Iso-bispectral potentials for Sturm-Liouville-type operators with small delay
aa r X i v : . [ m a t h . SP ] F e b ISO-BISPECTRAL POTENTIALS FOR STURM–LIOUVILLE-TYPEOPERATORS WITH SMALL DELAYNebojˇsa Djuri´c and Sergey Buterin Abstract.
In recent years, there appeared a considerable interest in the inverse spectral theory forfunctional-differential operators with constant delay. In particular, it is well known that, for each fixed ν ∈ { , } , the spectra of two operators generated by one and the expression − y ′′ ( x ) + q ( x ) y ( x − a )and the boundary conditions y ( ν ) (0) = y ( j ) ( π ) = 0 , j = 0 , , uniquely determine the complex-valuedsquare-integrable potential q ( x ) vanishing on (0 , a ) as soon as a ∈ [ π/ , π ) . For a < π/ , the mainequation of the corresponding inverse problem is nonlinear, and it actually became the basic question of the inverse spectral theory for Sturm–Liouville operators with constant delay whether the uniquenessholds also in this nonlinear case. A few years ago, a positive answer was obtained for a ∈ [2 π/ , π/ . Recently, the authors gave, however, a negative answer for a ∈ [ π/ , π/
5) by constructing infinite familiesof iso-bispectral potentials. Meanwhile, the question remained open for the most difficult nonlinear case a ∈ (0 , π/ , allowing the parameter a to approach the classical situation a = 0 , in which the uniquenessis well known. In the present paper, we address this gap and give a negative answer in this remarkablecase by constructing appropriate iso-bispectral potentials. Key words:
Sturm–Liouville operator with small delay, functional-differential operator, inverse spectralproblem, iso-bispectral potentials
1. Introduction
Inverse problems of spectral analysis consist in recovering operators from their spectral characteristics.One of the first results in the inverse spectral theory says that the spectra of two boundary value problemsfor one and the same Sturm–Liouville equation with one common boundary condition: − y ′′ ( x ) + q ( x ) y ( x ) = λy ( x ) , y (0) = y ( j ) ( π ) = 0 , j = 0 , , (1)uniquely determine the potential q ( x ) , see [1], where also local solvability and, actually, stability ofthis inverse problem were established in the class of real-valued q ( x ) ∈ L (0 , π ) . Later on, these resultswere refined and generalized to other classes of potentials and boundary conditions [2–9]. Moreover, thereappeared methods, which gave global solution for the inverse Sturm–Liouville problem as well as for inverseproblems for other classes of differential operators (see, e.g., monographs [7–10]).In recent years, there appeared a considerable interest in inverse problems also for nonlocal Sturm–Liouville-type operators with deviating argument (see [11–31] and references therein), which are oftenmore adequate for modelling various real-world processes frequently possessing a nonlocal nature. Aspecial place in this direction is occupied by the inverse problems for operators with constant delay.Fix ν ∈ { , } . For j = 0 , , consider the boundary value problem L ν,j ( a, q ) of the form − y ′′ ( x ) + q ( x ) y ( x − a ) = λy ( x ) , < x < π, (2) y ( ν ) (0) = y ( j ) ( π ) = 0 (3)with delay a ∈ (0 , π ) and a complex-valued potential q ( x ) ∈ L (0 , π ) such that q ( x ) = 0 on (0 , a ) . Denote by { λ n,j } n ≥ the spectrum of L ν,j ( a, q ) and consider the following inverse problem. Inverse Problem 1.
Given the spectra { λ n, } n ≥ and { λ n, } n ≥ , find the potential q ( x ) . Alternatively, one can consider the cases of Robin boundary conditions y ( ν ) (0) − νhy (0) = y ′ ( π ) + H j y ( π ) = 0 , j = 0 , , h, H , H ∈ C , H = H , (4)which, however, can be easily reduced to conditions (3), while all involved coefficients h and H alongwith H are uniquely determined by the two spectra (see [25]). Faculty of Architecture, Civil Engineering and Geodesy, University of Banja Luka, [email protected] Department of Mathematics, Saratov State University, [email protected] q ( x ) as soon as a ≥ π/ , when the inverse problem is even overdetermined (see [16]). For a < π/ , the dependence ofthe characteristic function of any problem consisting of (2) and (3) on the potential is nonlinear, whichfinally results in a nonlinear equation for recovering q ( x ) , which is referred as the main equation of theinverse problem. Specifically, for a ∈ [ π/ ( N + 1) , π/N ) with some N ≥ , the main equation will containnonlinear integral terms of the form: C k α ,k Z α ,k . . . α k,k Z α k,k q ( τ ) . . . q ( τ k ) dτ . . . dτ k , k = 1 , N , C N = 0 , (5)with certain limits of integration α lp,k = α lp,k ( x, τ , . . . , τ p − ) , where 1 ≤ p ≤ k ≤ N and l = 0 , . Actually, it became a basic question of the inverse spectral theory for functional-differential operatorswith constant delay whether the uniqueness holds also in the nonlinear case a ∈ (0 , π/ . A few yearsago, a positive answer in the case a ∈ [2 π/ , π/
2) was given in [19] for ν = 0 and independently in [20]for ν = 1 . Moreover, the overdetermination remains. However, recent authors’ papers [30] and [31] gavea negative answer as soon as a ∈ [ π/ , π/
5) for the cases ν = 0 and ν = 1 , respectively. Specifically,for each such a and ν, we constructed infinite families of different iso-bispectral potentials q ( x ) , i.e. forwhich both problems consisting of (2) and (3) possess one and the same pair of spectra. This appeared quiteunexpected, in particular, because of the inconsistence with Borg’s classical uniqueness result for a = 0 . The general strategy of constructing such iso-bispectral potentials involved establishing a special Fred-holm linear integral operator M h generated by the restriction h ( x ) of q ( x ) to some proper subinterval.Then an appropriate eigenfunction of M h was used as a part of the iso-bispectral potentials on some otherproper subinterval, while the variety of the potentials was achieved by multiplying the eigenfunction witharbitrary complex constants. Meanwhile, the case of a Neumann boundary condition at zero, i.e. when ν = 1 , appeared to be more difficult than the case of a Dirichlet one ( ν = 0) since the former requiredfinding such an operator M h that would possess an eigenfunction with the zero mean value. Even thoughthe existence of such an operator was not in doubt, finding its concrete example appeared to be a quitedifficult task. After a series of computational experiments we constructed several numerical examples, oneof which fortunately admitted a precise elementary implementation. The numerical simulation allowed usto construct also elementary iso-bispectral W -potentials for a ∈ ( π/ , π/
5) and ν = 0 (see [31]).However, the works [30] and [31] left unclear whether it is also impossible to uniquely recover the po-tential by the two spectra in the most difficult nonlinear case a ∈ (0 , π/ , when the main equation wouldcontain nonlinear terms of the form (5) with the nonlinearity parameter N > , becoming unboundedlylarge for arbitrarily small a. At the same time, the case of small a is especially interesting and importantsince it approximates the classical case a = 0 , in which the uniqueness is well known.In the present paper, we address this gap by giving a negative answer also in the remarkable case a ∈ (0 , π/
3) for both types of boundary conditions at zero, i.e. for ν = 0 and for ν = 1 . The paper is organized as follows. In the next section, we study sine- and cosine-type solutions ofequation (2) and find some appropriate representations for the characteristic functions of the problems L ν,j ( a, q ) for ν, j = 0 , . The main results of the paper are provided in Section 3.
2. Sine- and cosine-type solutions. The characteristic functions
For ν = 0 , , denote by y ν ( x, λ ) the unique solution of equation (2) under the initial conditions y ( j ) ν (0 , λ ) = δ ν,j , j = 0 , , where δ ν,j is the Kronecker delta. Here and below, f ( j ) as well as f ′ denotethe corresponding derivatives of f with respect to the first argument. Put ρ = λ and denote ω ( x ) = x Z a q ( t ) dt, ω ( x ) = x Z a q ( t ) ω ( t − a ) dt, y , ( x, λ ) = cos ρx, y , ( x, λ ) = sin ρxρ . The functions y ( x, λ ) and y ( x, λ ) are called cosine- and sine-type solutions of equation (2), which, inthe case of the zero potential, coincide with y , ( x, λ ) and y , ( x, λ ) , respectively.2or any pair of the parameters ν, j ∈ { , } , eigenvalues of the boundary value problem L ν,j ( a, q )with account of their multiplicities coincide with zeros of the entire function ∆ ν,j ( λ ) = y ( j )1 − ν ( π, λ ) , whichis called characteristic function of the problem L ν,j ( a, q ) . For each ν = 0 , , Lagrange’s method of variation of parameters leads to the integral equation y ν ( x, λ ) = y ν, ( x, λ ) + x Z a sin ρ ( x − t ) ρ q ( t ) y ν ( t − a, λ ) dt. Solving this equation by the method of successive approximations, we arrive at the series y ν ( x, λ ) = ∞ X k =0 y ν,k ( x, λ ) , y ν,k ( x, λ ) = x Z a sin ρ ( x − t ) ρ q ( t ) y ν,k − ( t − a, λ ) dt, k ≥ . (6)Clearly, y ν, ( x, λ ) = 0 for x ∈ [0 , a ] . Moreover, the induction gives y ν,k ( x, λ ) = 0 for x ∈ [0 , ka ] ∩ [0 , π ] . In particular, we have y ν,k ( x, λ ) ≡ k > N, where N ∈ N is such that a ∈ [ π/ ( N + 1) , π/N ) . Hence, formulae (6) take the form y ν ( x, λ ) = N X k =0 y ν,k ( x, λ ) , y ν,k ( x, λ ) = x Z ka sin ρ ( x − t ) ρ q ( t ) y ν,k − ( t − a, λ ) dt, k ≥ . (7)The following lemma gives explicit formulae for the terms y ν,k ( x, λ ) for ν = 0 , k = 1 , . Lemma 1.
For ν = 0 , , the following representations hold: y ν, ( x, λ ) = ω ( x )2( − λ ) ν y − ν, ( x − a, λ ) + 12 λ ν x Z a q ( t ) y − ν, ( x − t + a, λ ) dt, a ≤ x ≤ π, (8) y ν, ( x, λ ) = 12 λ ν x − a Z a P ν ( x, t ) y − ν, ( x − t + a, λ ) dt, a ≤ x ≤ π, (9) where P ν ( x, t ) = x Z t + a q ( τ ) dτ t − a Z a q ( ξ ) dξ + ( − ν x − t + a Z a q ( τ ) dτ x Z t + τ − a q ( ξ ) dξ, a ≤ t ≤ x − a ≤ π − a . (10) Proof.
Fix ν ∈ { , } . Applying the obvious unified trigonometric relationsin ρ ( x − t ) ρ y ν, ( ξ, λ ) = 12 λ ν (cid:16) ( − ν y − ν, ( x − t + ξ, λ ) + y − ν, ( x − t − ξ, λ ) (cid:17) , (11)with ξ = t − a to the second formula in (7) for k = 1 , we arrive at representation (8).Further, substituting (8) into the second formula in (7) for k = 2 , we obtain the relation y ν, ( x, λ ) = 12 λ ν x Z a sin ρ ( x − t ) ρ q ( t ) (cid:16) ( − ν ω ( t − a ) y − ν, ( t − a, λ ) + t − a Z a q ( τ ) y − ν, ( t − τ, λ ) dτ (cid:17) dt. Using (11) for ξ = t − a and for ξ = t − τ, we get y ν, ( x, λ ) = − ω ( x )4 λ y ν, ( x − a, λ ) + ( − ν λ x − a Z a q ( t + a ) y ν, ( x − t, λ ) dt t Z a q ( τ ) dτ ( − ν λ x Z a q ( t ) dt t − a Z a q ( τ ) y ν, ( x − τ, λ ) dτ + 14 λ x Z a q ( t ) dt t − a Z a q ( t − τ ) y ν, ( x − τ, λ ) dτ. After changing the order of integrations in the last two terms, we obtain the representation y ν, ( x, λ ) = − ω ( x )4 λ y ν, ( x − a, λ ) + ( − ν λ x − a Z a R ν ( x, t ) y ν, ( x − t, λ ) dt, (12)where R ν ( x, t ) = q ( t + a ) t Z a q ( τ ) dτ − q ( t ) x Z t + a q ( τ ) dτ + ( − ν x Z t + a q ( τ ) q ( τ − t ) dτ. Integrating with respect to the second argument, we get x − a Z t R ν ( x, τ ) dτ = x Z t + a q ( τ ) dτ τ − a Z a q ( ξ ) dξ − x − a Z t q ( τ ) dτ x Z τ + a q ( ξ ) dξ + ( − ν x − a Z t dτ x − τ Z a q ( ξ + τ ) q ( ξ ) dξ. Changing the order of integrations in the last two terms, we arrive at x − a Z t R ν ( x, τ ) dτ = x Z t + a q ( τ ) dτ t Z a q ( ξ ) dξ + ( − ν x − t Z a q ( τ ) dτ x Z t + τ q ( ξ ) dξ = P ν (cid:16) x, t + a (cid:17) , where the function P ν ( x, t ) is determined by (10). In particular, this yields P ν (cid:16) x, a (cid:17) = ( − ν x − a Z a q ( t ) dt x Z a + t q ( τ ) dτ = ( − ν x Z a q ( t ) dt t − a Z a q ( τ ) dτ = ( − ν ω ( x ) . Hence, the integration by parts in (12) along with the relations y ′ ν, ( x, λ ) = ( − λ ) − ν y − ν, ( x, λ ) , ν = 0 , , (13)gives (9). (cid:3) Corollary 1.
The following representations hold: y ′ ν, ( x, λ ) = ω ( x )2 y ν, ( x − a, λ ) + ( − ν x Z a q ( t ) y ν, ( x − t + a, λ ) dt, a ≤ x ≤ π, (14) y ′ ν, ( x, λ ) = ( − ν x − a Z a P ν ( x, t ) y ν, ( x − t + a, λ ) dt, a ≤ x ≤ π. (15) Proof.
By virtue of (13) along with the relation w ′ ( x ) = q ( x ) , formula (14) can be easily obtained bydifferentiating (8). Analogously, differentiating (9), we get y ′ ν, ( x, λ ) = A ν ( x, λ ) + B ν ( x, λ ) + ( − ν x − a Z a P ν ( x, t ) y ν, ( x − t + a, λ ) dt, a ≤ x ≤ π, (16)where A ν ( x, λ ) = ( − ν P ν (cid:16) x, x − a (cid:17) y ν, (2 a − x, λ ) , B ν ( x, λ ) = 12 λ ν x − a Z a ∂∂x P ν ( x, t ) y − ν, ( x − t + a, λ ) dt. P ν ( x, x − a/
2) = 0 , we have A ν ( x, λ ) ≡ ν = 0 , . Further, we obtain ∂∂x P ν ( x, t ) = q ( x ) t − a Z a q ( τ ) dτ + ( − ν x − t + a Z a q ( τ ) dτ . Thus, making the change of the integration variable t → ( x + a − t ) / , we arrive at B ν ( x, λ ) = q ( x )4 λ ν x − a Z a − x G ν ( x, t, λ ) dt, G ν ( x, t, λ ) = x − t Z a q ( τ ) dτ + ( − ν x + t Z a q ( τ ) dτ y − ν, ( t, λ ) . It remains to note that for each fixed x ∈ (2 a, π ] and λ ∈ C , both functions G ν ( x, t, λ ) , ν = 0 , , areodd with respect to t ∈ (2 a − x, x − a ) . Hence, B ν ( x, λ ) ≡ ν = 0 , , and we arrive at (15). (cid:3) In our target case a ∈ (0 , π/ , besides y ν, ( x, λ ) and y ν, ( x, λ ) , the series in (7), generally speaking,possesses also y ν,k ( x, λ ) for k = 3 , N , which causes the appearance of nonlinear integral terms having theform (5) for k > q ( x ) = 0 a.e. on (3 a, π ) , we arrive at the representations y ν ( x, λ ) = y ν, ( x, λ ) + y ν, ( x, λ ) + y ν, ( x, λ ) , ν = 0 , , (17)which allow us to derive convenient representations for the characteristic functions. Lemma 2.
Let q ( x ) = 0 a.e. on (3 a, π ) . Then, for ν, j = 0 , , the following representations hold ∆ ν,ν ( λ ) = ( − λ ) ν (cid:16) sin ρπρ − ω cos ρ ( π − a )2 λ + ( − ν λ a Z a w ν ( x ) cos ρ ( π − x + a ) dx (cid:17) , ∆ ν,j ( λ ) = cos ρπ + ω sin ρ ( π − a )2 ρ + ( − j ρ a Z a w ν ( x ) sin ρ ( π − x + a ) dx, ν = j, (18) where ω = ω ( π ) and the functions w ν ( x ) are determined by the formula w ν ( x ) = q ( x ) , x ∈ (cid:16) a, a (cid:17) ∪ (cid:16) a , a (cid:17) ,q ( x ) + Q ν ( x ) , x ∈ (cid:16) a , a (cid:17) , (19) while Q ν ( x ) = a Z x + a q ( t ) dt x − a Z a q ( τ ) dτ − ( − ν a − x Z a q ( t ) dt a Z x + t − a q ( τ ) dτ, x ∈ (cid:16) a , a (cid:17) . (20) Proof.
Under the hypothesis of the lemma, formula (10) gives P ν ( x, t ) = 0 for t ≥ a/ ν = 0 , . Thus, substituting x = π into (17) and taking (8)–(10) and (14), (15) into account, we arrive at formulae(18) and (19) with the functions Q ν ( x ) , ν = 0 , , determined by the formula Q ν ( x ) = P − ν ( π, x ) = π Z x + a q ( t ) dt x − a Z a q ( τ ) dτ − ( − ν π − x + a Z a q ( t ) dt π Z x + t − a q ( τ ) dτ, where, besides the obvious possibility of replacing both upper limits of integration π with 3 a, it re-mains to note that the last internal integral vanishes as soon as x + t − a/ ≥ a. Thus, the upper limitof integration in the last external integral π − x + a/ a/ − x, which gives (20). (cid:3) . The main results In this section, we establish non-uniqueness of the solution of Inverse Problem 1 for both cases ν = 0and ν = 1 as soon as a ∈ (0 , π/ . Specifically, for each such ν we construct an infinite family ofiso-bispectral potentials q ( x ) , i.e. for which both problems L ν, ( a, q ) and L ν, ( a, q ) possess one andthe same pair of spectra. According to the previous section, the spectrum of any problem L ν,j ( a, q )does not depend on q ( x ) ∈ B for some subset B ⊂ L (0 , π ) as soon as neither does the correspondingcharacteristic function ∆ ν,j ( λ ) . It is sufficient to restrict oneself with potentials vanishing on the interval(3 a, π ) , which allows one to use representations of the characteristic functions obtained in Lemma 2. Remark 1.
The difference between the cases ν = 0 and ν = 1 consists in the following. Since thefunctions ∆ ν,j ( λ ) are entire in λ, the first representation in (18) for ν = 0 implies ω = π Z a w ( x ) dx, (21)which can also be checked directly using (19) and (20) for ν = 0 . Thus, for ν = 0 , the iso-bispectralityof any subset B ⊂ L (0 , π ) requires only w ( x ) ’s independence of q ( x ) ∈ B. However, for ν = 1 , thereis no relation analogous to (21). In other words, the constant ω is not determined by w ( x ) . Thus, bothfunctions ∆ , ( λ ) and ∆ , ( λ ) may depend on q ( x ) ∈ B even when w ( x ) does not.Fix a ∈ (0 , π/
3) and consider the integral operator M h f ( x ) = a − x Z a K h (cid:16) x + t − a (cid:17) f ( t ) dt, a < x < a, where K h ( x ) = a Z x h ( τ ) dτ, (22)with a nonzero real-valued function h ( x ) ∈ L (5 a/ , a ) . Thus, M h is a nonzero compact Hermitianoperator in L (3 a/ , a ) and, hence, it has at least one nonzero eigenvalue η. Fix ν ∈ { , } and put h ν ( x ) := ( − ν h ( x ) /η. Then ( − ν is an eigenvalue of M h ν . Let e ν ( x ) be a related eigenfunction, i.e. M h ν e ν ( x ) = ( − ν e ν ( x ) , a < x < a. (23)Consider the one-parametric family of potentials B ν := { q α,ν ( x ) } α ∈ C determined by the formula q α,ν ( x ) = , x ∈ (cid:16) , a (cid:17) ,αe ν ( x ) , x ∈ (cid:16) a , a (cid:17) , − αK h ν (cid:16) x + a (cid:17) x − a Z a e ν ( t ) dt, x ∈ (cid:16) a, a (cid:17) ,h ν ( x ) , x ∈ (cid:16) a , a (cid:17) . (24) Lemma 3.
For each fixed ν ∈ { , } , the function w ν ( x ) constructed by the formulae (19) and (20)with q ( x ) = q α,ν ( x ) determined by (24) is independent of α. Proof.
Fix ν ∈ { , } and let q ( x ) = 0 on ( a, a/ . Then (20) takes the form Q ν ( x ) = − ( − ν a − x Z a q ( t ) dt a Z x + t − a q ( τ ) dτ, x ∈ (cid:16) a , a (cid:17) , a Z x + a q ( t ) dt x − a Z a q ( τ ) dτ, x ∈ (cid:16) a, a (cid:17) , w ν ( x ) = , x ∈ (cid:16) a, a (cid:17) ,q ( x ) − ( − ν M q q ( x ) , x ∈ (cid:16) a , a (cid:17) ,q ( x ) + K q (cid:16) x + a (cid:17) x − a Z a q ( t ) dt, x ∈ (cid:16) a, a (cid:17) ,q ( x ) , x ∈ (cid:16) a , a (cid:17) . (25)Substituting q ( x ) = q α,ν ( x ) determined by (24) into (25) and taking (23) into account, we arrive at w ν ( x ) = , x ∈ (cid:16) a, a (cid:17) ,h ν ( x ) , x ∈ (cid:16) a , π (cid:17) , which finishes the proof. (cid:3) Lemma 3 along with (18) for ν = 0 and relation (21) gives the following theorem. Theorem 1.
For j = 0 , , the spectrum of the problem L ,j ( a, q α, ) is independent of α. Thus, B is a set of iso-bispectral potentials in the case ν = 0 , i.e. Inverse Problem 1 is not uniquelysolvable for ν = 0 and a ∈ (0 , π/ . However, according to Remark 1, the set B , generally speaking,does not consist of iso-bispectral potentials for ν = 1 . Nevertheless, the following theorem holds.
Theorem 2.
For j = 0 , , the spectrum of the problem L ,j ( a, q α, ) is independent of α as soon as a Z a e ( x ) dx = 0 . (26) Proof.
According to (18) for ν = 1 and Lemma 3, it is sufficient to prove that the value ω = π Z a q α, ( x ) dx is independent of α when (26) holds. Integrating the third line in (24) for ν = 1 , we get I := a Z a K h (cid:16) x + a (cid:17) dx x − a Z a e ( t ) dt = a Z a K h (cid:16) x + a (cid:17) dx x − a Z a e ( x + a − t ) dt. Changing the order of integration and then the internal integration variable, we calculate I = a Z a dx a Z x + a K h (cid:16) t + a (cid:17) e ( t + a − x ) dt = a Z a dx a − x Z a K h (cid:16) x + t − a (cid:17) e ( t ) dt, which along with the first equality in (22) as well as (23) for ν = 1 and (26) implies I = a Z a M h e ( x ) dx = − a Z a e ( x ) dx = 0 . ν = 1 , relation (26) gives ω = π Z a q α, ( x ) dx = a Z a h ( x ) dx, i.e. the value ω does not depend on α, which finishes the proof. (cid:3) This theorem reduces the construction of iso-bispectral potentials for the problems L , ( a, q ) and L , ( a, q ) to the question of finding a function h ( x ) ∈ L (5 a/ , a ) such that the operator M h has atleast one eigenfunction possessing the zero mean value but related to a nonzero eigenvalue. An answer tothis question is given by the following assertion. Proposition 1.
Put h ( x ) := 6 π a cos π √ (cid:16) − xa (cid:17) , e ( x ) := cos 4 πxa − cos 2 πxa . (27) Then relation (23) for ν = 1 as well as equality (26) are fulfilled.Proof. Let us start with (26), which can be checked by the direct substitution: a Z a e ( x ) dx = a π (cid:16)
12 sin 4 πxa − sin 2 πxa (cid:17)(cid:12)(cid:12)(cid:12) ax = a = 0 . Further, according to (22) and (27), we have M h e ( x ) = 3 πa √
10 ( A − A ) , (28)where A j = 2 a − x Z a cos 2 πjta · sin π √ (cid:16) − x + ta (cid:17) dt, j = 1 , . For each j ∈ { , } , we calculate: A j = X k =0 a − x Z a sin π (cid:16) √ (cid:16) − xa (cid:17) − √
10 + 2 j ( − k a t (cid:17) dt. Fulfilling the integration, we arrive at the relation A j = 1 π X k =0 a √
10 + 2 j ( − k cos π (cid:16) √ (cid:16) − xa (cid:17) − √
10 + 2 j ( − k a t (cid:17)(cid:12)(cid:12)(cid:12) a − xt = a = 1 π X k =0 a √
10 + 2 j ( − k (cid:16) cos 2 πj (cid:16) − xa (cid:17) − cos π (cid:16) √ (cid:16) − xa (cid:17) − j ( − k (cid:17)(cid:17) = a √ π (cid:16) cos π √ (cid:16) − xa (cid:17) − cos 2 πjxa (cid:17) , j = 1 , . Substituting the obtained representations for A and A into (28), we arrive at (23) for ν = 1 . (cid:3) Theorem 2 and Proposition 1 imply that the family B constructed by using the functions h ( x ) and e ( x ) determined in (27) consists of iso-bispectral potentials for the problems L , ( a, q ) and L , ( a, q ) . Thus, Inverse Problem 1 is not uniquely solvable also in the case ν = 1 as soon as a ∈ (0 , π/ . Acknowledgement.
The first author is supported by Project 19.032/961-103/19 of the Republic ofSrpska Ministry for Scientific and Technological Development, Higher Education and Information Society.The second one is supported by Grants 19-01-00102, 20-31-70005 of Russian Foundation for Basic Research.8 eferences [1] Borg G.
Eine Umkehrung der Sturm–Liouvilleschen Eigenwertaufgabe , Acta Math. 78 (1946) 1–96.[2] Karaseva T.M.
On the inverse Sturm–Liouville problem for a non-Hermitian operator , Mat. Sbornik32 (1953) no.74, 477–484.[3] Marchenko V.A. and Ostrovskii I.V.
A characterization of the spectrum of the Hill operator , Mat. Sb.97 (1975), 540–606; English transl. in Math. USSR-Sb. 26 (1975) no.4, 493–554.[4] Hryniv R.O. and Mykytyuk Ya.V.
Inverse spectral problems for Sturm–Liouville operators with sin-gular potentials, II. Reconstruction by two spectra , in: V. Kadets, W. Zelazko (Eds.), FunctionalAnalysis and Its Applications, in: North-Holland Math. Stud., vol. 197, North-Holland Publishing,Amsterdam (2004), 97–114.[5] Savchuk A.M. and Shkalikov A.A.
Inverse problems for Sturm–Liouville operators with potentials inSobolev spaces: Uniform stability , Funk. Anal. i ego Pril. 44 (2010) no.4, 34–53; English transl. inFunk. Anal. Appl. 44 (2010) no.4, 270–285.[6] Buterin S.A. and Kuznetsova M.A.
On Borg’s method for non-selfadjoint Sturm–Liouville operators ,Anal. Math. Phys. 9 (2019) 2133–2150.[7] Marchenko V.A.
Sturm–Liouville Operators and Their Applications , Naukova Dumka, Kiev, 1977;English transl.: Birkh¨auser, Basel, 1986.[8] Levitan B.M.
Inverse Sturm–Liouville Problems , Nauka, Moscow, 1984; English transl.: VNUSci.Press, Utrecht, 1987.[9] Freiling G. and Yurko V.A.
Inverse Sturm–Liouville Problems and Their Applications , NOVA SciencePublishers, New York, 2001.[10] Yurko V.A.
Method of Spectral Mappings in the Inverse Problem Theory , Inverse and Ill-posed Prob-lems Series. VSP, Utrecht, 2002.[11] Pikula M.
Determination of a Sturm–Liouville-type differential operator with delay argument fromtwo spectra , Mat. Vestnik 43 (1991) no.3-4, 159–171.[12] Albeverio S., Hryniv R.O. and Nizhnik L.P.
Inverse spectral problems for non-local Sturm–Liouvilleoperators , Inverse Problems 23 (2007) 523–535.[13] Freiling G. and Yurko V.A.
Inverse problems for Sturm–Liouville differential operators with a constantdelay , Appl. Math. Lett. 25 (2012) 1999–2004.[14] Yang C.-F.
Inverse nodal problems for the Sturm–Liouville operator with a constant delay , J. Differ-ential Equations 257 (2014) no.4, 1288–1306.[15] Vladiˇci´c V. and Pikula M.
An inverse problem for Sturm–Liouville-type differential equation with aconstant delay , Sarajevo J. Math. 12 (2016) no.1, 83–88.[16] Buterin S.A. and Yurko V.A.
An inverse spectral problem for Sturm–Liouville operators with a largeconstant delay , Anal. Math. Phys. 9 (2019) no.1, 17–27.[17] Buterin S.A., Pikula M. and Yurko V.A.
Sturm–Liouville differential operators with deviating argu-ment , Tamkang J. Math. 48 (2017) no.1, 61–71.[18] Ignatiev M.Yu.
On an inverse Regge problem for the Sturm–Liouville operator with deviating argu-ment , J. Samara State Tech. Univ., Ser. Phys. Math. Sci. 22 (2018) no.2, 203–211.[19] Bondarenko N. and Yurko V.
An inverse problem for Sturm–Liouville differential operators withdeviating argument , Appl. Math. Lett. 83 (2018) 140–144.920] Pikula M., Vladiˇci´c V. and Vojvodi´c B.
Inverse spectral problems for Sturm–Liouville operators with aconstant delay less than half the length of the interval and Robin boundary conditions , Results Math.(2019) 74:45.[21] Djuri´c N. and Vladiˇci´c V.
Incomplete inverse problem for Sturm–Liouville type differential equationwith constant delay , Results Math. (2019) 74:161.[22] Sat M. and Shieh C.-T.
Inverse nodal problems for integro-differential operators with a constant delay ,J. Inverse Ill-Posed Probl. 27 (2019) no.4, 501–509.[23] Wang Y.P., Shieh C.T. and Miao H.Y.
Reconstruction for Sturm–Liouville equations with a constantdelay with twin-dense nodal subsets , Inverse Probl. Sci. Eng. 27 (2019) no.5, 608–617.[24] Djuri´c N.
Inverse problems for Sturm–Liouville-type operators with delay: symmetric case , AppliedMathematical Sciences 14 (2020) no.11, 505–510.[25] Yurko V.A.
Solution of Inverse Problems for Differential Operators with Delay . In: Trends in Math-ematics: Transmutation Operators and Appl., Birkh¨auser, Basel, 2020. P.467–475.[26] Buterin S.A., Malyugina M.A. and Shieh C.-T.
An inverse spectral problem for second-orderfunctional-differential pencils with two delays , arXiv:2010.14238 [math.SP] (2020) 25pp.[27] Buterin S. and Kuznetsova M.
On the inverse problem for Sturm–Liouville-type operators with frozenargument: rational case , Comp. Appl. Math. (2020) 39:5, 15pp.[28] Buterin S. and Hu Y.-T.
Inverse spectral problems for Hill-type operators with frozen argument , Anal.Math. Phys. (in press, arXiv:2008.07147 [math.SP])[29] Wang Y.P., Zhang M., Zhao W. and Wei X.
Reconstruction for Sturm–Liouville operators with frozenargument for irrational cases , Appl. Math. Lett. 111 (2021) 106590.[30] Djuri´c N. and Buterin S.
On an open question in recovering Sturm–Liouville-type operators with delay ,Appl. Math. Lett. 113 (2021) 106862.[31] Djuri´c N. and Buterin S.