On non-uniqueness of recovering Sturm-Liouville operators with delay and the Neumann boundary condition at zero
aa r X i v : . [ m a t h . SP ] J a n ON NON-UNIQUENESS OF RECOVERING STURM–LIOUVILLE OPERATORSWITH DELAY AND THE NEUMANN BOUNDARY CONDITION AT ZERONebojˇsa Djuri´c and Sergey Buterin Abstract.
As is known, for each fixed ν ∈ { , } , the spectra of two operators generated by − y ′′ ( x )+ q ( x ) y ( x − a ) and the boundary conditions y ( ν ) (0) = y ( j ) ( π ) = 0 , j = 0 , , uniquely determine thecomplex-valued square-integrable potential q ( x ) vanishing on (0 , a ) as soon as a ∈ [2 π/ , π ) . Meanwhile,it actually became the main question of the inverse spectral theory for Sturm–Liouville operators withconstant delay whether the uniqueness holds also for smaller values of a. Recently, a negative answerwas given by the authors [Appl. Math. Lett. 113 (2021) 106862] for a ∈ [ π/ , π/
5) in the case ν = 0by constructing an infinite family of iso-bispectral potentials. Moreover, an essential and dramatic reasonwas established why this strategy, generally speaking, fails in the remarkable case when ν = 1 . Here weconstruct a counterexample giving a negative answer for ν = 1 , which is an important subcase of theRobin boundary condition at zero. We also refine the former counterexample for ν = 0 to W -potentials.Key words: Sturm–Liouville operator with delay, inverse spectral problem, iso-bispectral potentials2010 Mathematics Subject Classification: 34A55 34K29
1. Introduction
In recent years, there appeared an interest in inverse spectral problems for Sturm–Liouville-type op-erators with deviating argument, see, e.g., papers [1–19], among which a big part is devoted to operatorswith delay. For j = 0 , , denote by { λ n,j } n ≥ the spectrum of the boundary value problem − y ′′ ( x ) + q ( x ) y ( x − a ) = λy ( x ) , < x < π, (1) U ( y ) = y ( j ) ( π ) = 0 (2)with delay a ∈ (0 , π ) and a complex-valued potential q ( x ) ∈ L (0 , π ) such that q ( x ) = 0 on (0 , a ) , while U ( y ) = y (0) or U ( y ) = y ′ (0) − hy (0) , h ∈ C . Such cases of U ( y ) correspond to Dirichlet andRobin boundary conditions at zero, to which we will refer as Case 1 and case
Case 2 , respectively.
Inverse Problem 1.
Given the spectra { λ n, } n ≥ and { λ n, } n ≥ , find the potential q ( x ) . Alternatively, one can consider the case of Robin boundary conditions also at the right end: U ( y ) = y ′ ( π ) + H j y ( π ) = 0 , j = 0 , , H , H ∈ C , H = H , (3)which, however, can be easily reduced to conditions (2). Moreover, in the cases of Robin boundaryconditions, the coefficients h and H , H are uniquely determined by the two spectra (see [15]).Various aspects of Inverse Problem 1 were studied in [1, 2, 4–6, 8–11, 14–16, 19] and other works. Inparticular, it is well known that the two spectra uniquely determine the potential as soon as a ≥ π/ . Moreover, the inverse problem is overdetermined (see [5]). For a < π/ , the dependence of the character-istic function of any problem of the form (1) and (2) on the potential is nonlinear. It became actually the main question of the inverse spectral theory for the operators with constant delay whether the uniquenessholds also for small a. Recently, a positive answer when a ∈ [2 π/ , π/
2) was given in [8] for Case 1 andindependently in [9] for Case 2. However, recent authors’ paper [19] gave a negative answer in Case 1as soon as a ∈ [ π/ , π/ . Specifically, for each such a, we constructed an infinite family of differentiso-bispectral potentials q ( x ) , i.e. for which both problems consisting of (1) and (2) possess one and thesame pair of spectra. This appeared quite unexpected because of the inconsistence with Borg’s classicaluniqueness result for a = 0 [20], and also in light of recent paper [15] announcing the uniqueness for a ∈ [ π/ , π ) in Case 2 for boundary conditions (3). In [19], we also established an essential and dramaticreason why the idea of that counterexample, generally speaking, fails in Case 2 (see Remark 2 in [19]).In the present paper, we return to Case 2 and construct a counterexample giving a negative answer forthe Neumann boundary condition at zero (i.e. when h = 0) , which, unfortunately, refutes the uniqueness Faculty of Architecture, Civil Engineering and Geodesy, University of Banja Luka, [email protected] Department of Mathematics, Saratov State University, [email protected] a ∈ [ π/ , π/ . For this purpose, we establish Theorem 1 (see the next section)first, which reduces finding a counterexample in Case 2 to constructing a Hermitian integral operator of aspecial form possessing an eigenfunction with the mean value zero. Even though the existence of such anoperator was highly believable, finding its concrete example appeared to be a quite difficult task. After aseries of computational experiments we constructed several numerical examples, one of which fortunatelyadmitted a precise elementary implementation (see Proposition 1).This new non-uniqueness result along with the one in [19] changes the further strategy of studyinginverse problems for the operators with delay (see also Remark 1 in [19]). In particular, there appearsthe relevance of finding various conditions on the class of potentials that would guarantee the uniquenessof recovering q ( x ) . That is especially important for justifying constructive procedures for solving InverseProblem 1 when a < π/ , otherwise the corresponding algorithms in [11] and [15] become indefinite. Byvirtue of (15), as such a condition one can impose holomorphy of q ( x ) on an appropriate part of ( a, π ) . Finally, we note that both our counterexamples involve discontinuous potentials. So it is also relevantto investigate the possibility of constructing iso-bispectral potentials in W k [0 , π ] with k ∈ N so large aspossible. Here we construct such potentials in Case 1 for k = 1 and a ∈ ( π/ , π/
5) (see Theorem 2).
2. The main results
For ν, j ∈ { , } , denote by L ν,j ( a, q ) the eigenvalue problem for equation (1) under the boundaryconditions y ( ν ) (0) = y ( j ) ( π ) = 0 . (4)Fix a ∈ [ π/ , π/ . Following the main idea of the work [19], we consider the integral operator M h f ( x ) = Z π − x + a a K h (cid:16) x + t − a (cid:17) f ( t ) dt, a < x < π − a, where K h ( x ) = Z πx h ( τ ) dτ, (5)with a nonzero real-valued function h ( x ) ∈ L (5 a/ , π ) . Thus, M h is a nonzero compact Hermitianoperator in L (3 a/ , π − a ) and, hence, it has at least one nonzero eigenvalue η. Further, fix ν ∈ { , } and put h ν ( x ) := ( − ν h ( x ) /η. Then ( − ν is an eigenvalue of the operator M h ν . Let e ν ( x ) be some related eigenfunction, i.e. M h ν e ν ( x ) = ( − ν e ν ( x ) , a < x < π − a. (6)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) ∪ ( π − a, a ) ∪ (cid:16) π − a , a (cid:17) ,αe ν ( x ) , x ∈ (cid:16) a , π − a (cid:17) , − αK h ν (cid:16) x + a (cid:17) Z x − a a e ν ( t ) dt, x ∈ (cid:16) a, π − a (cid:17) ,h ν ( x ) , x ∈ (cid:16) a , π (cid:17) . (7)In [19], it was established that, for j = 0 , , the spectrum of the problem L ,j ( a, q α, ) is independentof α for any function h ( x ) conditioned above. This means that B is an iso-bispectral set (of potentials)for these two problems, i.e. the solution of Inverse Problem 1 in Case 1 is not unique.Moreover, in [19], it was noted that acting in the analogous way but for the problems L ,j ( a, q ) ,j = 0 , , would lead to the family of potentials B . However, an essential reason was established why B , generally speaking, does not form an iso-bispectral set for these two problems (see also Remark 1 inSection 3). In this paper, we find a concrete example when it is. We begin with the following theorem. Theorem 1.
For j = 0 , , the spectrum of the problem L ,j ( a, q α, ) is independent of α as soon as Z π − a a e ( x ) dx = 0 . (8)2hus, the problem of constructing iso-bispectral potentials for the problems L , ( a, q ) and L , ( a, q )is reduced to the question of finding a function h ( x ) ∈ L (5 a/ , π ) such that the operator M h has atleast one eigenfunction possessing the zero mean value but related to a nonzero eigenvalue. The answerto this question is given by the following assertion. Proposition 1.
Put h ( x ) := 6 π (2 π − a ) cos π √ π − x )2 π − a , e ( x ) := cos 2 π (2 x − a )2 π − a + cos π (2 x − a )2 π − a . (9) Then relation (6) for ν = 1 as well as equality (8) are fulfilled. Theorem 1 and Proposition 1 imply that the family B constructed by using the functions h ( x )and e ( x ) that are determined by (9) 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 Case 2 .Finally, we construct iso-bispectral potentials in W [0 , π ] in Case 1. For this purpose, we consider e ( x ) := sin 2 π (2 x − a )2 π − a + 2 sin π (2 x − a )2 π − a (10)and introduce the family of potentials ˜ B := { q α ( x ) } α ∈ C determined by the formula q α ( x ) = , x ∈ h , a (cid:17) ∪ [ π − a, a ) ,αe ( x ) , x ∈ h a , π − a (cid:17) , − αK h (cid:16) x + a (cid:17) Z x − a a e ( t ) dt, x ∈ h a, π − a (cid:17) ,g ( x ) , x ∈ h π − a , a (cid:17) ,h ( x ) , x ∈ h a , π i , (11)with h ( x ) = h ( x ) , where h ( x ) is, in turn, determined in (9), while g ( x ) is an arbitrary fixed functionin W [ π − a/ , a/
2] obeying the boundary conditions g (cid:16) π − a (cid:17) = 0 , g (cid:16) a (cid:17) = 6 π (2 π − a ) cos π √ . (12) Theorem 2.
For j = 0 , , the spectrum of the boundary value problem L ,j ( a, q α ) is independentof α. Moreover, the inclusion ˜ B ⊂ W [0 , π ] holds as soon as a ∈ ( π/ , π/ . The proofs of Theorems 1 and 2 as well as Propositions 1 are provided in the next section.
3. The proofs
Denote by y ( x, λ ) and y ( x, λ ) the unique solutions of equation (1) under the initial conditions y ( j ) ν (0 , λ ) = δ ν,j , ν, j = 0 , , where δ ν,j is the Kronecker delta. For any pair of ν, j ∈ { , } , eigenvaluesof the boundary value problem L ν,j ( a, q ) with account of multiplicity coincide with zeros of the entirefunction ∆ ν,j ( λ ) = y ( j )1 − ν ( π, λ ) , which is called characteristic function of the problem L ν,j ( a, q ) . Thus,the spectrum of any problem L ν,j ( a, q ) does not depend on q ( x ) ∈ B for some subset B ⊂ L (0 , π ) assoon as neither does the corresponding characteristic function ∆ ν,j ( λ ) . Put ρ = λ and denote ω := Z πa q ( x ) dx. (13)Before proving Theorem 1, we provide necessary information from [19]. For ν, j = 0 , , we have∆ ν,ν ( λ ) = ( − λ ) ν (cid:16) sin ρπρ − ω cos ρ ( π − a )2 λ + ( − ν λ Z πa 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 Z πa w ν ( x ) sin ρ ( π − x + a ) dx, ν = j, (14)3here the functions w ν ( x ) are determined by the formula w ν ( x ) = q ( x ) , x ∈ (cid:16) a, a (cid:17) ∪ (cid:16) π − a , π (cid:17) ,q ( x ) + Q ν ( x ) , x ∈ (cid:16) a , π − a (cid:17) , (15)while Q ν ( x ) = Z x − a a q ( t ) dt Z πx + a q ( τ ) dτ − ( − ν Z π − x + a a q ( t ) dt Z πx + t − a q ( τ ) dτ. (16) Remark 1.
As was established [19], the difference between the cases ν = 0 and ν = 1 is as folows.Since the functions ∆ ν,j ( λ ) are entire in λ, the first representation in (14) for ν = 0 implies ω = Z πa w ( x ) dx, (17)which can also be checked directly using (15) and (16) for ν = 0 . Thus, for ν = 0 , the iso-bispectralityof B requires only w ( x ) ’s independence of q ( x ) ∈ B. However, for ν = 1 , there is no relationanalogous to (17). In other words, the constant ω is not determined by w ( x ) . Thus, both functions∆ , ( λ ) and ∆ , ( λ ) may depend on q ( x ) ∈ B even when w ( x ) does not.Let q ( x ) = 0 on ( a, a/ . Hence, formulae (15) and (16) give w ν ( x ) = , x ∈ (cid:16) a, a (cid:17) ,q ( x ) − ( − ν M h q ( x ) , x ∈ (cid:16) a , π − a (cid:17) ,q ( x ) , x ∈ ( π − a, a ) ,q ( x ) + K h (cid:16) x + a (cid:17) Z x − a a q ( t ) dt, x ∈ (cid:16) a, π − a (cid:17) ,q ( x ) , x ∈ (cid:16) π − a , a (cid:17) ,h ( x ) , x ∈ (cid:16) a , π (cid:17) , (18)where h ( x ) = q ( x ) on (5 a/ , π ) , while M h and K h ( x ) are determined by (5). Proof of Theorem 1.
Substituting q ( x ) = q α, ( x ) into (18) for ν = 1 , where q α, ( x ) is determinedby (7) with ν = 1 , and taking (6) for ν = 1 into account, we arrive at w ( x ) = 0 , a < x < a , w ( x ) = h ( x ) , a < x < π. Thus, the function w ( x ) is independent of α. Hence, it remains to prove that so is also the value ω determined by formula (13) with q ( x ) = q α, ( x ) . Integrating the third line in (7) for ν = 1 , we get I := Z π − a a K h (cid:16) x + a (cid:17) dx Z x − a a e ( t ) dt = Z π − a a K h (cid:16) x + a (cid:17) dx Z x − a a e ( x + a − t ) dt. Changing the order of integration and then the internal integration variable, we calculate I = Z π − a a dx Z π − a x + a K h (cid:16) t + a (cid:17) e ( t + a − x ) dt = Z π − a a dx Z π − x + a a K h (cid:16) x + t − a (cid:17) e ( t ) dt, which along with the first equality in (5) as well as (6) for ν = 1 and (8) implies I = Z π − a a M h e ( x ) dx = − Z π − a a e ( x ) dx = 0 . ν = 1 , the assumption (8) of the theorem gives ω = Z πa q α, ( x ) dx = Z π a h ( x ) dx, i.e. the value ω does not depend on α, which finishes the proof. (cid:3) For shortening the remaining proofs, we provide the following auxiliary assertion.
Proposition 2.
For each fixed ν ∈ { , } , relation (6) is equivalent to the relation m χ ν ǫ ν ( ξ ) = ( − ν ǫ ν ( ξ ) , < ξ < , m χ f ( ξ ) := Z − ξ f ( η ) dη Z ξ + η χ ( θ ) dθ, (19) as soon as ǫ ν ( ξ ) = e ν (cid:16) a (cid:16) π − a (cid:17) ξ (cid:17) , χ ν ( θ ) = (cid:16) π − a (cid:17) h ν (cid:16) a (cid:16) π − a (cid:17) θ (cid:17) . (20) Proof.
Making in (6) the change of variable ξ := (2 x − a ) /A, where A = 2 π − a, we obtain M h ν e ν (cid:16) a (cid:16) π − a (cid:17) ξ (cid:17) = ( − ν ǫ ν ( ξ ) , < ξ < . Using (5) and successively making the changes η := (2 t − a ) /A and θ := (2 τ − a ) /A, we get( − ν ǫ ν ( ξ ) = (cid:16) π − a (cid:17) Z − ξ ǫ ν ( η ) dη Z π a +( π − a )( ξ + η ) h ν ( τ ) dτ = Z − ξ ǫ ν ( η ) dη Z ξ + η χ ν ( θ ) dθ, which coincides with (19). (cid:3) Proof of Proposition 1.
Let us start with (8), which can be checked by the direct substitution: Z π − a a e ( x ) dx = 2 π − a π (cid:16)
12 sin 2 π (2 x − a )2 π − a + sin π (2 x − a )2 π − a (cid:17)(cid:12)(cid:12)(cid:12) π − ax = a = 0 . According to Proposition 2, it remains to prove relation (19) for ν = 1 with the functions ǫ ( ξ ) = cos πξ + cos 2 πξ, χ ( θ ) = 3 π π √ − θ )2 , which are determined by (9) and (20) for ν = 1 . Indeed, it is easy to calculate m χ ǫ ( ξ ) = 3 π √
10 ( A + A ) , A j = 2 Z − ξ cos πjη · sin π √ − ξ − η )2 dη = X k =0 Z − ξ sin (cid:16) π √ − ξ )2 − π (cid:16) √
102 + ( − k j (cid:17) η (cid:17) dη = 1 π √ − j (cid:16) cos πj (1 − ξ ) − cos π √ − ξ )2 (cid:17) , where j = 1 , , which leads to (19) for ν = 1 . (cid:3) Proof of Theorem 2.
Using Proposition 2 as in the preceding proof, one can establish (6) for ν = 0with h ( x ) = h ( x ) determined in (9) and e ( x ) determined by (10). Further, substituting q ( x ) = q α ( x )into (18) for ν = 0 , where q α ( x ) is determined by (11), and taking (6) for ν = 0 into account, we get w ( x ) = 0 , a < x < π − a , w ( x ) = g ( x ) , π − a < x < a , w ( x ) = h ( x ) , a < x < π. Thus, the function w ( x ) is independent of α. Moreover, according to (14) and (17), for j = 0 , , thecharacteristic function ∆ ,j ( λ ) of the problem L ,j ( a, q α ) is independent of α. Finally, we note that, for any α ∈ C and a ∈ ( π/ , π/ , the inclusion q α ( x ) ∈ W [0 , π ] followsfrom (9)–(12) along with the last equality in (5). (cid:3) Acknowledgement.
The first author was supported by the Project 19.032/961-103/19 of the Repub-lic of Srpska Ministry for Scientific and Technological Development, Higher Education and InformationSociety. The second author was supported by Grants 19-01-00102 and 20-31-70005 of the Russian Foun-dation for Basic Research. 5 eferences [1] Pikula M.
Determination of a Sturm–Liouville-type differential operator with delay argument fromtwo spectra , Mat. Vestnik 43 (1991) no.3-4, 159–171.[2] Freiling G. and Yurko V.A.
Inverse problems for Sturm–Liouville differential operators with a constantdelay , Appl. Math. Lett. 25 (2012) 1999–2004.[3] 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.[4] 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.[5] Buterin S.A. and Yurko V.A.
An inverse spectral problem for Sturm–Liouville operators with a largedelay , Anal. Math. Phys. 9 (2019) no.1, 17–27.[6] 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.[7] 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.[8] Bondarenko N. and Yurko V.
An inverse problem for Sturm–Liouville differential operators withdeviating argument , Appl. Math. Lett. 83 (2018) 140–144.[9] 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.[10] Djuri´c N. and Vladiˇci´c V.
Incomplete inverse problem for Sturm–Liouville type differential equationwith constant delay , Results Math. (2019) 74:161.[11] Yurko V.A.
An inverse spectral problem for second order differential operators with retarded argument ,Results Math. (2019) 74:71.[12] 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.[13] 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.[14] Djuri´c N.
Inverse problems for Sturm–Liouville-type operators with delay: symmetric case , AppliedMathematical Sciences 14 (2020) no.11, 505–510.[15] 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.[16] 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.[17] 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.[18] 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.[19] Djuri´c N. and Buterin S.
On an open question in recovering Sturm–Liouville-type operators with delay ,Appl. Math. Lett. 113 (2021) 106862.[20] Borg G.