ ( λ )Θ ( µ ) + Θ ( λ )Θ ( µ ) , for details see [13, Proposition 2]. Dividing on λ − µ both parts and taking the limit at µ → λ, we obtainΘ ( λ )Θ ′ ( λ ) − Θ ′ ( λ )Θ ( λ ) = Z b N + M a ϕ ( σ ( t ) , λ )∆ t > , λ = λ n , since ϕ ( a , λ ) = 0 . Then the function M − ( λ ) = − Θ ( λ ) / Θ ( λ ) is increasing on everyinterval ( λ n − , , λ n ) , n ≥ , where we put λ := −∞ . It is easy to see that lim λ → λ n ± M − ( λ ) = ∓∞ , n ≥ . Using formulae (6) and (8), we obtainlim λ →−∞ M − ( λ ) = − a − a , a = b , −∞ , a < b . Then from the monotonicity of M − ( λ ) it follows that λ n ∈ ( λ n − , , λ n ) , n ≥ , whichfinishes the proof.The properties described in Proposition 1 and Theorem 1 are well known for the classicalself-adjoint Sturm–Liouville operator, see [3]. In [26] alternation of two spectra was proved inthe case of a general time scale but under different boundary conditions. Further propertiesof the eigenvalues and the weight numbers, namely asymptotic formulae, can be found inSection 4.From now on, we assume that q ∈ W [ a l k , b l k ] , k = 1 , N . In [13] we proved that spectraldata of three types uniquely determine the potential:1. M ( λ );2. { λ nj } n ≥ , j = 0 , { λ n } n ≥ and { α n } n ≥ . Given the spectral data of one type, we can recover them of any other one, see [13]. By thisreason, it is sufficient to provide only an algorithm solving the inverse problem with M ( λ ) asthe input data: Inverse problem 1.
Given M ( λ ) , find q ( x ) on T . We should note that together with the Weyl function the stucture of T is known. Therecovery of T along with the potential requires a separate investigation.The main result of the paper is an algorithm for solving Inverse problem 1. We give it inrecursive style, which means that on the m -th step, m = 1 , N + M − µ − , we reduce therecovery of q on T m to its recovery on T m +1 , where T m = N + M [ l = m [ a l , b l ] , m = 1 , N + M − µ . (11)The initial step is to recover the potential on the first segment [ a , b ] or in the first isolatedpoint a of the time scale. Consider the following auxiliary local inverse problem. Inverse problem 2.
Given M ( λ ) , find q on [ a , b ] . In Sections 5 and 6, we obtain two algorithms for solving this local inverse problem. Thereare different approaches according to whether a < b or a = b . On the interval thepotential is recovered by the method of spectral mappings, see Algorithm 1. In isolated pointwe use asymptotic relations to find q ( a ) , see Algorithm 2. Further in Section 6, we obtainAlgorithm 3 for solution of Inverse problem 1, which is based on the Algorithms for InverseProblem 2. In this section, we provide asymptotic formulae for the eigenvalues and the weight numbers aswell as for the functions C ( x, λ ) and Φ( x, λ ) , which will be used below for solving Inverseproblem 2 in the case a < b . First, we establish asymptotic formulae for C ( x, λ ) and Φ( x, λ ) , x ∈ ( a , b ) , if a < b . Asymptotics for Φ( x, λ ) are obtained via decomposition into Birkhoff solutions and thensolving the obtained linear system.
Lemma 1.
Let a < b . Then for every fixed δ > the estimates C ( x + a , λ ) = cos ρx + O (cid:18) exp( τ x ) ρ (cid:19) ,C ′ ( x + a , λ ) = − ρ sin ρx + O (exp( τ x )) , x ∈ (0 , d ] , ρ ∈ G δ , (12)Φ( x + a , λ ) = f ( xρd − ρ ) ρf ( ρ ) + O (cid:18) exp( − τ x ) ρ (cid:19) , Φ ′ ( x + a , λ ) = ( − δ +1 f ( xρd − ρ ) f ( ρ ) + O (cid:18) exp( − τ x ) ρ (cid:19) , x ∈ [0 , d ) , ρ ∈ G δ (13) hold, where G δ := (cid:26) ρ : (cid:12)(cid:12)(cid:12) ρ − πn d k (cid:12)(cid:12)(cid:12) ≥ δ, k = 1 , N , n ∈ Z (cid:27) . Proof.
Since C ( x, λ ) is the cosine-type solution of the first equation in (3), equalities (12)are obvious.Let us prove (13). It is known (see, for example, [3]) that for k = 1 , N there exists thefundamental system of solutions { Y k ( x, ρ ) , Y k ( x, ρ ) } , x ∈ [ a l k , b l k ] , of the k -th equation in(3) having the asymptotics Y ( j )1 k ( x + a l k , ρ ) = ( iρ ) j exp( iρx )[1] , Y ( j )2 k ( x + a l k , ρ ) = ( − iρ ) j exp( − iρx )[1] , j = 0 , , (14)where [1] := 1 + O ( ρ − ) uniformly for x ∈ [0 , d k ] as ρ → ∞ . Expanding Φ( x, λ ) for x ∈ [ a l k , b l k ] with respect to the systems { Y k ( x, ρ ) , Y k ( x, ρ ) } , k = 1 , N , we getΦ( x, λ ) = A k − ( ρ ) Y k ( x, ρ ) + A k ( ρ ) Y k ( x, ρ ) , x ∈ [ a l k , b l k ] , k = 1 , N , (15)where A := (cid:0) A l ( ρ ) (cid:1) Nl =1 is a solution of a certain linear system.To write this system, denote by D the following matrix: D := r s . . . p q r s . . . p q r s . . . p q r s . . . p q r s . . . . . . p N q N r N s N . . . p N q N r N s N . . . p N +1 , q N +1 , with the coefficients given for ν = 1 , p mν := β l m − ν ( λ ) Y ,m − ( b l m − , ρ ) + β l m − ν ( λ ) Y ′ ,m − ( b l m − , ρ ) , m = 2 , N + 1 ,q mν := β l m − ν ( λ ) Y ,m − ( b l m − , ρ ) + β l m − ν ( λ ) Y ′ ,m − ( b l m − , ρ ) , m = 2 , N + 1 ,r mν := − Y ( ν − m ( a l m , ρ ) , s mν := − Y ( ν − m ( a l m , ρ ) , m = 1 , N , The other elements are equal to zero.Substituting (15) into (5) and (9), we obtain the linear system D A = ( − , , . . . , T (16)with respect to the vector A. Further, using this system, we estimate the coefficients A ( ρ )and A ( ρ ) . Consider the case
N > . Solving system (16) by Cramer’s formulae, we get A ( ρ ) = q det D − q det D det D , A ( ρ ) = p det D − p det D det D , (17)where D j is the submatrix of D including columns with the numbers 3 , , . . . , N − , N and rows with the numbers 2 + j, , . . . , N − , N for j = 0 , . For j = 0 , , denote by D l j ( λ ) the characteristic function of L j ( T l ) , where T m isdetermined in (11). One can show thatdet D = ( − iρ ) N [1]Θ ( λ ) , det D j = ( − j +1 ( − iρ ) N − [1] D l j , j = 0 , . (18)From (8) we obtainΘ ( λ ) = ( − − δ ρ µ N − Y k =1 β l k , ( λ ) β l N , µ ( λ ) N Y k =2 f k ( ρ ) f ( ρ )[1] , ρ ∈ G δ . Since D l j is the object that plays for T l the same role as Θ j ( λ ) does for T, we have D l j = ( − j (1 − δ ) ρ µ + j − N − Y k =2 β l k , ( λ ) β l N , µ ( λ ) N Y k =3 f k ( ρ ) f j ( ρ )[1] , ρ ∈ G δ , j = 0 , D det D = − iρ sin ρd β l ( λ ) [1] , det D det D = O (cid:18) ρd β l ( λ ) ρ (cid:19) , ρ ∈ G δ . Using these estimates along with (6), (14), and (17), we obtain A ( ρ ) = exp( − iρd )2 ρ sin ρd [1] , A ( ρ ) = exp( iρd )2 ρ sin ρd [1] , ρ ∈ G δ . These formulae along with (15) for k = 1 yield (13) for N > . The case N = 1 can betreated analogously.In order to write the asymptotic formulae for the eigenvalues and the weight numbers, weintroduce the constants δ jk := δ ( δ k , j ) and c k := 12 Z b lk a lk q ( t ) dt + min( l k +1 ,N + M ) X l = l k +1 ( a l − b l − ) − , z k := 1 π (cid:16) c k + l k − X l = max (1 ,l k − ( a l +1 − b l ) − (cid:17) (19)for k = 1 , N , j = 0 , . The following two theorems are Theorems 3 and 4, respectively, from [13]. From now on, { κ n } n ≥ denotes different sequences from l . Theorem 2.
Fix j ∈ { , } . Then the spectrum of L j ( T ) consists of N + 1 parts: { λ nj } n ≥ = Λ j [ N [ k =1 (cid:8) ( ρ knj ) (cid:9) n ≥ , j = 0 , , (20) where Λ j contains N + M − j (1 − µ )sign( N − µ ) − µ elements. Assume that d k = rx k , x k ∈ N , k = 1 , N , for some r independent of k. (21) Then for the sequences (cid:8) ρ knj (cid:9) n ≥ the following asymptotic formulae are fulfilled: ρ knj = π ( n − δ jδ (1 ,l k ) k ) d k + z k n − δ jδ (1 ,l k ) k + κ n n , k = 1 , N , n ∈ N , (22) where z k are real constants given in (19) . Theorem 3.
In accordance with (20) , the sequence { α n } n ≥ consists of N + 1 parts: { α n } n ≥ = A [ N [ k =1 (cid:8) α kn (cid:9) n ≥ , α kn := Res λ =( ρ kn ) M ( λ ) , A := n Res λ = z M ( λ ) : z ∈ Λ o . If (21) holds, and z l d l = z ν d ν if l = ν, l, ν = 1 , N , (23) then the following asymptotic formulae are fulfilled: α kn = d k (cid:16) κ n n (cid:17) , k = 1 , a = b ,κ n n , otherwise . (24)0We need asymptotic formulae (22) and (24) to apply the method of spectral mappings inthe following section. Remark 1.
Further, we consider implicitly the eigenvalues and the weight numbers ofthe boundary value problems L j ( T l k ) , k = 1 , N . For these boundary value problems, (21)and the following condition guarantee the analogs of asymptotic formulae (22) and (24) withresidual summands κ n /n : c k πd k / ∈ n z s d s o Ns = k +1 and z l d l = z ν d ν if l = ν, l, ν = k + 1 , N , for each k ∈ , N − . (25) a < b a < b . Let M ( λ ) and T be given. We explain how q can be found on [ a , b ] by the method of spectral mappings. For simplicity, we assume that(21) and (23) are fulfilled. These conditions allow us to use the simplest form of the methodof spectral mappings, see [3, Ch. 1].Together with the boundary value problem L ( T ) we consider a problem ˜ L ( T ) of thesame form but with another potential ˜ q. If an object γ is related to L ( T ) , we denote by˜ γ the analogous object for ˜ L ( T ) . Put ξ n := (cid:12)(cid:12)(cid:12)p λ n − q ˜ λ n (cid:12)(cid:12)(cid:12) + | α n − ˜ α n | , n ∈ N . It follows from Theorems 2 and 3 that { ξ n } n ∈ N ∈ l . We introduce for x ∈ [ a , b ] the function D ( x, λ, µ ) := h C ( x, λ ) , C ( x, µ ) i λ − µ = Z xa C ( t, λ ) C ( t, µ ) dt, (26)where h y, z i := yz ′ − y ′ z and the classical derivatives are taken with respect to the firstargument. The second equality in formula (26) follows from the relations ℓ C ( t, λ ) = λC ( t, λ )and ℓ C ( t, µ ) = µC ( t, µ ) on [ a , b ] (see (3) for the definition of ℓ k ).Introduce the following designations: θ n := λ n , θ n := ˜ λ n , α n := α n , α n := ˜ α n , C ni ( x ) := C ( x, θ ni ) , ˜ C ni ( x ) := ˜ C ( x, θ ni ) ,R ni,kj ( x ) := α kj D ( x, θ ni , θ kj ) , ˜ R ni,kj ( x ) := α kj ˜ D ( x, θ ni , θ kj ) , i, j = 0 , , n, k ∈ N . Then, in particular, we have ξ n = |√ θ n − √ θ n | + | α n − α n | . In order to obtain the solution of Inverse problem 2, we need the following lemmas. Fromnow on, C denotes sufficiently large positive constants. Lemma 2.
The following estimates are valid for x ∈ [ a , b ] , n, k ∈ N , i, j = 0 , | C ni ( x ) | ≤ C, | C n ( x ) − C n ( x ) | ≤ Cξ n , (27) | R ni,kj ( x ) | ≤ C |√ θ n − √ θ k | + 1 , | R ni,k ( x ) − R ni,k ( x ) | ≤ Cξ k |√ θ n − √ θ k | + 1 , | R n ,kj ( x ) − R n ,kj ( x ) | ≤ Cξ n |√ θ n − √ θ k | + 1 , | R n ,k ( x ) − R n ,k ( x ) − R n ,k ( x ) + R n ,k ( x ) | ≤ Cξ n ξ k |√ θ n − √ θ k | + 1 . (28)1 The analogous estimates are also valid for ˜ C ni ( x ) , ˜ R ni,kj ( x ) . The proof of this lemma is standard, see [3, Lemma 1.6.2]. In the proof, we also obtainedthe inequalities | D ( x, λ, θ ) | ≤ C exp( τ ( x − a )) | ρ − √ θ | + 1 , | Im θ | ≤ r, | D ( x, λ, θ k ) − D ( x, λ, θ k ) | ≤ Cξ k exp( τ ( x − a )) | ρ − √ θ k | + 1 . (29) Lemma 3.
The following quantities are finite: ∞ X k =1 | ρ − √ θ k | + 1) ≤ ∞ , λ ∈ C , (30)sup n ∈ N ∞ X k =1 |√ θ n − √ θ k | + 1) ≤ ∞ . (31) Proof.
Let us prove (31); for (30) the proof is analogous. For simplicity, we assume thatall θ n ≥ , which can be achieved by shifting the potential ˜ q in Eq. (2).Consider the numbers R B := π (cid:16) B D + 14 D (cid:17) , B ∈ N , R := 0 , D := x . . . x N r, (32)where x j and r are determined in (21). By formula (20) and (22), the set J B := { n ∈ N : R B − ≤ θ n < R B } contains no more than N elements for a suffiently large B. Thenthe number of elements in each J B is bounded by some constant K. Consequently, for any n, j ∈ N we have the inequality | p θ n − p θ m | ≥ j D , m ∈ N : j + 1 ≤ | m − n | K < j + 2 . (33)For n ∈ N , we can write ∞ X k =1 |√ θ n − √ θ k | + 1) = n − K X k =1 |√ θ n − √ θ k | + 1) + n + K − X k =max(1 ,n − K +1) |√ θ n − √ θ k | + 1) + ∞ X k = n + K p | θ k − √ θ n | + 1) . Using inequality (33), we obtain ∞ X k =1 |√ θ n − √ θ k | + 1) ≤ K ⌈ ( n − K ) /K ⌉ X j =1 j D + 1) + (2 K −
1) + K ∞ X j =1 j D + 1) , where ⌈·⌉ is the ceiling function. This concludes the proof.Let P ( x, λ ) = [ P jk ( x, λ )] j,k =1 , be the matrix determined by the formula P ( x, λ ) (cid:20) ˜ C ( x, λ ) ˜Φ( x, λ )˜ C ′ ( x, λ ) ˜Φ ′ ( x, λ ) (cid:21) = (cid:20) C ( x, λ ) Φ( x, λ ) C ′ ( x, λ ) Φ ′ ( x, λ ) (cid:21) , x ∈ [ a , b ] . h C, Φ i = 1 , the matrix P ( x, λ ) exists and the following formulae hold: P j ( x, λ ) = C ( j − ( x, λ ) ˜Φ ′ ( x, λ ) − Φ ( j − ( x, λ ) ˜ C ′ ( x, λ ) ,P j ( x, λ ) = Φ ( j − ( x, λ ) ˜ C ( x, λ ) − C ( j − ( x, λ ) ˜Φ( x, λ ) . Using these formulae for P ij ( x, λ ) along with (12) and (13), we obtain the estimates | P ij ( x, λ ) − δ ( i, j ) | ≤ C | ρ | i − j − , ρ ∈ G δ , x ∈ [ a , b ] , i, j = 1 , . (34)Fix δ > x ∈ [ a , b ] . In the λ -plane we consider closed contours γ B , B ∈ N , (with counterclockwise circuit) of the form γ B = γ + B ∪ γ − B ∪ γ ′ ∪ (Γ B \ Γ ′ B ) , where γ ± B := n λ : ± Im λ = δ, Re λ ≥ θ ′ , | λ | ≤ R B o , θ ′ := min n ∈ N , i =0 , θ ni ,γ ′ := n λ : λ − θ ′ = δ exp( iα ) , α ∈ (cid:16) π , π (cid:17)o , Γ ′ B := Γ B ∩ { λ : | Im λ | ≤ δ, Re λ > θ ′ } , Γ B := n λ : | λ | = R B o , while R B are determined in (32). Applying Cauchy’s integral formula on the contours γ B forthe elements of P ( x, λ ) , analogously to the case of the classical Sturm–Liouville equation [3,Sect. 1.6] one can prove the following relations:˜ C ( x, λ ) = C ( x, λ ) + ∞ X k =1 (cid:16) α k ˜ D ( x, λ, θ k ) C k ( x ) − α k ˜ D ( x, λ, θ k ) C k ( x ) (cid:17) , (35) D ( x, λ, µ ) − ˜ D ( x, λ, µ ) + ∞ X k =1 (cid:16) α k ˜ D ( x, λ, θ k ) D ( x, θ k , µ ) − α k ˜ D ( x, λ, θ k ) D ( x, θ k , µ ) (cid:17) = 0 , (36)where the series converge absolutely and uniformly with respect to x ∈ [ a , b ] and λ, µ oncompact sets. Absolute and uniform convergence follows from estimates (29) and (30).It follows from the definition of ˜ R ni,kj ( x ) , R ni,kj ( x ) and formulae (35), (36) that˜ C ni ( x ) = C ni ( x ) + ∞ X k =1 (cid:0) ˜ R ni,k ( x ) C k ( x ) − ˜ R ni,k ( x ) C k ( x ) (cid:1) , (37) R ni,lj ( x ) − ˜ R ni,lj ( x ) + ∞ X k =1 (cid:0) ˜ R ni,k ( x ) R k ,lj ( x ) − ˜ R ni,k ( x ) R k ,lj ( x ) (cid:1) = 0 , (38)where i, j = 0 , n, l ∈ N . The series in (37) and (38) converge absolutely and uniformlywith respect to x ∈ [ a , b ] . For each fixed x ∈ [ a , b ] , relation (37) can be treated as a system of linear equations withrespect to C ni ( x ) , n ∈ N , i = 0 , . But the series therein converges only ”with brackets”,i.e. the terms in them cannot be dissociated. For this reason, it is inconvenient to use(37) for solving Inverse problem 2. Further we transform (37) into a linear equation in thecorresponding Banach space of sequences, see formula (40) below.Let V be the set of indices u = ( n, i ) , n ≥ , i = 0 , . For each fixed x ∈ [ a , b ] , wedefine the vector ψ ( x ) = [ ψ u ( x )] u ∈ V = (cid:20) ψ n ( x ) ψ n ( x ) (cid:21) n ∈ N (cid:20) ψ n ( x ) ψ n ( x ) (cid:21) := (cid:20) χ n − χ n (cid:21) (cid:20) C n ( x ) C n ( x ) (cid:21) , χ n := (cid:26) ξ − n , ξ n = 0 , , ξ n = 0 , n ∈ N . We also consider the block matrix H ( x ) = [ H u,v ( x )] u,v ∈ V = (cid:20) H n ,k ( x ) H n ,k ( x ) H n ,k ( x ) H n ,k ( x ) (cid:21) n,k ∈ N , u = ( n, i ) , v = ( k, j ) , determined in the following way: (cid:20) H n ,k ( x ) H n ,k ( x ) H n ,k ( x ) H n ,k ( x ) (cid:21) := (cid:20) χ n − χ n (cid:21) (cid:20) R n ,k ( x ) R n ,k ( x ) R n ,k ( x ) R n ,k ( x ) (cid:21) (cid:20) ξ k − (cid:21) . Analogously we define ˜ ψ ( x ) and ˜ H ( x ) by replacing C ni ( x ) by ˜ C ni ( x ) and R ni,kj ( x ) by˜ R ni,kj ( x ) in the previous definitions.Let us consider the Banach space B of bounded sequences α = [ α u ] u ∈ V with the norm k α k B := sup u ∈ V | α u | . It follows from (27) that ψ ( x ) , ˜ ψ ( x ) ∈ B . Using (28), for n, k ∈ N and i, j = 0 , | H ni,kj ( x ) | ≤ Cξ k |√ θ n − √ θ k | + 1 , | ˜ H ni,kj ( x ) | ≤ Cξ k |√ θ n − √ θ k | + 1 . (39)Consider the linear operators on B associated with the matrices H ( x ) and ˜ H ( x ) : H ( x ) α := [ y u ] u ∈ V , y u := X v ∈ V H u,v ( x ) α v , ˜ H ( x ) α := [˜ y u ] u ∈ V , ˜ y u := X v ∈ V ˜ H u,v ( x ) α v . Due to (39), (31), and { ξ n } n ∈ N ∈ l , for each fixed x the operators H ( x ) and ˜ H ( x ) arelinear bounded operators in the space B . It is easy to see from (37) that for each fixed x ∈ [ a , b ] the vector ψ ( x ) satisfies theequation ˜ ψ ( x ) = ( I + ˜ H ( x )) ψ ( x ) (40)in the Banach space B , where I is the identity operator. From (38) it follows that I − H ( x )is the inverse operator to I + ˜ H ( x ) . The existence of the inverse operator means that equation(40) is uniquely solvable.Thus, we obtain the following algorithm for solving Inverse problem 2 in the case a < b . Algorithm 1.
Let the function M ( λ ) be given.1) Find the sequences { α n } ∞ n =1 and { λ n } ∞ n =1 as the residues and the poles of M ( λ ) , re-spectively.2) Choose any model boundary value problem ˜ L ( T ) . Construct ˜ ψ ( x ) and ˜ H ( x ) for x ∈ [ a , b ] .
3) Find ψ ( x ) by solving equation (40), x ∈ [ a , b ] .
4) Find C ( x, λ ) , x ∈ [ a , b ] , from (35).5) Calculate q ( x ) = ( C ′′ ( x, λ ) + λC ( x, λ )) C − ( x, λ ) , x ∈ [ a , b ] . So, we can find the potential on [ a , b ] by the method of spectral mappings if a < b . The case a = b will be treated in the next section. Remark 2.
If (23) does not hold, Algorithm 1 can be obtained as well. In this case, wehave no asymptotic formulae (24) for individual weight numbers. Instead of this, formulae canbe found for the sums of the weight numbers in the groups of asymptotically close eigenvalues;see [23, 24, 27], where the analogous situations occur. Further, one should apply the form ofthe method of spectral mappings developed in [27].4 a = b . Solu-tion of Inverse problem 1
Further, we use the recursive structure of T : T = [ a , b ] ∪ T with the time scale T consisting only of isolated points and segments, see (11) for thedefinition of T m . Assume that N + M − µ > , then the time scale T contains at leasttwo points. Let us provide several relations between some objects for T and the analogousones for T . Consider the solution S ( x, λ ) , x ∈ T , of the Sturm–Liouville equation (2) on T satisfying the initial conditions S ( a , λ ) = 0 , S ∆2 ( a , λ ) = 1 . If T is the empty set, then the time scale T is the union of two isolated points a and a . In this case all values of S ( x, λ ) are completely determined by the initial conditions.Denote D ( λ ) := S ( b N + M , λ ) . This function is the characteristic functions of the bound-ary value problem L ( T ) . We introduce the function Φ ( x, λ ) , x ∈ T , which is the solution of equation (2) underthe boundary conditions Φ ∆2 ( a , λ ) = 1 , Φ ( b N + M , λ ) = 0 . We also consider M ( λ ) := Φ ( a , λ ) which is the Weyl function for the Sturm–Liouvilleboundary value problems L j ( T ) , j = 0 , . It is easy to see that M ( λ ) = Φ( a , λ )Φ ∆ ( a , λ ) . (41)Clearly, the functions Φ ( x, λ ) and M ( λ ) are analogues of the functions Φ( x, λ ) and M ( λ ) , respectively.Now we are in position to solve the local inverse problem in the case a = b , whichconsists in recovering the value q ( a ) given M ( λ ) . For this purpose we use the followingrelation obtained in [13]: D ( λ ) = α ( λ )Θ ( λ ) − α ( λ )Θ ( λ ) = Θ ( λ ) − ( a − b )Θ ( λ ) . (42)The conditions a = b and a ∈ T guarantee that N + M − µ > ( λ ) D ( λ ) = ( a − a ) ( q ( a ) − λ ) − ( a − a ) ρi + 1 + o (1) , a < b , ( a − a ) ( q ( a ) − λ ) + a − a a − a + 1 + o (1) , a = b , λ → −∞ . (43)Note that formula (43) can not be proved under the weaker assumptions on the potential than q ∈ W [ a l k , b l k ] , k = 1 , N . This is due to the fact that information about the values of q in the isolated points can not be extracted from the leading terms in the polynomials β lij ( λ ) . The details of the proof can be found in [13].Taking (43) into account, we get the following algorithm for solving Inverse problem 2 inthe case a = b . Algorithm 2.
Let the function M ( λ ) be given.1) Construct Θ ( λ ) and Θ ( λ ) . Find D ( λ ) with formula (42).2) Find q ( a m ) from (43).Algorithms 1 and 2 give the complete solution of the local inverse problem. Now weare ready to formulate the recursive algorithm for solving Inverse problem 1. Assume thatconditions (21) and (25) are fulfilled. Algorithm 3.
Given the Weyl function M ( λ ) .
1) Construct q ( x ) on [ a , b ] using Algorithm 1 or Algorithm 2.2) If T = [ a , b ] , then terminate the algorithm.3) Calculate C ∆ ν ( b , λ ) and S ∆ ν ( b , λ ) for ν = 0 , .
4) Find Φ ∆ ν ( b , λ ) for ν = 0 , a , λ ) and Φ ∆ ( a , λ ) via jump conditions (4).6) Calculate M ( λ ) via (41). Apply Algorithm 3 to T given the Weyl function M ( λ ) . On step 6), we run the Algorithm 3 to the time scale T given the corresponding data M ( λ ) . Actually, we should re-designate T = T and consider all other objects for this new T ( = T ). Then Algorithm 3 is repeated with the same notations. Call every its launch aniteration: the initial launch is the first iteration and so on.Conditions (21) and (25) formulated in the initial terms guarantee that the asymptoticformulae for the eigenvalues and the weight numbers necessary for Algorithm 1 hold on everyiteration as soon as the corresponding new T begins with a segment. However, condition(25) can be eliminated by the way mentioned in Remark 2.After the k -th iteration, the potential is found on the set S kl =1 [ a l , b l ] . Thus, the recon-struction of the potential will be fully completed after a finite number of iterations.