A uniqueness theorem on inverse spectral problems for the Sturm--Liouville differential operators on time scales
aa r X i v : . [ m a t h . SP ] M a r A uniqueness theorem on inverse spectral problems for theSturm–Liouville differential operators on time scales
M.A. Kuznetsova Abstract.
In the paper, Sturm–Liouville differential operators on time scales consistingof a finite number of isolated points and segments are considered. Such operators unifydifferential and difference operators. We obtain properties of their spectral characteristicsincluding asymptotic formulae for eigenvalues and weight numbers. Uniqueness theorem isproved for recovering the operators from the spectral characteristics.
Keywords: differential operators; Sturm–Liouville equation; time scales; closed sets; inversespectral problems.
AMS Mathematics Subject Classification (2010):
1. Introduction
Time scale theory unifies discrete and continuous calculus. It has important applicationsin natural sciences, engineering, economics and in other fields; for examples see [1–3, 21, 22].Models of processes in these cases include differential equations on a time scale, i.e. closedsubset of the real line. Various aspects of differential equations on time scales includingboundary value problems were considered in [1–10].In this paper, we study inverse spectral problems for the Sturm–Liouville operator on timescales. Such problems consist in recovering operators from given spectral characteristics. Forthe classical Sturm–Liouville operators on an interval, inverse problems have been studiedfairly completely; the classical results can be found in [17–19]. However, nowadays there areonly few works on inverse problem theory for differential operators on time scales because thestatement and the study of inverse spectral problems essentially depend on the structure ofthe time scale. In particular, in [4] an Ambarzumian type theorem was obtained for Sturm–Liouville operators on time scales.We consider bounded time scales T consisting of N < ∞ segments and M < ∞ isolatedpoints: 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)where l k denotes the indice corresponding to the k -th segment. The case N = 1 , M = 0corresponds to the classical Sturm–Liouville operator.If T consists only of isolated points, i.e. N = 0 , we have a difference operator. Inversespectral problems for the difference operators were studied in [11–15] and other works. In [11]the coefficients of finite discrete Sturm–Liouville type bondary value problem are recoveredfrom the spectrum and the set of normalization constants or from two spectra. The works[12,13] are devoted to the discrete analogues of inverse scattering problems on semiaxis and thewhole axis. In [14] V. A. Yurko studied the so-called operators of triangular structure, whichgeneralize the difference ones, and proved the uniqueness theorem and obtained the algorithmfor recovery from the Weyl matrix. In [15] the uniqueness theorem for the inverse problem fromthe eigenvalues and the weight numbers of Sturm–Liouville type difference operator on a finite Department of Mathematics, Saratov State University, Astrakhanskaya 83, Saratov 410012, Russia, email:[email protected] set of integers is proved. Let us note that this result is the particular case of Theorem 5 in thepresent paper. Moreover, some numerical methods for solving inverse problems for ordinarydifferential operators are based on their approximations by difference operators (see [16] andreferences therein).The paper is organized as follows. The Sturm–Liouville operator on the time scale T isintroduced in Section 2. We study the following its spectral characteristics: the spectra oftwo boundary value problems with one common boundary condition, the weight numbers andthe Weyl function. In Section 3, we establish their asymptotical behavior (Theorems 1–4). InSection 4, we study three inverse problems of recovering the potential of the Sturm–Liouvilleoperator from the given Weyl function, the two spectra or the spectrum along with the weightnumbers. The uniqueness theorem for these inverse problems is proved, see Theorem 5. Wealso offer Algorithm 1, which allows one to recover the potential of the difference Sturm–Liouville operator (i.e. when N = 0 ).
2. Sturm–Liouville operators on time scales
For convenience of the reader here we provide necessary notions of the time scale theory(see [1, 2] for more details). Let T be an arbitrary closed subset of R , which we refer to asthe time scale. We define the so-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 is left-isolated, and T := T, otherwise. We alsodenote by C ( B ) the class of functions continuous on the 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. The following proposition gives conditions of ∆ -differentiability at points of differenttypes.
Proposition 1.
1) If f ( t ) is ∆ -differentiable at t, then f ( t ) is continuous in t.
2) Let t ∈ T be a right-isolated point. Then f is ∆ -differentiable at t, if and only if f is continuous in t. In this case we have f ∆ ( t ) = f ( σ ( t )) − f ( t ) σ ( t ) − t .
3) Let t ∈ T be a right-dense point. Then f is ∆ -differentiable at t, if and only ifthere exists the limit lim s → t, s ∈ T f ( t ) − f ( s ) t − s =: f ∆ ( t ) . In particular, if ( t − ε, t + ε ) ⊂ T for some ε > , then f is ∆ -differentiable at t, if andonly if f is differentiable at t. In this case the equality f ∆ ( t ) = f ′ ( t ) holds. 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 − , inturn, 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 ≥ C n ( T ) the class of functions f for which there exists the n -th ∆ -derivative f ∆ n and f ∆ n ∈ C ( T n ) . From now on, f ∆ ν ( x , . . . , x n ) denotes the ν -th ∆ -derivative of the function f ( x , . . . , x n ) with respectto the first argument, and f ( ν ) ( x , . . . , x n ) denotes its classical ν -th derivative with respectto the first argument.A function F ( t ) is called antiderivative of f ( t ) , if there exists F ∆ ( t ) = f ( t ) for all t ∈ T . In [1, Section 1.4] it was established that any function from C ( T ) has antiderivatives,which differ by constant. For any a, b ∈ T, the formula Z ba f ( t ) ∆ t := F ( b ) − F ( a )defines the definite ∆ -integral of a function f ( t ) on T ∩ [ a, b ] . Consider the following Sturm–Liouville equation on T : ℓy := − y ∆∆ ( x ) + q ( x ) y ( σ ( x )) = λy ( σ ( x )) , x ∈ T . (2)Here λ is the spectral parameter, and q ( x ) ∈ C ( T ) is a real-valued function. A function y is called a solution of equation (2), if y ∈ C ( T ) and equality (2) is fulfilled.For definiteness, we restrict ourself to time scales T of the form (1). We consider N > M ≥ , otherwise equation (2) degenerates. Let us additionaly assume that q ∈ W [ a l k , b l k ] , k = 1 , N . Note that the last condition is equivalent to the belongingness of q to the corresponding Sobolev-type space on T (see [5]).For T of the form (1) some concepts of time scale theory can be clarified. In particular,if a, b ∈ S N + Mk =1 { a k , b k } , a ≤ b, then by additivity of ∆ -integral and [1, Theorem 1.79] Z ba f ( t ) ∆ t = X k : b k ∈ [ a,b ) f ( b k )( a k +1 − b k ) + X k : a ≤ a k
Let T = { l } l =0 , i.e. N = 0 , M = 4 . Then q ( t ) is defined in t = 0 and t = 1 . Using (7), one can compute the characteristic functionsΘ ( λ ) = λ − λ + 3 , Θ ( λ ) = λ − λ + 1 . Thus, we have λ = 1 , λ = 3 , λ = −√ , λ = √ . Example 2.
Consider T = [0 , , i.e. N = 1 , M = 0 . In this case (2) coincides with theclassical Sturm-Liouville equation, andΘ ( λ ) = sin ρρ , Θ ( λ ) = cos ρ, λ nj = π (cid:16) n − j (cid:17) , n ≥ , j = 0 , . Example 3.
Consider T = [0 , S [2 , , i.e. N = 2 , M = 0 . ThenΘ ( λ ) = cos ρ + 2 − λρ cos ρ sin ρ − sin ρ, Θ ( λ ) = ( λ −
1) sin ρ + cos ρ − ρ sin ρ cos ρ. With the standard method involving Rouche’s theorem it can be established that { λ nj } n ≥ = n ( πn + o (1)) o n ≥ [ n(cid:16) π (cid:0) n − − j (cid:1) + o (1) (cid:17) o n ≥ − j , j = 0 , . In these examples we can conclude that both spectra are finite if and only if N = 0 . Inthe last case each one contains M − N > . To obtain the other necessary properties of the eigenvalues, we introduce the notion of theWronskian-type determinant W ( ϕ, ψ ) := ϕ ( t ) ψ ∆ ( t ) − ϕ ∆ ( t ) ψ ( t ) , where ϕ ( t ) and ψ ( t ) aresolutions of equation (4). By virtue of Theorem 3.13 in [1], we have W ( ϕ, ψ ) ≡ const on T . Proposition 2.
1. The sequences { λ n } n ≥ and { λ n } n ≥ have no common elements.2. All zeros of Θ j ( λ ) , j = 0 , , are real and simple.Proof.
1. It is obvious that W ( C, S ) = 1 for all t ∈ T and λ ∈ C . Suppose that S ( b N + M , λ ) = C ( b N + M , λ ) = 0 for some λ . If a N + M < b N + M , then { C ( t, λ ) , S ( t, λ ) } ,t ∈ ( a N + M , b N + M ) , is a fundamental system of solutions of the N -th equation (5) in λ = λ , and we arrive at the contradiction. In the case when a N + M = b N + M , from (7) follows thelinear dependence of the vectors( S ( b N + M − , λ ) , C ( b N + M − , λ )) T , ( S ∆ ( b N + M − , λ ) , C ∆ ( b N + M − , λ )) T , where T is the transposition sign. The latter contradicts to W ( C, S ) = 1 .
2. Consider the case j = 1 . Let Θ ( λ ) = 0 and t ∈ T . From (2) for y = C ( t, λ ) andfor y = C ( t, λ ) one can obtain − C ∆∆ ( t, λ ) C ( σ ( t ) , λ ) + C ∆∆ ( t, λ ) C ( σ ( t ) , λ ) = ( λ − λ ) C ( σ ( t ) , λ ) C ( σ ( t ) , λ ) . The relation ( f ( t ) g ( t )) ∆ = f ∆ ( t ) g ( σ ( t )) + f ( t ) g ∆ ( t ) yields that( − C ∆ ( t, λ ) C ( t, λ ) + C ∆ ( t, λ ) C ( t, λ )) ∆ = ( λ − λ ) C ( σ ( t ) , λ ) C ( σ ( t ) , λ ) . Denote t r := max T . Note that t r = b N + M when b N + M is left-dense and t r = b N + M − in the opposite case. Integrating both sides of the previous relation and using the initialconditions (8), we get( λ − λ ) Z t r a C ( σ ( t ) , λ ) C ( σ ( t ) , λ ) ∆ t = C ∆ ( t r , λ ) C ( t r , λ ) − C ( t r , λ ) C ∆ ( t r , λ ) . (9)Due to real-valuedness of q, since λ is an eigenvalue, the number λ is also an eigenvaluewith the eigenfunction C ( t, λ ) = C ( t, λ ) . Substituting λ = λ into (9), we have − λ Z t r a | C ( σ ( t ) , λ ) | ∆ t = C ∆ ( t r , λ ) C ( t r , λ ) − C ( t r , λ ) C ∆ ( t r , λ ) . Then the following relation is obvious in the case t r = b N + M : − λ Z t r a | C ( σ ( t ) , λ ) | ∆ t = 0 . (10)The formula is also valid when t r = b N + M since C ∆ ( t r , λ ) = C ( b N + M ,λ ) − C ( t r ,λ ) b N + M − t r in this case.From (3) it follows that Z t r a | C ( σ ( t ) , λ ) | ∆ t = X k : b k
0; when a = b the relation C ( a , λ ) = C ( a , λ )+ C ∆ ( a , λ )( a − a ) =1 is fullfiled, and since a = b < t r , we have X k : b k
3. Properties of the spectral characteristics
Let us put d k := b l k − a l k , k = 1 , N , where l k are determined in (1). Without loss ofgenerality, 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 ( 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 , where β l ( λ ) are determined for k = 1 , N + µ and s = 1 , l k − l k − as follows: β l k − s ( λ ) := α l k − ( λ ) . . . α l k − s ( λ ); β l N ( λ ) := (1 , , l N = N + M. By virtue of (7), we have (cid:0) y ( a l k ) , y ∆ ( a l k ) (cid:1) T = β l k − s ( λ ) (cid:0) y ( b l k − s ) , y ∆ ( b l k − s ) (cid:1) T , k = 1 , N , s = 1 , l k − l k − ,y ( a l N +1 ) = β l N +1 − s ( λ ) (cid:0) y ( b l N +1 − s ) , y ∆ ( b l N +1 − s ) (cid:1) T , s = 1 , l N +1 − l N . Further we establish asymptotic formulae for the elements of β l ( λ ) . Lemma 1.
For k = 1 , N + µ , s = 1 , l k − l k − the following asymptotic formulae are ful-filled: β l k − sij ( λ ) = a l k − sij ( λ s − i + b l k − sij λ s − i + O ( λ s − i )) , i = 1 , − δ ( k, N + 1) , j = 1 , , (15) where a l k − sij = ( − s − i ( a l k − s +1 − b l k − s ) j − ( a l k − b l k − ) i − l k − Y l = l k − s ( a l +1 − b l ) , (16) b l k − sij = − l k − i X l = l k − s +2 − j ( a l +1 − b l ) − + l k − X l = l k − s +1 ( a l +1 − b l ) − ( a l − b l − ) − + l k − i X l = l k − s q ( b l ) ! . (17) Proof.
Fix any k ∈ , N + µ . In the case s = 1 formulae (15)–(17) are checked directly.Let (15)–(17) be fulfilled for some s = µ ∈ [1 , l k − l k − ) . Then β l k − µ − ij ( λ ) = β l k − µi ( λ ) α l k − µ − j ( λ ) + β l k − µi ( λ ) α l k − µ − j ( λ ) , i = 1 , − δ ( k, N + 1) , j = 1 , . Using this formulae and the induction assumption, we get (15) for s = µ + 1 with thecoefficients a l k − µ − ij = − ( a l k − µ − b l k − µ − ) j a l k − µi and b l k − µ − ij = − q ( b l k − µ − ) − a l k − µi ( a l k − µ − b l k − µ − ) a l k − µi + 1 − j ( a l k − µ − b l k − µ − ) + b l k − µi . Then representations (16) and (17) in s = µ yield (16) and (17) in s = µ + 1 . Thus, byinduction (15)–(17) is proved for s = 1 , l k − l k − . Let us split T into the union of the sets T m := N + M [ k = m [ a k , b k ] , T m, := m − [ k =1 [ a k , b k ] , m = 1 , N + M − µ , and consider the solutions S m ( x, λ ) , C m ( x, λ ) of Sturm–Liouville equation (2) on T m satis-fying the initial conditions S m ( a m , λ ) = S ∆ m ( a m , λ ) − C m ( a m , λ ) − C ∆ m ( a m , λ ) = 0 . If T m T T = ∅ , then the functions S m and C m are completely determined by these initialconditions.The functions S l k ( x + a l k , λ ) , C l k ( x + a l k , λ ) , x ∈ [0 , d k ] , k = 1 , N , can be obtained asthe solutions of the following integral equations: S l k ( x + a l k , λ ) = sin ρxρ + Z x sin ρ ( x − t ) ρ S l k ( t + a l k , λ ) q k ( t ) dt,C l k ( x + a l k , λ ) = cos ρx + Z x sin ρ ( x − t ) ρ C l k ( t + a l k , λ ) q k ( t ) dt, (18)where q k ( x ) := q ( a l k + x ) , x ∈ [0 , d k ] . Obviously, q k ∈ W [0 , d k ] . Substituting the standardasymptotic formulae for S l k ( x + a l k , λ ) and C l k ( x + a l k , λ ) (see [19, Sect.1.1]) into (18), for x = d k we obtain S l k ( b l k , λ ) = sin ρd k ρ " A k ρ − cos ρd k ρ ω k − ρ Z d k q ′ k ( t ) sin ρ (2 t − d k ) dt + O ( e | τ | d k ) ρ , (19) S ′ l k ( b l k , λ ) = cos ρd k " A k ρ + sin ρd k ρ ω k + 14 ρ Z d k q ′ k ( t ) cos ρ (2 t − d k ) dt + O ( e | τ | d k ) ρ , (20) C l k ( b l k , λ ) = cos ρd k " A k ρ + sin ρd k ρ ω k − ρ Z d k q ′ k ( t ) cos ρ (2 t − d k ) dt + O ( e | τ | d k ) ρ , (21) C ′ l k ( b l k , λ ) = − ρ sin ρd k " A k ρ +cos ρd k ω k − ρ Z d k q ′ k ( t ) sin ρ (2 t − d k ) dt + O ( e | τ | d k ) ρ . (22)Here τ := Im ρ, ω k := 12 Z d k q k ( t ) dt, ˜ A ik := ( − [( i − / q k (0)4 + ( − i − q k ( d k )4 − ω k ,i = 1 , , k = 1 , N , and [ x ] denotes the integer part of x. Denote D m ( λ ) := S m ( b N + M , λ ) , D m ( λ ) := C m ( b N + M , λ ) , m = 1 , N + M − µ . In par-ticular, Θ j ( λ ) = D j ( λ ) , j = 0 , . We also introduce the functions Φ m ( x, λ ) , x ∈ T m , which are solutions of equation (2), m = 1 , N + M − µ , satisfying the boundary condi-tions Φ ∆ m ( a m , λ ) = 1 , Φ m ( b N + M , λ ) = 0 . One can obtain the following formulae, which areanalogues of (13), (14): Φ m ( x, λ ) = S m ( x, λ ) + M m ( λ ) C m ( x, λ ) , (23)where M m ( λ ) = − D m ( λ ) D m ( λ ) . (24) Lemma 2.
For j = 0 , the following representations hold: D lj ( λ ) = β l , − j ( λ ) , l = l N + 1 , N + M − , (25) D l k − sj ( λ ) = ρ µ − β l k − s , − j ( λ ) N − Y i = k β l i ( λ ) β l N , µ ( λ ) N Y l = k g l ( ρ ) + O ( e | τ | γ k ) ρ ! ,s = 1 , l k − l k − − δ ( k, µ , (26) D l k j ( λ ) = ( − j (1 − δ k ) ρ µ + j − N − Y i = k β l i ( λ ) β l N , µ ( λ ) N Y l = k +1 g l ( ρ ) v kj ( ρ ) + O ( e | τ | γ k ) ρ ! , (27) where k = 1 , N . For these k and j = 0 , we denoted γ k := P Nl = k d l , δ k := δ ( l k , N + M ) ,g k ( ρ ) := v k ( ρ ) + ( − δ k v k ( ρ ) ρ ( a l k − b l k − ) , l k > ,v kj ( ρ ) := f kj ( ρ ) (cid:18) A kj ρ (cid:19) + f k, − j ( ρ ) c k ( − j + δ k ρ + ( − δ k ρ Z d k f kj ((2 t/d k − ρ ) q ′ k ( t ) dt, (28) where 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 ,c k and A kj are some constants, which can be expressed from q k : c k := ( ω k , δ k = 1 ,ω k + a lk +1 − b lk , δ k = 0 , A kj := ( ˜ A k, j +1 , δ k = 1 , ˜ A k, j +2 − ω k a lk +1 − b lk , δ k = 0 . Proof.
We will prove these formulae by induction. For k = N + M − µ the formulae(25) or (27) is fulfilled: (25) follows from the jump conditions (7) while (27) follows from (19),(21). Let D m +1 j ( λ ) be given by formulae (27) for some l k = m + 1 > . We consider twopossible cases. First, let l i = l k − , i = k − > . Using (7) we expand Y := S m and Y := C m with respect to the system { C m +1 , S m +1 } on T m +1 : D mj ( λ ) = ( α m ( λ ) Y j ( b m , λ ) + α m ( λ ) Y ′ j ( b m , λ )) D m +11 ( λ )+( α m ( λ ) Y j ( b m , λ ) + α m ( λ ) Y ′ j ( b m , λ )) D m +10 ( λ ) , j = 0 , . (29)From (19)–(22) and the definition of β l ( λ ) it follows that α m ( λ ) Y j ( b m , λ ) + α m ( λ ) Y ′ j ( b m , λ ) = ( − j ρ j β m ( λ ) (cid:18) v k − ,j ( ρ ) + O ( e | τ | d k − ) ρ (cid:19) ,α m ( λ ) Y j ( b m , λ ) + α m ( λ ) Y ′ j ( b m , λ ) = ( − j ρ j β m ( λ ) (cid:18) v k − ,j ( ρ ) + O ( e | τ | d k − ) ρ (cid:19) = ( − j +1 ρ j β m ( λ ) (cid:18) λ − v k − ,j ( ρ ) a l k − b l k − + O ( e | τ | d k − ) ρ (cid:19) . (30)These relations with (29) and the induction assumption (27) give formulae (27) for k = i. Second, let a m = b m . Expanding S m , C m with respect to the system { S m +1 , C m +1 } on T m +1 , we get D mj ( λ ) = α m , − j ( λ ) D m +11 ( λ ) + α m , − j ( λ ) D m +10 ( λ ) , j = 0 , . (31)Bracing α m , − j ( λ ) with (15) and applying the induction assumption, we prove (26) in s = 1 . The other cases are operated with the same technique. (cid:3)
The previous lemma yields thatΘ 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 , , (32)0where 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 ) . By the standard method involving Rouche’s theorem [19], from (32) the following structureof the spectra can be established.
Theorem 1.
Each spectrum consists of N + 1 parts: { λ nj } n ≥ = Λ j [ (cid:18) N [ k =1 (cid:8) ( ρ ( k ) nj ) (cid:9) n ≥ (cid:19) , j = 0 , , where Λ j contains N + M + j (1 − µ )sign( N − µ ) − µ − elements and for thesubsequences (cid:8) ( ρ ( k ) nj ) (cid:9) n ≥ the following asymptotic formulae are fulfilled: ρ ( k ) nj = π n − δ jδ (1 ,k )(1 − µ ) k d k + o (1) , δ k := 12 δ ( δ k , , δ k := 12 − δ k , ≤ k ≤ N. (33)The main parts of eigenvalues’ roots in (33) from different subsequences can occur arbitrar-ily close to each other, which causes the difficulty in the further refinement of these asymptoticformulae. To overcome it, we make the following additional assumption: d k = rx k , x k ∈ Q , k = 1 , N , for some r > , (34)which means commensurability of the segments. Analogous commensurability assumptionsappear also in other situations, e.g. for studying spectral properties of differential operatorson geometrical graphs (see, e.g., [20]). Assumption (34) is needed for Theorems 2–4 and is notused anywhere else. This assumption yields that for any fixed s, k ∈ , N and j, ν ∈ { , } for all l, n ∈ N we have the following alternatives: d k d s = l − δ jk n − δ νs or (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d k d s − l − δ jk n − δ νs (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ ( Cn ) − , where and in the sequel C denotes different sufficiently large constants. Then we have f kj (cid:18) π n − δ νs d s (cid:19) = 0 or C > (cid:12)(cid:12)(cid:12)(cid:12) f kj (cid:18) π n − δ νs d s (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) > C − . (35)Denote by η kj ( ρ ) the multiplicity of ρ as a zero of the function N Y l = k +1 f l ( ρ ) f kj ( ρ ) , k = 1 , N . For briefness denote different sequences from l by one and the same symbol { κ n } n ≥ . Wealso use { κ n ( z ) } n ≥ to designate different sequences of functions which are continuous insome circle | z | ≤ R with n max | z |≤ R | κ n ( z ) | o n ≥ ∈ l . The following theorem refines formulae (33) under the additional condition (34).1
Theorem 2. If (34) is fulfilled, for the subsequences (cid:8) ( ρ ( k ) nj ) (cid:9) n ≥ we have ρ ( k ) nj = π ( n − δ jδ (1 ,l k ) k ) d k + O (cid:18) n (cid:19) , k = 1 , N , n ∈ N . (36) Proof.
We plan to use the formulae of Lemma 2. In the case N = 1 the computations areanalogous to the classical case of the Sturm–Liouville equation on interval since the problemof close eigenvalues’ roots does not arise. Therefore, we consider only the case N > . For definiteness we consider ρ ( N ) nj = K nj + z nj := π n − δ N d N + z nj , z nj = o (1) , n → ∞ . We substitute ρ = ρ ( N ) nj into D lν ( ρ ) , ν = 0 , , l = 1 , N + M − µ . The following formulaecan be proved by induction: D l k ν (( ρ ( N ) nj ) ) = ( ρ ( N ) nj ) µ + ν − β l N , µ (( ρ ( N ) nj ) ) N − Y i = k β l i (cid:0) ( ρ ( N ) nj ) (cid:1) × c kνnj z η kν ( K nj ) nj + η kν ( K nj ) − X l =0 z lnj O ( n l − η kν ( K nj ) ) ! , (37) D l k − sν (( ρ ( N ) nj ) ) = ( ρ ( N ) nj ) µ − β l N , µ (( ρ ( N ) nj ) ) N − Y i = k β l i (cid:0) ( ρ ( N ) nj ) (cid:1) β l k − s , − j (cid:0) ( ρ ( N ) nj ) (cid:1) × c k nj z η k ( K nj ) nj + η k ( K nj ) − X l =0 z lnj O ( n l − η k ( K nj ) ) ! , (38)where | c kνnj | ≥ C − η kν ( K nj ) for sufficiently large n, s = 1 , l k − l k − − δ ( k, µ . Theirproof is conducted according the scheme of the one of Lemma 2; for formula (37) one shouldconsider two cases f kν ( K nj ) = 0 and C > | f kν ( K nj ) | > C − due to (35). In the first case v kν ( ρ ( N ) nj ) = ( ± d k + o (1)) z nj + O ( n − ) and the degree η kν ( K nj ) = η k +1 , ( K nj ) + 1 , in thesecond case C > | v kν ( ρ ( N ) nj ) | > C − and η kν ( K nj ) = η k +1 , ( K nj ) . We note that D j (( ρ ( N ) nj ) ) = 0 . Thus, from (37) for µ = 0 or from (38) for µ = 1 wehave | z nj | N ≤ C N P Nl =0 | z nj | l O ( n l − N ) , which yields | y nj | N ≤ C N +1 N − X l =0 | y nj | l for y nj = nz nj and sufficiently large C. From the last inequality it follows that y nj = O (1) . Indeed, if | y nj | > , we can estimate | y nj | ≤ C N +1 N − X l =0 | y nj | l +1 − N ≤ C N +1 , and, hence, { y nj } ∞ n =1 is bounded.Thus, we proved (36) for k = N. The other formulae can be proved analogously.2Asymptotic formulae (36) can be refined as well. Namely, the following theorem holds.
Theorem 3.
Denote z k := 1 π (cid:16) c k + l k − X l = max (1 ,l k − ( a l +1 − b l ) − (cid:17) , k = 1 , N , where c k are defined in Lemma 2. If (34) holds, then ρ ( k ) nj = π ( n − δ jδ (1 ,l k ) k ) d k + z k n − δ jδ (1 ,l k ) k + r ( k ) nj n , r ( k ) nj = o (1) , k = 1 , N , n ∈ N . (39) Provided all z k /d k , k = 1 , N , are distinct, we have r ( k ) nj = κ n /n. Proof.
By the same reason as for the proof of Theorem 2, we consider only the case
N > . Any point in the vicinity of ρ ( N ) nj can be represented in the form ρ ( N ) nj ( z ) = π n − δ N d N + zn − δ N =: K nj + zn − δ N , | z | ≤ C. Substituting ρ = ρ ( N ) nj ( z ) into the D lν ( ρ ) , ν = 0 , , l = 1 , N + M − µ , analogously to(37) and (38) one can prove that D l k ν (( ρ ( N ) nj ( z )) ) = ( − ν (1 − δ k ) ρ µ + ν − N − Y i = k β l i (( ρ ( N ) nj ( z )) ) β l N , µ (( ρ ( N ) nj ( z )) ) × N Y l = k +1 g l ( ρ ( N ) nj ( z )) v kν ( ρ ( N ) nj ( z )) + κ n ( z ) K η kν ( K nj )+1 nj ! , k = 1 , N , (40) D l k − sν (( ρ ( N ) nj ( z )) ) = ρ µ − β l k − s , − j (( ρ ( N ) nj ( z )) ) N − Y i = k β l i (( ρ ( N ) nj ( z )) ) β l N , µ (( ρ ( N ) nj ( z )) ) × N Y l = k g l ( ρ ( N ) nj ( z )) + κ n ( z ) K η k ( K nj )+1 nj ! , k = 1 , N , s = 1 , l k − l k − − δ ( k, µ . (41)For the proof it is sufficient to obtain the analogue of (30) with κ n ( z ) instead of O ( e | τ | d k )in the case when f kν ( K nj ) = 0 using Lemma 1; the other computations are similar to theproof of Theorem 2.Denote I := (cid:26) { m : f m ( K nj ) = 0 , m > } S { } , f ,j (1 − µ ) ( K nj ) = 0 , { m : f m ( K nj ) = 0 , m > } , f ,j (1 − µ ) ( K nj ) = 0 . Consider µ = 0 and (40) for k = 1 (for µ = 1 one uses (41)). Using the condition (35)we obtainΘ j (( ρ ( N ) nj ( z )) ) = D j (( ρ ( N ) nj ( z )) ) = ( ρ ( N ) nj ( z )) µ + j − β N , µ (( ρ ( N ) nj ( z )) ) N − Y i =1 β l i (cid:0) ( ρ ( N ) nj ( z )) (cid:1) × C nj Y m ∈ I (cid:18) d m zn − δ N − d N z m n − δ N (cid:19) + κ n ( z )( n − δ N ) η j ( K nj )+1 ! , (42)3where | C nj | ≥ C η j ( K nj ) − N . With Rouche’s theorem we obtain that Θ j (( ρ ( N ) nj ( z )) ) has η ,j (1 − µ ) ( K nj ) zeros ρ ( N ) nj ( z ) = π n − δ N d N + zn − δ N with z = d N z m /d m + o (1) , m ∈ I. Thismeans that Θ j ( ρ ) has the following η ,j (1 − µ ) ( K nj ) zeros which are close to K nj : π n − δ N d N + d N z m + o (1) d m ( n − δ N ) = π l − δ jδ (1 ,l m ) m d m + z m + o (1) l − δ jδ (1 ,l m ) m , m ∈ I, such that d N /d m = ( n − δ N ) / ( l − δ jδ (1 ,l m ) m ) for some l ∈ N . In particular, we have (39) when m = N ∈ I. Using (42) in the vicinity | z − z N | < δ for a sufficiently small δ it is easy to prove (39)for k = N with r ( k ) nj = κ n /n. The other formulae can be proved analogously.Let us obtain asymptotic formulae for the weight numbers. For them one can prove theanalogues of Theorems 1–3. However, for briefness we provide only formulae under the con-ditions of Theorem 3.
Theorem 4.
The sequence { α n } n ≥ consists of N + 1 parts: { α n } n ≥ = A [ (cid:18) N [ k =1 (cid:8) α kn (cid:9) n ≥ (cid:19) , α kn := Res λ =( ρ ( k ) n ) M ( λ ) , A := { Res λ = z M ( λ ) : z ∈ Λ } . (43) If (34) is fulfilled and all z k /d k , k = 1 , N , are distinct, the following asymptotic formulaehold: α kn = d (cid:16) κ n n (cid:17) , k = 1 , µ = 0 ,κ n n , all the other cases . (44) Proof.
First of all, we note that (43) follows from Theorem 1.Let us prove (44). Consider the case k = 1 , µ = 0 , N > . Then δ = 0 , δ = 0 ,z = c , f ( x ) = sin d x, f ( x ) = cos d x. Let δ > δ < | z l /d l − z /d | , l = 2 , N . Denote ρ n ( z ) := π nd + z + zn =: K n + z + zn , | z | ≤ δ. By Cauchy’s residue theorem we obtain α n = 12 πi Z | z | = δ ρ n ( z ) n M ( ρ n ( z )) dz (45)for sufficiently large n. Further, for | z | ≤ δ by (40) we get M ( ρ n ( z )) = N Y l =2 g l ( ρ n ( z )) v ( ρ n ( z )) + κ n ( z ) n η ( K n )+1 ρ n ( z ) N Y l =2 g l ( ρ n ( z )) v ( ρ n ( z )) + κ n ( z ) n η ( K n )+1 ! . (cid:12)(cid:12)(cid:12)Q Nl =2 g l ( ρ n ( z )) (cid:12)(cid:12)(cid:12) ≥ C − n − η ( K n )+1 on | z | = δ and | η ( K n ) − η ( K n ) | ≤ M ( ρ n ( z )) = v ( ρ n ( z )) + κ n ( z ) n ρ n ( z ) (cid:16) v ( ρ n ( z )) + κ n ( z ) n (cid:17) . Substituting Taylor’s formulae of sin and cos into (28), we write v ( ρ n ( z )) = ( − n d n (cid:18) z + κ n ( z ) n (cid:19) , v ( ρ n ( z )) = ( − n (cid:18) κ n ( z ) n (cid:19) . Using (45) and the subsequent formulae, we get α n = 2 d πi Z | z | = δ κ n ( z ) n z + κ n ( z ) n dz = 2 d πi Z | z | = δ (cid:16) z + κ n ( z ) n (cid:17) dz. From this equation we obtain (44) for k = 1 , µ = 0 . The other cases can be operated analogously.Now let us study the asymptotical behavior of the functions C l k ( x, λ ) and Φ l k ( x, λ ) inthe case N > , k = 1 , N . For our purposes it is sufficient to consider ρ ∈ Ω δ := { z : arg z ∈ [ δ, π − δ ] } and x ∈ ( a l k , b l k ) . From (21)–(22) it follows that C ( ν ) l k ( x + a l k , λ ) = ( − iρ ) ν − iρx )[1] , x ∈ (0 , d k ] , ρ ∈ Ω δ , ν = 0 , . (46)Using the standard approach (see, for example, [10]) one can prove the following formulae:Φ ( ν ) l k ( x + a l k , λ ) = ( iρ ) ν − exp( iρx )[1] , x ∈ [0 , d k ) . (47)
4. Inverse problems
Consider the following three inverse problems.
Inverse problem 1.
Given M ( λ ) , find q ( x ) on T . Inverse problem 2.
Given { λ nj } n ≥ , j = 0 , , find q ( x ) on T . Inverse problem 3.
Given { λ n } n ≥ , { α n } n ≥ , find q ( x ) on T . First, we show that these inverse problems are equalent, i.e. their input data uniquely de-termine each other. Since Θ ( λ ) and Θ ( λ ) have no common zeros, { λ n } n ≥ and { λ n } n ≥ are determined as zeros and poles of the Weyl function. Conversely, Hadamard’s factorizationtheorem gives 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 is a non-zero complex constants, while s j is the multiplicity of the zero eigenvaluein the spectrum { λ nj } n ≥ . By virtue of (32), the following limits exist:lim λ → i ∞ Θ j ( λ ) F j ( λ ) = 1 , j = 0 , , C j = lim λ → i ∞ F j ( λ ) p j ( λ ) . Thus, the characteristic functions Θ j ( λ ) are uniquely determined by their zeros { λ nj } n ≥ . Taking into account formula (14) we conclude that two spectra uniquely determine the Weylfunction as well.Using Lemmas 1 and 2, one can prove by technique analogous to [19] that the weightnumbers and the poles uniquely determine the Weyl function by the formula M ( λ ) = − µ ( a − δ ( N + M, − a ) + ∞ X n =1 α n λ − λ n , N > , − ( a − a ) + N − X n =1 α n λ − λ n , N = 0 . Thus, given the input data of one inverse problem, we can recover them of any other one.Moreover, both characteristic functions are determined by specifying the input data of anyInverse problem 1–3.Further, using the ideas of the method of spectral mappings [19] we prove the uniquenesstheorem for the solutions of the inverse problems. For this purpose together with the boundaryvalue problem L we consider a problem ˜ L of the same form but with another potential˜ q. In this section we agree that if a certain symbol γ denotes an object related to L , thenthis symbol with tilde ˜ γ will denote the analogous object related to ˜ L . Theorem 5.
If one of the following conditions is fulfilled, then q = ˜ q on T : M ( λ ) = ˜ M ( λ ); { λ nj } n ≥ = { ˜ λ nj } n ≥ , j = 0 , { λ n } n ≥ = { ˜ λ n } n ≥ and { α n } n ≥ = { ˜ α n } n ≥ . Thus, specification of the spectral data of any type uniquely determines the potential q. Proof.
I. At first, fix m ∈ , N + M such that [ a m , b m ] ⊆ T and suppose that D mj ( λ ) ≡ ˜ D mj ( λ ) , j = 0 , . It follows from (24) that M m ( λ ) ≡ ˜ M m ( λ ) . Let us prove that q and ˜ q coincide on [ a m , b m ] . First, consider the case a m < b m . For x ∈ ( a m , b m ) we define the functions P j ( x, λ ) = ( − j (Φ m ( x, λ ) ˜ C (2 − j ) m ( x, λ ) − ˜Φ (2 − j ) m ( x, λ ) C m ( x, λ )) , j = 1 , . By virtue of the relation C m ( x, λ )Φ ′ m ( x, λ ) − C ′ m ( x, λ )Φ m ( x, λ ) ≡ , we have P ( x, λ ) ˜ C m ( x, λ ) + P ( x, λ ) ˜ C ′ m ( x, λ ) = C m ( x, λ ) . (48)It also follows from (46), (47) that for each fixed x ∈ ( a m , b m ) P ( x, λ ) = 1 + O (cid:16) ρ (cid:17) , P ( x, λ ) = O (cid:16) ρ (cid:17) , ρ → ∞ , ρ ∈ Ω δ . (49)On the other hand, using (23) and the coinsidence of the Weyl functions, we get P j ( x, λ ) = ( − j ( S m ( x, λ ) ˜ C (2 − j ) m ( x, λ ) − ˜ S (2 − j ) m ( x, λ ) C m ( x, λ )) , j = 1 , . x ∈ ( a m , b m ) , the functions P ( x, λ ) and P ( x, λ ) are entirein λ of order 1 / . By the Phragmen–Lindel¨of theorem and Liouville’s theorem, asymptotics(49) imply P ( x, λ ) ≡ P ( x, λ ) ≡ , which along with (48) give C m ( x, λ ) = ˜ C m ( x, λ )for x ∈ ( a m , b m ) and, by continuity, for x ∈ [ a m , b m ] . Then q ( x ) = ˜ q ( x ) for x ∈ [ a m , b m ] . Now let a m = b m . The relation a m ∈ T means m < N + M and m < N + M − µ = 1 . If we prove that D mj ( λ ) , j = 0 , , uniquely determine q ( a m ) , this will yield q ( a m ) = ˜ q ( a m ) . Solving system (31) with respect to D m +10 ( λ ) and D m +11 ( λ ) , we get D m +10 ( λ ) = α m ( λ ) D m ( λ ) − α m ( λ ) D m ( λ ) = D m ( λ ) − ( a m +1 − b m ) D m ( λ ) , (50) D m +11 ( λ ) = α m ( λ ) D m ( λ ) − α m ( λ ) D m ( λ ) . (51)Thus, by (50) the function D m +10 ( λ ) can be computed. Let us write the asymptotic formulaefor D m ( λ ) / D m +10 ( λ ) , ρ ∈ Ω δ , λ → ∞ . There are two possible cases:Case 1. a m +1 < b m +1 . Then l k = m + 1 for some k ∈ , N . Use (26) for l k − s = m and(27) for l k = m + 1 : D m ( λ ) D m +10 ( λ ) = α m ( λ ) g k ( ρ ) + o (exp( | τ | d k ) ρ − ) v k ( ρ ) + o (exp( | τ | d k ) ρ − ) . Dividing the numerator and the denominator on f k ( ρ ) and using the estimate | f k ( ρ ) | ≥ C − e | τ | d k for ρ ∈ Ω δ , we get by (28) that D m ( λ ) D m +10 ( λ ) = α m ( λ ) 1 + P ( ρ ) + P ( ρ ) + o ( ρ − )1 + P ( ρ ) + o ( ρ − ) , (52)where P ( ρ ) := ( − δ k f k ( ρ ) ρf k ( ρ ) c k + A k ρ + ( − δ k ρ f k ( ρ ) Z d k f k (cid:18) tρd k − ρ (cid:19) q ′ k ( t ) dt = O (cid:18) ρ (cid:19) ,P ( ρ ) := ( − δ k ρ ( a m +1 − a m ) (cid:18) f k ( ρ ) f k ( ρ ) − ( − δ k c k ρ (cid:19) = O (cid:18) ρ (cid:19) . One can also prove that ( − δ k f k ( ρ ) f k ( ρ ) = i + o ( ρ − ) , ρ ∈ Ω δ . Then from the equalities(1 + P ( ρ ) + o ( ρ − )) − = 1 − P ( ρ ) + P ( ρ ) + o ( ρ − )and (52) we obtain D m ( λ ) D m +10 ( λ ) = ( a m +1 − a m ) ( q ( a m ) − λ ) − ( a m +1 − a m ) ρi + 1 + o (1) . From this formula one can compute the quantity q ( a m ) . Case 2. a m +1 = b m +1 . Let N m be the number of the indices in { l s } Ns =1 which are greaterthen m. Then the functions D m ( λ ) and D m +10 ( λ ) are given by (25) or by (26) dependingon whether N m is zero or not respectively. Then D m ( λ ) D m +10 ( λ ) = β m − δ ( N m , , ( λ ) β m +12 − δ ( N m , , ( λ ) (cid:18) o (cid:18) ρ (cid:19)(cid:19) . D m ( λ ) D m +10 ( λ ) = ( a m +1 − a m ) × (cid:18) − λ + 1( a m +1 − a m ) + 1( a m +1 − a m )( a m +2 − a m +1 ) + q ( a m ) (cid:19) + o (1) . (53)With this relation q ( a m ) can be computed.II. Let us prove by induction that the spectral data uniquely determine the potential q ( x ) . From the assumption of the theorem we have D j ( λ ) = ˜ D j ( λ ) , j = 0 , , and, by part I, q ≡ ˜ q on [ a , b ] . Let m ∈ , N + M − a m +1 , b m +1 ] ⊂ T and D mj ( λ ) ≡ ˜ D mj ( λ ) , j = 0 , , q ( x ) = ˜ q ( x ) , x ∈ T m +1 , . (54)In the case a m +1 = b m +1 we obtain D m +1 j ( λ ) ≡ ˜ D m +1 j ( λ ) , j = 0 , , from formulae (50)and (51). In the case a m +1 < b m +1 the functions D m +10 ( λ ) and D m +11 ( λ ) are solutions ofnon-degenerate systems (29). Moreover, by (54) the functions ˜ D m +10 ( λ ) and ˜ D m +11 ( λ ) aresolutions of the same system, which yields D m +1 j ( λ ) ≡ ˜ D m +1 j ( λ ) , j = 0 , . By virtue of part I,we conclude that q ≡ ˜ q on [ a m +1 , b m +1 ] , and the theorem is proved by induction.Developing the ideas of the method of spectral mapping [19], one can obtain the algorithmfor the recovery of the potential. Here we restrict ourself the case of N = 0 (i.e. the caseof the difference Sturm–Liouville operator) since it is sufficient to have the computations ofTheorem 5 for the algorithm. Algorithm 1.
Let the functions D j ( λ ) = Θ j ( λ ) , j = 0 , , be given. To recover q ( a l ) ,l = 1 , M − , for m = 1 , M − D m +10 ( λ ) using (50) and the known functions D m ( λ ) , D m ( λ ) .
2) Find q ( a m ) from the relation (53).3) If m < M − , construct the function D m +11 ( λ ) using (51) and the found value q ( a m ) . Example 4.
Let us consider the time scale T and the characteristic functions Θ j ( λ ) , j =0 , , from Example 1 and apply Algorithm 1. First, we compute D ( λ ) = Θ ( λ ) − Θ ( λ ) = 2 − λ, Θ ( λ ) D ( λ ) = − λ + 2 + 1 λ − . By formula (53), 2 = 2 + q (0) and q (0) = 0 . With (51) we find D ( λ ) = (1 − λ ) D m ( λ ) + λ D m ( λ ) = 1 − λ. Further, D ( λ ) = D ( λ ) − D ( λ ) = 1 and D ( λ ) / D ( λ ) = 2 − λ. Applying (53), weconclude that 2 = 2 + q (1) and q (1) = 0 . Thus, the potential q is recovered. Acknowledgment.
This is a pre-print of an article published in Results in Mathematics.The final authenticated version is available online at: https://doi.org/10.1007/s00025-020-1171-z .This work was supported by Grant 19-71-00009 of the Russian Science Foundation.8