STURM-LIOUVILLE DIFFERENTIAL OPERATORS WITH DEVIATING ARGUMENT

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1 - TAMKANG JOURNAL OF MATHEMATICS Volume 48, Number 1, 61-71, Mrch 017 doi: /j.tkjm This pper is vilble online t STURM-LIOUVILLE DIFFERENTIAL OPERATORS WITH DEVIATING ARGUMENT S. A. BUTERIN, M. PIKULA AND V. A. YURKO Abstrct. Non-selfdjoint second-order differentil opertors with constnt dely re studied. We estblish properties of the spectrl chrcteristics nd investigte the inverse problem of recovering opertors from their spectr. For this inverse problem the uniqueness theorem is proved. 1. Introduction We study n inverse spectrl problem for non-selfdjoint Sturm-Liouville differentil opertors on finite intervl with constnt dely nd with complex-vlued potentils. Inverse spectrl problems consist in recovering opertors from their spectrl chrcteristics. The gretest success in the inverse spectrl theory hs been chieved for the clssicl Sturm Liouville opertor see the monogrphs [1 5] nd the references therein nd fterwrds for higher-order differentil opertors nd other clsses of differentil opertors nd systems [4] [7]. The clssicl methods of inverse spectrl theory trnsformtion opertor method [1] [4] nd method of spectrl mppings [3] [6], which llow obtining globl solutions of inverse problems for differentil opertors, re not pplicble for differentil opertors with deviting rgument s well s for other clsses of nonlocl opertors such s integrodifferentil, integrl nd other opertors. Therefore, the generl inverse spectrl theory for nonlocl opertors hs not yet been constructed nd there re only isolted results in this direction not forming the generl picture [8] [1]. the form In the present pper we consider the boundry vlue problems L j = L j q,h, j = 0,1, of y x qxyx = λyx, x 0,π, 1 y 0 hy0 = y j π=0, Received July 1, 016, ccepted September, Mthemtics Subject Clssifiction. 34A55 34K9 47E05. Key words nd phrses. Differentil opertors, deviting rgument, spectrl properties, inverse spectrl problems. Corresponding uthor: S. A. Buterin. 61

2 6 S. A. BUTERIN, M. PIKULA AND V. A. YURKO where λ is the spectrl prmeter, 0,π, h is complex number, qx is complex-vlued function, qx L,π, nd qx=0 for x [0, ]. The following inverse problem is studied: given the spectr of the problems L j, j = 0,1, find the potentil qx nd the coefficient h. Differentil equtions with dely rise in vrious problems of mthemtics s well s in pplictions see the monogrphs [] [5] nd the references therein. Some results on the spectrl theory of differentil opertors with dely cn be found in [10, 18, 4, 6] nd other works. The presence of dely in mthemticl model produces serious qulittive chnges in the study of spectrl problems. Therefore, up to now there re no comprehensive results in the inverse problem theory for opertors with dely. In the next section we study spectrl properties of the boundry vlue problems 1, in prticulr, properties of the chrcteristic functions nd the eigenvlues of L j. In Section 3 we consider the inverse spectrl problem of recovering the potentil qx nd the coefficient h from the given two spectr of the boundry vlue problems L 0 nd L 1. We provide uniqueness result for this inverse problem. More precisely, we prove tht if the eigenvlues of L j q,h re the sme s for the zero potentil, then q cn be only zero.. Chrcteristic functions nd spectr Let N N be such tht N < π N1, i.e. [π/n1,π/n. Let C x,λ, Sx,λ nd ϕx,λ be solutions of Eq. 1 under the initil conditions C 0,λ=S 0,λ=ϕ0,λ=1, S0,λ= C 0,λ=0, ϕ 0,λ=h. For ech fixed x, nd ν=0,1 the functions C ν x,λ, S ν x,λ nd ϕ ν x,λ re entire in λ of order 1/, nd ϕx,λ= C x,λhsx,λ. 3 Let λ= nd = σiτ, i.e. σ=re, τ= Im. Lemm 1. The following representtions hold ϕx,λ = ϕ 0 x,λϕ 1 x,λ...ϕ N x,λ, Sx,λ=S 0 x,λs 1 x,λ... S N x,λ, 4 where ϕ 0 x,λ = cosx h sinx, S 0 x,λ= sinx, x 0, sinx t ϕ k x,λ= qt ϕ k 1 t,λd t, k x sinx t S k x,λ= qt S k 1 t,λd t, k 5

3 STURM-LIOUVILLE DIFFERENTIAL OPERATORS WITH DEVIATING ARGUMENT 63 for x k, nd ϕ k x,λ=s k x,λ=0 for x k. Moreover, for, ν=0,1, uniformly in x the following estimtes hold: ϕ ν k x,λ= Oν k exp τ x k, S ν k x,λ= Oν k 1 exp τ x k. 6 Proof. The functions ϕx,λ nd Sx,λ re the solutions of the integrl equtions ϕx,λ=cosx h sinx Sx,λ= sinx 0 0 sinx t qt ϕt,λd t, sinx t qt St,λd t. 7 Solving integrl equtions 7 by the method of successive pproximtions we rrive t 4, where the functions ϕ k x,λ nd S k x,λ, k = 1, N, re defined by 5. Moreover, it follows from 5 tht ϕ k x,λ = cosx t qt ϕ k 1 t,λd t, k S k x,λ = cosx t qt S k 1 t,λd t, k for x k. Clerly, ϕ ν 0 x,λ = Oν exp τ x, S ν 0 x,λ = Oν 1 exp τ x. Using 5 nd 8, we rrive t 6 by induction. 8 Corollry 1. The following representtion holds C x,λ = C 0 x,λc 1 x,λ...c N x,λ, 9 C 0 x,λ = cosx, x 0, sinx t C k x,λ = qt C k 1 t,λd t, 10 k for x k, nd C k x,λ = 0 for x k. Moreover, for, ν = 0,1, uniformly in x the following estimte holds: C ν k x,λ= Oν k exp τ x k. 11 Reltions 9, 10 nd 11 follow from 4, 5 nd 6, respectively, with h= 0. Remrk 1. The functions C 1 x,λ,s 1 x,λ nd ϕ 1 x,λ depend linerly on the potentil q. In

4 64 S. A. BUTERIN, M. PIKULA AND V. A. YURKO prticulr, tking 5 nd 10 into ccount, we clculte for x : sinx C 1 x,λ = C 1 x,λ = cosx cosx S 1 x,λ = S 1 sinx x,λ = qt d t 1 qt d t 1 qt d t 1 qt d t 1 qt sinx t d t, qt cosx t d t, qt cost x d t, qt sint x d t. 1 Denote j λ := ϕ j π,λ, j = 0,1. The functions j λ re entire in λ of order 1/. The zeros of j λ coincide with the eigenvlues of the boundry vlue problems L j q,h counting with multiplicities. The function j λ is clled the chrcteristic function for L j q,h. Lemm. For the following symptoticl formule re vlid: 0 λ = cosπh sinπ sinπ cosπ 1 λ = sinπh cosπ Proof. Using 9, 11 nd 1, we obtin Since it follows tht sinπ C π,λ = cosπ O exp τ π. qt sinπ t d t = 1 π π sinπ C π,λ=cosπ qt d t o exp τ π, 13 qt d t 1 qt d t o exp τ π. 14 qt sinπ t d t qπ ξ/sinξdξ= oexp τ π, qt d t o 1 exp τ π. 15 Anlogously we clculte C cosπ π,λ = sinπ Sπ,λ= sinπ cosπ S sinπ π,λ = cosπ qt d t oexp τ π, qt d t o exp τ π, qt d t o 1 exp τ π. 16

5 STURM-LIOUVILLE DIFFERENTIAL OPERATORS WITH DEVIATING ARGUMENT 65 By virtue of 3, one hs j λ= C j π,λhs j π,λ, j = 0,1. 17 Substituting into 17, we rrive t Lemm 3. The boundry vlue problem L j hs countble set of eigenvlues {λ n j } n 0, j = 0,1 counting with multiplicities nd for n : n0 := λ n0 = n 1 h πn n1 := λ n1 = n h πn Proof. It follows from 14 tht cos n πn cosn 1/ πn qt d t o qt d t o, 18 n n λ= sinπ g λ, g λ C exp τ π. 0 Here nd below, the symbol C denotes vrious positive constnts in estimtes. Fix δ > 0. Denote G δ := { : k δ, k Z}. Since sinπ C exp τ π for G δ, it follows from 0 tht sinπ > g λ, G δ,, 1 for sufficiently lrge. Let Γ n := {λ : λ = n 1/ }. Using 0, 1 nd Rouché s theorem [7, p.15], we conclude tht the number of zeros of 1 λ inside Γ n is equl to n 1. Thus, in the circle λ <n 1/ there exist exctly n 1 eigenvlues of the boundry vlue problem L 1 : λ 01,...,λ n1. Applying now Rouche s theorem to the circle γ n δ := { : n δ}, we conclude tht for sufficiently lrge n, in γ n there is exctly one zero of, nmely n1 = λn1. Since δ>0 is rbitrry, it follows tht Since 1 n1 =0, it follows from 14 nd tht n1 = n ε n, ε n = o1, n. nd, consequently, n1 sin n1 π= h cos n1 π cos n1π qt d t o1, n, This yields cos n n sinε n π= h ε n = h cos n πn πn qt d t o1, n. qt d t o, n, n nd we rrive t 19. Reltion 18 cn be obtined by similr rguments.

6 66 S. A. BUTERIN, M. PIKULA AND V. A. YURKO Lemm 4. The specifiction of the spectrum {λ n j } n 0 uniquely determines the chrcteristic function j λ by the formule λ n0 λ 0 λ= n 1/, 1λ=πλ 01 λ n=0 n=1 λ n1 λ n. 3 Proof. By Hdmrd s fctoriztion theorem [7, p.89], 1 λ is uniquely determined up to multiplictive constnt by its zeros: 1 λ= C n=0 1 λ λ n1 the cse when 1 0= 0 requires minor modifictions. Consider the function Then 1 λ= sinπ= λπ 1 λ 1 λ = C λ λ 01 λ 01 πλ Tking 14 nd 19 into ccount we clculte nd hence 1 λ lim λ 1 λ = 1, n n=1 n=1 λ n1 n=1 lim λ n=1 C = πλ 01 1 λ n. 1 λ n1 n n λ 1 λ n1 n n=1 λ n1 n. n λ. = 1, Substituting this into 4 we rrive t 3 for 1 λ. For the function 0 λ, the rguments re similr. 4 Denote L := 1 λi 0 λ. 5 The function L is entire in of exponentil type, nd L is the chrcteristic function for the Redge-type boundry vlue problem for Eq. 1 with the boundry conditions y 0 h y0= y πiyπ=0. Clerly, L=L 0 L 1...L N, L k =ϕ k π,iϕ kπ,. 6 In prticulr, nd consequently, L 0 = sinπh cosπi cosπih sinπ, L 0 =i hexpi π.

7 STURM-LIOUVILLE DIFFERENTIAL OPERATORS WITH DEVIATING ARGUMENT 67 Let us clculte L 1. For this purpose we use 5, 10, 1 nd 6: L 1 =C 1 π,λhs 1 π,λic 1π,λhS 1 π,λ cosπ π = qt d t 1 qt cosπ t d t h sinπ π qt d t h qt sint π d t i sinπ π qt d t i qt sinπ t d t i h cosπ π qt d t i h qt cost π d t. Therefore, L 1 = h i h i expi π expi π qt d t qt exp i t d t. 7 Lemm 5. For τ 0,, k 1, the following estimte holds L k = O 1 k 1 Proof. Using 6, 5 nd 8, we clculte nd consequently, L k = k L k = k qt exp i t π k d t. 8 cosπ t i sinπ t qt ϕ k 1 t,λd t, On the other hnd, it follows from 6 tht k expi π t qt ϕ k 1 t,λd t, k 1. 9 ϕ k 1 t,λ C 1 k exp i t k, τ Substituting 30 into 9, we rrive t The inverse problem In this section we consider the following inverse problem: given two spectr {λ n j } n 0, j = 0,1, find qx nd h. In order to formulte uniqueness result for this inverse problem, we consider together with L j the boundry vlue problems L j := L j q, h of the sme form but with different q nd h. We gree tht if certin symbol β denotes n object relted to L j, then β will denote the nlogous object relted to L j.

8 68 S. A. BUTERIN, M. PIKULA AND V. A. YURKO Let { λ n j } n 0, j = 0,1, be the eigenvlues of the boundry vlue problems L j with qx 0. Let j λ be the chrcteristic functions of L j, nd L= 1 λi 0 λ. 31 Theorem 1. If λ n j = λ n j for ll n 0, j = 0,1, then qx= qx.e. on,π nd h= h. Proof. 1 By virtue of Lemm 4, one hs 0 λ= 0 λ, 1 λ= 1 λ. Using 5 nd 31 we get L= L. 3 Since qx 0, it follows from nd 9 tht In prticulr, 33 nd 7 yield qt d t = 0, h= h, L L 0 =i hexpi π. 33 L 1 = h expi π i qt exp i t d t. 34 Denote L := L...L N for N, nd L 0 for N = 1. Using 6, 3 nd 33, we obtin L 1 L. 35 First of ll, we note tht if qx = 0.e. on, π, then qx = 0.e. on, π. Indeed, under this ssumption we hve L 0, nd ccording to 35 we infer L 1 0. Using 34 we obtin qt exp i t d t 0, nd consequently, qx=0.e. on,π. In prticulr, this finishes the proof for N = 1, since in this cse we hve π nd utomticlly L 0. 3 Let N. Fix ν=0,n 3. Let us prove tht i f qx=0.e. on Indeed, it follows from 8 tht π ν,π, t hen qx=0.e. on π ν1 1 π ν/ L = O qt exp i t π d t, τ 0,.,π. 36

9 STURM-LIOUVILLE DIFFERENTIAL OPERATORS WITH DEVIATING ARGUMENT 69 Let π ν/>, otherwise we rrive t the sitution in nd the proof is finished. Then in the integrl we hve π t π π ν, which yields L =O exp i π ν, τ 0,. 37 For k >, the functions L k hve less growth thn in 37. This mens tht It follows from 34, 35 nd 38 tht expi π L = O exp i π ν, τ 0,. 38 ν/ or, which is the sme, Moreover, one hs expi π ν1 ν1/ Let us introduce the function qt exp i t d t = O exp i π ν, τ 0,, ν/ qt exp i t d t = O, τ 0,. 39 qt exp i t d t = O exp i π ν1, τ 0,. 40 F := expi π ν1 ν/ π ν1/ qt exp i t d t. The function F is entire in. Clerly, F =O1 for τ 0. On the other hnd, it follows from 39 nd 40 tht F =O1 for τ 0. By Liouville s theorem [7, p.77], F C const. Since F =o1 for rel,, it follows tht F 0, i.e. ν/ π ν1/ qt exp i t d t 0. This yields qx=0.e. on the intervl π ν1/,π ν/, i.e. 36 is proved. Applying proposition 36 successively for ν=0,1,...,n 3, we obtin qx= 0.e. on the intervl π N 1,π,π. According to we get qx=0.e. on,π. Theorem 1 is proved. Acknowledgement This work ws supported in prt by Grnts K, /PCh of the Ministry of Eduction nd Science of RF nd by Grnts , of Russin Foundtion for Bsic Reserch.

10 70 S. A. BUTERIN, M. PIKULA AND V. A. YURKO References [1] V. A. Mrchenko, Sturm-Liouville Opertors nd Their Applictions. Nukov Dumk, Kiev, 1977; English trnsl., Birkhäuser, [] B. M. Levitn, Inverse Sturm-Liouville Problems, Nuk, Moscow, 1984 Russin; English trnsl., VNU Sci.Press, Utrecht, [3] G. Freiling nd V. A. Yurko, Inverse Sturm-Liouville Problems nd Their Applictions. NOVA Science Publishers, New York, 001. [4] V. A. Yurko, Introduction to the Theory of Inverse Spectrl Problems, Moscow, Fizmtlit, 007. [5] V. A. Yurko, Method of Spectrl Mppings in the Inverse Problem Theory, Inverse nd Ill-posed Problems Series. VSP, Utrecht, 00. [6] V. A. Yurko, Inverse Spectrl Problems for Differentil Opertors nd Their Applictions, Gordon nd Brech Science Publishers, Amsterdm, 000. [7] R. Bels, P. Deift nd C. Tomei, Direct nd Inverse Scttering on the Line, Mthemtic Surveys nd Monogrphs, 8. AMS, Providence, RI, [8] V. A. Yurko, An inverse problem for integrl opertors, Mt. Zmetki, , no.5, ; English trnsl. in Mth. Notes no 5-6, [9] M. S. Eremin, An inverse problem for second-order integro-differentil eqution with singulrity, Diff. Urvn., , no., [10] M. Pikul, Determintion of Sturm-Liouville-type differentil opertor with dely rgument from two spectr, Mt. Vesnik, , no.3-4, Russin. [11] V. A. Yurko, An inverse problem for integro-differentil opertors, Mtem. Zmetki, , no.5, Russin; English trnsl. in Mth. Notes, , no.5-6, [1] K. V. Krvchenko, On differentil opertors with nonlocl boundry conditions, Diff. Urvn., , no. 4, ; English trnsl. in Diff. Eqns., , no.4, [13] S. A. Buterin, The inverse problem of recovering the Volterr convolution opertor from the incomplete spectrum of its rnk-one perturbtion, Inverse Problems, 006, [14] S. A. Buterin, Inverse spectrl reconstruction problem for the convolution opertor perturbed by onedimensionl opertor, Mtem. Zmetki, , no.5, Russin; English trnsl. in Mth. Notes, , no.5, [15] S. A. Buterin, On n inverse spectrl problem for convolution integro-differentil opertor, Results in Mth., , no.3-4, [16] Yu. V. Kuryshov, The inverse spectrl problem for integro-differentil opertors, Mtem. Zmetki, , no. 6, Russin; English trnsltion in Mth. Notes, , no. 5-6, [17] S. A. Buterin, On the reconstruction of convolution perturbtion of the Sturm-Liouville opertor from the spectrum, Diff. Urvn., , Russin; English trnsl. in Diff. Eqns., , [18] G. Freiling nd V. A. Yurko, Inverse problems for Sturm-Liouville differentil opertors with constnt dely, Appl. Mth. Lett., 5 01, [19] V. A. Yurko, An inverse spectrl problems for integro-differentil opertors, Fr Est J. Mth. Sci., 9 014, no., [0] S. A. Buterin nd A. E. Choque Rivero, On inverse problem for convolution integro-differentil opertor with Robin boundry conditions, Appl. Mth. Lett., , [1] N. P. Bondrenko nd S. A. Buterin, On recovering the Dirc opertor with n integrl dely from the spectrum, Res. Mth. 016, DOI /s [] A. D. Myshkis, Liner Differentil Equtions with Dely Argument, Moscow, Nuk, [3] R. Bellmn nd K. L. Cooke, Differentil-Difference Equtions, The RAND Corp. R-374-PR, [4] S. B. Norkin, Second Order Differentil Equtions with Dely Argument, Moscow, Nuk, [5] J. Hle, Theory of Functionl-Differentil Equtions, Springer-Verlg, New York, [6] V. V. Vlsov, On the solvbility nd properties of solutions of functionl-differentil equtions in Hilbert spce, Mt. Sb., , no.8, 67 9 Russin; English trnsl. in Sb. Mth., , no. 8,

11 STURM-LIOUVILLE DIFFERENTIAL OPERATORS WITH DEVIATING ARGUMENT 71 [7] J. B. Conwy, Functions of One Complex Vrible, nd ed., vol.i, Springer-Verlg, New York, Deprtment of Mthemtics, Srtov University, Astrkhnsky 83, Srtov 41001, Russi. E-mil: Deprtment of Mthemtics, Srtov University, Astrkhnsky 83, Srtov 41001, Russi. E-mil: Deprtment of Mthemtics, Informtics nd Physics, University Est Srjevo, Alekse Šntic 1, Est Srjevo, Bosni nd Herzegovin E-mil: