Inverse spectral problems for first order integro-differential operators

Size: px

Start display at page:

Download "Inverse spectral problems for first order integro-differential operators"

Frank Stewart Poole
5 years ago
Views:

1 Yurko Boundary Value Problems :98 DOI /s R E S E A R C H Open Access Inverse spectral problems for first order integro-differential operators Vjacheslav Yurko * * Correspondence: yurkova@info.sgu.ru Department of Mathematics, Saratov State University, Astrakhanskaya 83, Saratov, 4112, Russia Abstract Inverse spectral problems are studied for the first order integro-differential operators on a finite interval. Properties of spectral characteristic are established, and the uniqueness theorem is proved for this class of inverse problems. MSC: 47G2; 45J5; 44A15 Keywords: integro-differential operators; inverse spectral problems; uniqueness theorem 1 Introduction The paper is devoted to studying inverse spectral problems for integro-differential operators of the form ly := iy +R Vtyt dt. Inverse problems of spectral analysis consist in recovering operators from their spectral characteristics. Such problems often appear in mathematics, mechanics, physics, electronics, geophysics, meteorology and other branches of natural sciences. Inverse problems also play an important role in solving nonlinear evolution equations in mathematical physics. Interest in this subject has been increasing permanently because of the appearance of new important applications, and nowadays the inverse problem theory is being developed intensively all over the world. The greatest success in the inverse problem theory has been achieved for the Sturm-Liouville operator see, e.g., [1 3] and afterwards for higher order differential operators [4 6] and other classes of differential operators. For integro-differential and other classes of nonlocal operators, inverse problems are more difficult to investigate, and the main classical methods transformation operator method and the method of spectral mappings either are not applicable to them or require essential modifications; and for such operators, the general inverse problem theory does not eist. At the same time, nonlocal and, in particular, integro-differential operators are of great interest, because they have many applications see, e.g., [7]. We note that some aspects of inverse problems for integro-differential and integral operators were studied in [8 12] and other works. In the present paper, we study the inverse spectral problem for the first order integro-differential operator l on a finite interval. Properties of spectral character- The Authors 217. This article is distributed under the terms of the Creative Commons Attribution 4. International License which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original authors and the source, provide a link to the Creative Commons license, and indicate if changes were made.

2 Yurko Boundary Value Problems :98 Page 2 of 7 istics are established, and the uniqueness theorem is proved for the inverse problem of recovering the function Rand Vfromthe given spectraldata. 2 Preliminary information Consider the integro-differential equation ly := iy +R Vtyt dt = λy, [, π], 1 where R, Vt are continuous comple-valued functions, and Rπ C α α, V D β β, +, C α D β. Let ϕ, λ be the solution of Eq. 1 with the condition ϕ, λ = 1. Then the following representation holds see [3]: ϕ, λ =ep iλ + K, t ep iλt dt, 2 where K, t is a continuous function, and K, =. Denote + := {λ :Imλ, δ := { λ : arg λ [π + δ,2π δ]. It follows from 2thatfor λ uniformly in [, π]: ϕ ν, λ= iλ ν ep iλ1 + o1, λ +, ν =,1, ϕ ν, λ=oλ ν, λ δ, ν =,1. 3 Denote ϕ ν, λ:= 1 ν ϕ, λ, ν, λ:=ϕπ, λ. ν! λ ν The function λ isentireinλ of eponential type, and its zeros := {λ n n 1 counting with multiplicities coincide with the eigenvalues of the boundary value problem L = LR, V foreq.1 with the condition yπ =.Letm n be the multiplicity of λ n λ n = λ n+1 = = λ n+mn 1. Denote S := {n : n 1 N, λ n 1 λ n {1, s n+ν :=ϕ ν, λ n, n S, ν = m n 1. The functions {s n n 1 are eigen and associated functions for L. Eample 1 Let λ 1 = λ 2 < λ 3 < λ 4 = λ 5 = λ 6 < λ 7 < λ 8 <.ThenS = {1,3,4,7,8,..., s 1 =ϕ, λ 1, s 2 =ϕ 1, λ 1, s 3 =ϕ, λ 3, s 4 =ϕ, λ 4, s 5 =ϕ 1, λ 4, s 6 = ϕ 2, λ 4, s 7 =ϕ, λ 7,... Let the function η, λ be the solution of the problem iη, λ R Vtηt, λ dt + R=λη, λ, ηπ, λ=. 4

3 Yurko Boundary Value Problems :98 Page 3 of 7 Denote θ, λ:=ηπ, λ. Then iθ, λ+r V tθt, λ dt R = λθ, λ, θ, λ=, 5 where R :=Rπ, V :=Vπ. It follows from 5that θ, λ= g, t, λr t dt, 6 where g, t, λ is Green s function of the Cauchy problem, and ig + t, t, λ λg + t, t, λ+r + t Vτ + tgτ + t, t, λ dτ =, gt, t, λ= i; and consequently, g + t, t, λ= iϕ, λ; t, where ϕ, λ; t is the solution of the Cauchy problem iϕ, λ; t+r + t Vτ + tϕτ, λ; t dτ = λϕ, λ; t, ϕ, λ; t=1. In view of 2weget ϕ, λ; t =ep iλ + K, τ; t ep iλτ dτ, where K, τ; t is a continuous function. This yields g, t, λ= iep iλ t i Substituting 7 into6, we obtain t K t, τ; t ep iλτ dτ. 7 θ, λ= where P, t= ir t i P, t epiλt dt, 8 t R τk τ, t; τ dτ. 9 Clearly, P, = ir, P, = ir. Using 8-9 and4, we conclude that for λ uniformly in [, π]: η ν, λ=oλ ν, λ +, ν =,1, η ν, λ=oλ ν epiλπ, λ δ, ν =,1. 1 Denote λ:=1 Vtηt, λ dt. 11

4 Yurko Boundary Value Problems :98 Page 4 of 7 Using 4and11, we calculate iη, λ+r λ+ Vtηt, λ dt = λη, λ, ηπ, λ=. 12 In particular, it follows from 12 that the zeros of the entire function λ coincide with the zeros of λ, and multiplicities of zeros of λ are not more than multiplicities of zeros of λ. Therefore the function λ/ λisentireinλ of eponential type. Denote 1 λ:= λ ep iλπ. Using 8, 9and11, by standard arguments see, e.g., [3], we obtain that for λ, the following asymptotical formulae hold: 1 λ=ep iλπ1 + o1, λ +, 1 λ=bλ γ o1, λ δ, 13 where B,γ := α + β + 1. The function Fλ:= λ/ 1 λ isentireinλ of eponential type. By virtue of 3, λ=ep iλπ1 + o1, λ +, λ=o1, λ δ. 14 Together with 13 this yields that Fλ 1, i.e., λ 1 λor λ λ ep iλπ. 15 Denote η ν, λ:= 1 η, λ, ν, ψ ν! λ ν n+ν :=η ν, λ n, n S, ν = m n 1. The functions {ψ n n 1 are eigen and associated functions for the boundary value problem L,and ψ n+ν = ν β n+ν j s n+j, n S, ν = m n j= The coefficients {β n n 1 are called Levinson s weight numbers, and the data {λ n, β n n 1 are calledthe spectral data forthe boundaryvalue problem L. We will consider the following inverse problem. Inverse problem 1 Given the spectral data {λ n, β n n 1, construct R and V. 3 The uniqueness theorem Below we will assume that R a.e. on,π. If this condition does not hold, then the specification of the spectral data does not uniquely determine L see Eample 2. Let us formulate the uniqueness theorem for this inverse problem. For this purpose, together with L we consider the boundary value problem L := L R, Ṽ ofthesameform but with different functions R, Ṽ t. We agree that in what follows if a certain symbol α denotes an object related to L,then α will denote the analogous object related to L.

5 Yurko Boundary Value Problems :98 Page 5 of 7 Theorem 1 Let { λ n, β n be the spectral data for the problem L = L R, Ṽ. If λ n = λ n, β n = β n for all n 1, then R R, V Ṽ, [, π]. Proof Using and Hadamard s factorization theorem, we get λ λ. Taking 16 into account, we deduce that the functions A j, λ= λ 1 epiλ ϕ, λη j 1, λ η, λϕ j 1, λ, j =1,2, are entire in λ of eponential type. Taking 3, 1 and13 into account, we obtain for λ A 1, λ=o1, A 2, λ=oλ, λ +, A 1, λ=o λ γ +1, A 2, λ=o λ γ +2, λ δ, and consequently, A 1, λ, A 2, λ A, 17 where the function A does not depend on λ.in particular,17yields ϕ, λη, λ η, λϕ, λ, 18 ϕ, λη, λ η, λϕ, λ A λ ep iλ. 19 Similarly, we obtain ϕ, λη, λ η, λϕ, λ A λ ep iλ, 2 where A does not depend on λ.using18wecalculate ϕ, λ ϕ, λη, λ η, λϕ, λ = ϕ, λ ϕ, λη, λ η, λϕ, λ. Together with 19-2thisyields ϕ, λa ϕ, λa. Taking 3 into account, we conclude that A A, and ϕ, λ ϕ, λ A. 21 Furthermore, using 18, 19 and Eqs.1 and4, we infer ia λ ep iλ = i ϕ, λη, λ i η, λϕ, λ = ϕ, λ R Vtηt, λ dt R + η, λr Vtϕt, λ dt.

6 Yurko Boundary Value Problems :98 Page 6 of 7 Hence, for λ, λ +,wegeta ir. In view of 21, one has ϕ, λ ϕ, λ R. 22 Since R a.e.on,π, it follows from 22that ϕ, λ ϕ, λ. By virtue of 18, η, λ η, λ. Then, according to 4, R R=R Vtηt, λ dt R Ṽtηt, λ dt. For λ this yields R R, [, π], and consequently V Ṽ, [, π]. Theorem 1 is proved. Eample 2 Fi a, π. Let R for [, a]andr for a, π. Put R R for [, π], and choose Vt, Ṽ t suchthatvt Ṽ t fort a, π, and Vt Ṽ tfort [, a]. Then ϕ, λ ϕ, λand η, λ η, λ; hence λ n = λ n, β n = β n for all n 1. Acknowledgements This work was supported by Grant of the Russian Science Foundation. Funding This work was supported by Grant of the Russian Science Foundation. Abbreviations Availability of data and materials Ethics approval and consent to participate Competing interests No potential conflict of interest was reported by the author. Consent for publication Author s contributions All authors read and approved the final manuscript. Author s information Vjacheslav Yurko is Full Professor, Head of the Faculty of Mathematical Physics. Publisher s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Received: 17 May 217 Accepted: 19 June 217

7 Yurko Boundary Value Problems :98 Page 7 of 7 References 1. Marchenko, VA: Sturm-Liouville Operators and Their Applications. Naukova Dumka, Kiev 1977 English transl., Birkhäuser, Basel, Levitan, BM: Inverse Sturm-Liouville Problems. Nauka, Moscow 1984 English transl., VNU Sci. Press, Utrecht, Freiling, G, Yurko, VA: Inverse Sturm-Liouville Problems and Their Applications. Nova Science Publishers, New York Beals, R, Deift, P, Tomei, C: Direct and Inverse Scattering on the Line. Math. Surveys and Monographs, vol. 28. Am. Math. Soc., Providence Yurko, VA: Method of Spectral Mappings in the Inverse Problem Theory. Inverse and Ill-Posed Problems Series. VSP, Utrecht Yurko, VA: Inverse Spectral Problems for Differential Operators and Their Applications. Gordon & Breach, New York 2 7. Lakshmikantham, V, Rao, HRM: Theory of Integro-Differential Equations. Stability and Control: Theory and Applications, vol. 1. Gordon & Breach, New York Yurko, VA: An inverse problem for integro-differential operators. Mat. Zametki 55, Russian; English transl. in Math. Notes 55-6, Kuryshova, Yu: An inverse spectral problem for differential operators with integral delay. Tamkang J. Math. 423, Buterin, SA: On the reconstruction of a convolution perturbation of the Sturm-Liouville operator from the spectrum. Differ. Uravn. Minsk 46, Russian; English transl. in Diff.Equ. 46, Yurko, VA: An inverse problem for integral operators. Mat. Zametki 375, Russian; English transl. in Math. Notes 375-6, Buterin, SA, Choque Rivero, AE: On inverse problem for a convolution integro-differential operator with Robin boundary conditions. Appl. Math. Lett. 48,

An inverse spectral problem for Sturm-Liouville operators with singular potentials on star-type graphs

An inverse spectral problem for Sturm-Liouville operators with singular potentials on star-type graphs G. Freiling, M. Ignatiev, and V. Yuro Abstract. Sturm-Liouville differential operators with singular