REPRESENTATION OF THE NORMING CONSTANTS BY TWO SPECTRA

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1 Electronic Journal of Differential Equations, Vol. 0000, No. 59, pp. 0. ISSN: URL: or ftp ejde.math.txstate.edu REPRESENTATION OF THE NORMING CONSTANTS BY TWO SPECTRA TIGRAN N. HARUTYUNYAN Abstract. The representation of the norming constants by spectra was studied by Levitan, Gasymov and others for the Sturm-Liouville problem with boundary conditions y0 cos α+y 0 sin α = 0, yπ cos β +y π sin β = 0, when sin α 0 and sin β 0. We investigate the representation by spectra without these restrictions.. Introduction and statement of the main result Let Lq, α, β denote the Sturm-Liouville S.-L. problem ly y + qxy = µy, x 0, π, µ C,. y0 cos α + y 0 sin α = 0, α 0, π],. yπ cos β + y π sin β = 0, β [0, π,.3 where q is a real-valued, summable on [0, π] function we write q L R [0, π]. By Lq, α, β we also denote the self-adjoint operator, generated by the problem.-.3, see [6, 8]. It is known, that the spectra of the operator Lq, α, β is discrete and consists of simple eigenvalues, which we denote by µ n q, α, β, n = 0,,,..., emphasizing the dependence of µ n on q, α and β. The eigenvalues µ n q, α, β are enumerated in increasing order; i.e., µ 0 q, α, β < µ q, α, β < < µ n q, α, β < µ n+ q, α, β <.... In what follows, for brevity, we often use the notation µ n = µ n α, β = µ n q, α, β. By ϕx, µ, γ and ψx, µ, δ we denote the solutions of., satisfying the initial conditions ϕ0, µ, γ = sin γ, ϕ 0, µ, γ = cos γ, γ C,.4 ψπ, µ, δ = sin δ, ψ π, µ, δ = cos δ, δ C,.5 correspondingly. The eigenvalues µ n = µ n q, α, β, n = 0,,,..., of Lq, α, β are the solutions of the equation Φµ = Φµ, α, β := ϕπ, µ, α cos β + ϕ π, µ, α sin β = 0, Mathematics Subject Classification. 34B4, 34B05, 34B09, 34L5. Key words and phrases. Sturm-Liouville problem; eigenvalues; norming constants. c 00 Texas State University - San Marcos. Submitted 4 May 00. Published November 4, 00.

2 T. N. HARUTYUNYAN EJDE-00/59 or the equation Ψµ = Ψµ, α, β := ψ0, µ, β cos α + ψ 0, µ, β sin α = 0..7 According to the Liouville formula, the Wronskian W x = W x, ϕ, ψ = ϕ ψ ϕ ψ of the solutions ϕ and ψ is constant. It follows that W 0 = W π and, consequently, Ψµ, α, β = Φµ, α, β. The functions ϕ n x = ϕx, µ n, α and ψ n x = ψx, µ n, β, n = 0,,,..., are the eigenfunctions, corresponding to the eigenvalue µ n. The squares of the L -norms of these eigenfunctions: a n = a n q, α, β = π 0 ϕ n x dx, b n = b n q, α, β = are called the norming constants. Our aim is to prove the following theorem. π 0 ψ n x dx,.8 Theorem.. For arbitrary ε 0, π, ε α, the following representations are true n = 0,,,... : a n q, α, β = sin α sin ε sinα ε µ n α, β µ n ε, β, µ k α, β µ n α, β µ k ε, β µ n α, β.9 if α 0, π, β [0, π. 4n a n q, π, β = π µ0π, β µ n π, β µ 0 ε, β µ n π, β µ n π, β µ n ε, β k= k µkπ, β µ n π, β µ k ε, β µ n π, β.0 if β 0, π, n 0. a 0 q, π, β = 4 π if β 0, π. µ 0 π, β µ 0 ε, β k= k µkπ, β µ 0 π, β µ k ε, β µ 0 π, β. a n q, π, 0 = π µ n π, 0 µ n ε, 0 For arbitrary η 0, π, η β, n = 0,,,... : b n q, α, β = sin β sin η sinβ η µ n α, β µ n α, η µkπ, 0 µ n π, 0 µ k ε, 0 µ n π, 0.. µ k α, β µ n α, β µ k α, η µ n α, β,.3

3 EJDE-00/59 REPRESENTATION OF THE NORMING CONSTANTS 3 if α 0, π], β 0, π. 4n b n q, α, 0 = π µ0α, 0 µ n α, 0 µ n α, η µ n α, 0 if α 0, π, n 0. b 0 q, α, 0 = if α 0, π. µ n α, 0 µ n α, η 4 π µ 0 α, 0 µ 0 α, η b n q, π, 0 = π µ n π, 0 µ n π, η k= k= k k µkα, 0 µ n α, 0 µ k α, η µ n α, 0,.4 µkα, 0 µ n α, 0 µ k α, η µ n α, 0,.5 µkπ, 0 µ n π, 0 µ k π, η µ n π, 0..6 The representation of the norming constants by two spectra was investigated in [,, 7, 0] with the purpose of using it for constructive solution of the inverse Sturm-Liouville problem by two spectra. They do it by reducing the inverse problem by spectra to the solution of the inverse problem by spectral function and solve it by Gelfand-Levitan method. In these papers the Sturm-Liouville problem Lq, α, β considered under conditions sin α 0 and sin β 0 and as the norming constants the authors consider in our notation ã n q, α, β = anq,α,β sin α. Levitan [7] studied the representation of ã n q, α, β by spectra {µ k q, α, β} and {µ k q, α, β }, β β, α, β, β 0, π, when q C R [0, π]. Levitan and Gasymov [] study the representation of ã n q, α, β by spectra {µ k q, α, β} and {µ kq, ε, β}, α ε, α, ε, β 0, π, when q L R [0, π]. The formula, obtained in [], coincides with.9. The cases, when sin α = 0 α = π or sin β = 0 β = 0, in [7] and [] are not considered. The representation of norming constants b n q, α, β or b n q, α, β = bnq,α,β are usually not considered since it is very similar to investigation of a n, and also it is not required in sin β their solution of inverse problem by Gelfand-Levitan method. One of our aims is the solution of inverse problem by the eigenvalues function EVF of the family of Sturm-Liouville operators see [3]. The statement of the problem and its solution will be the subject of our future articles. However in this solution we will need to simultaneously use the representations for both a n and b n, and this is the main reason for such a detailed formulation of the Theorem of this article. Technically the consideration of the cases sin α 0, sin β 0 is distinguished by the fact that in these cases the rough asymptotic behavior of the eigenvalues is and thus µ n q, α, β = n + O,.7 µ n q, α, β µ n q, ε, β = O,.8 µ n q, α, β µ n q, α, β = O..9

4 4 T. N. HARUTYUNYAN EJDE-00/59 These relations are important for the convergence in the infinite product.9 and.3. Meanwhile, when sin α = 0 α = π or/and sin β = 0 β = 0 the rough asymptotic behavior of the eigenvalues have the form n in [3] we have obtained the common formula for the asymptotic behavior of the eigenvalues µ n q, α, β in all the cases α 0, π] and β [0, π, from which.7,.0-. follow as particular cases: µ n q, π, β = + O, when sin β 0.0 µ n q, α, 0 = + O, when sin α 0. µ n q, π, 0 = + O.. The relations.8 and.9 do not hold in the general case for example, µ n q, π, β µ n q, α, β = On when α 0, π. Thus to obtain the representation of a n and b n for all the cases by two spectra we need some improvement of the method in [7] and []. These improvement are the causes that the formulae.9 and.0-. similarly.3 and.4-.6 are different and depend on the cases: α, β 0, π, α = π, β 0, π, α 0, π, β = 0, α = π, β = 0. Whether it is possible to join all these formulae in one may be more complicated, but one, today we do not know. It was obtained in [3], that the lowest eigenvalue µ 0 q, α, β has the property: for arbitrary β [0, π, lim α 0 µ 0 q, α, β = and for arbitrary α 0, π], lim β π µ 0 q, α, β =. In combination with the property a n q, α, β = nq, α, β α and b = nq,α,β nq,α,β β [3] this leads to the fact that the formulae. and.5 for a 0 q, π, β and b 0 q, α, 0 have a special form.. Proof of the main theorem The proof of the main theorem is based on the following lemmas. Here we give the statements of these lemmas and after that we prove the Theorem. The proofs of the lemmas are presented in 3. Lemma.. As t, we have the following: Φ t, α, β = teπt [sin α sin β + O ], t when α, β 0, π,. Φ t, π, β = eπt [sin β + O ], t when β 0, π,. Φ t, α, 0 = eπt [sin α + O ], t when α 0, π,.3 Φ t, π, 0 = eπt t [ + O ]. t.4

5 EJDE-00/59 REPRESENTATION OF THE NORMING CONSTANTS 5 Lemma.. The specification of the spectra {µ n q, α, β} n=0 uniquely determines the characteristic functions Φµ, α, β by the formulae µ k α, β µ Φµ, α, β = π µ 0 α, β µ sin α sin β k,.5 when α, β 0, π. Φµ, π, β = sin β Φµ, α, 0 = sin α k= µ k π, β µ, when β 0, π,.6 µ k α, 0 µ, when α 0, π,.7 Φµ, π, 0 = π µ k π, 0 µ..8 In what follows, by fx, µ we denote the derivative by µ; i.e., fx, µ = fx,µ δµ. Lemma.3. The following formulae hold when α, β 0, π. Φµ, α, β µ=µ0α,β = π sin α sin β k= Φµ, α, β µ=µnα,β = π n [µ 0α, β µ n α, β] when α, β 0, π and n 0. when β 0, π. when α 0, π. sin α sin β k= Φµ, π, β = sin β µ=µnπ,β Φµ, α, 0 = sin α µ=µnα,0 Φµ, π, 0 = π µ=µnπ,0 µ k α, β µ k,.9 µ k α, β µ n α, β k,.0 µ k π, β µ n π, β,. µ k α, 0 µ n α, 0,. µ k π, 0 µ n π, 0..3 We consider also the meromorphic functions see.6 and.7: m α,β,ε µ := n α,β,δ µ := Ψµ, α, β Ψµ, ε, β = ψ0, µ, β cos α + ψ 0, µ, β sin α ψ0, µ, β cos ε + ψ 0, µ, β sin ε,.4 Φµ, α, β Φµ, α, δ = ϕπ, µ, α cos β + ϕ π, µ, α sin β ϕπ, µ, α cos δ + ϕ π, µ, α sin δ..5

6 6 T. N. HARUTYUNYAN EJDE-00/59 Lemma.4. For arbitrary ε 0, π, ε α, and for arbitrary δ 0, π, δ β, m α,β,ε µ = a nq, α, β µ=µnα,β sinα ε,.6 n α,β,δ µ = b nq, α, β µ=µnα,β sinβ δ..7 As it is noted in, Ψµ, α, β = Φµ, α, β. So,.4 we can be rewritten as m α,β,ε µ = It is easy follows from the.8 and.5 that Φµ, α, β Φµ, ε, β..8 m α,β,ε µ = Φµ n α, β, α, β µ=µnα,β Φµ n α, β, ε, β, n α,β,δ µ = Φµ n α, β, α, β µ=µnα,β Φµ n α, β, α, δ. Then, from.6 and.7, we obtain a n q, α, β = sinα ε Φµ n α, β, α, β Φµ n α, β, ε, β,.9 b n q, α, β = sinβ δ Φµ n α, β, α, β Φµ n α, β, α, δ..0 Substituting corresponding formulae from.9-.3 and.5-.8 in.9 and.0, we obtain the formulae.9-. and.3-.6 of Theorem. Thus, our Theorem is proved if the lemmas.-.4 hold. 3. Proofs of lemmas Proof of the lemma.. Let us denoted by y x, λ and y x, λ the solutions of the equation y + qxy = λ y, satisfying the initial conditions y 0, λ =, y 0, λ = 0, y 0, λ = 0, y 0, λ =. For y x, λ and y x, λ it is well known [, 6, 8, 9] the case q L C [0, π] is considered in detail in [4] the asymptotic formulae when λ e Imλ π y π, λ = cos λπ + O, y λ π, λ = λ sin λπ + O e Imλ π, sin λπ e Imλ π e y π, λ = + O λ λ, y π, Imλ π λ = cos λπ + O. λ Since see.4, 3. ϕx, λ, α = y x, λ sin α y x, λ cos α 3.

7 EJDE-00/59 REPRESENTATION OF THE NORMING CONSTANTS 7 and Φµ, α, β is defined by.6, if we substitute in 3., 3. and.6 µ = λ = it = t, we obtain for Φ t, α, β the formula Φ t, α, β = cosiπt + O e πt siniπt sin α cos β + O e πt cos α cos β t it t + it siniπt + Oe πt sin α cos β cosiπt + O e πt cos α sin β. t Taking into account the formulae cosiπt = eπt +e πt, siniπt = e πt e πt i obtain the assertions of lemma.. Proof of the lemma.. The case α, β 0, π was considered in [,, 5]; see [, formula..6] for details. In this case sin α 0, sin β 0 they consider as the characteristic function Φµ,α,β sin α sin β in our notation. The remaining three results are presented in [], but without a proof, and in the case α = π, β = 0 they write µ k π, 0 µ Φµ, π, 0 = π k, k= since they begin the enumeration for µ n π, 0 from n =, but not from n = 0. We use Hadamard s factorization theorem it is similar to method of [] to represent the entire function Φµ, α, β of order / in the form Φµ, α, β = c we µ. 3.3 µ k α, β If Φ0, α, β = 0, i.e. for some k 0 µ k0 α, β = 0, then we must change the k 0 -th factor by µ and it is easy to show that we can take µ k0 α, β = 0 = µ 0 α, β change the enumeration for finite number of eigenvalues and correspondingly for finite factors in Φµ, α, β = c µ k k 0 µ µ k α, β = c µ k= µ. 3.4 µ k α, β Detailed computations show that results for the cases 3.3 and 3.4 are the same. In the case α = π, β 0, π sin β 0 we use the formula and consider the relation Φλ, π, β cos λπ = c cos λπ = λ µ k π,β k+ λ k+ λ = c According to.0, in this case the infinite product µ k π, β λ λ = 3.5 µ k π, β µkπ, β λ λ µ kπ, β λ

8 8 T. N. HARUTYUNYAN EJDE-00/59 converges uniformly because λ = t, ]. The infinite product µ k π, β also converses. So we can write the right hand side of 3.6 in the form c µ k π, β µ k π, β λ λ. Thus, if we pass to the limit in 3.6 when λ = it, t, then in the left hand side of 3.6, according to lemma., we obtain Φ t e, α, β tπ lim = lim t cositπ t [sin β + O t ] = sin β, e πt +e πt and in the right hand side we obtain c k+ µ k π,β ; i.e., c = sin β µ k π,β. k+ Substituting this value of c in 3.3, we obtain formula.6. In the case α 0, π, β = 0 we follow the same procedure and obtain formula.7. In case α = π, β = 0 we consider the relation of Φλ, π, 0 = c λ µ k π, 0 and sinλπ/λ, which we can write in the form sin λπ λ = π λ. According to., the infinite product k+ µ k π,0 converges and the product µ k π, 0 λ λ = [ + µ kπ, 0 ] λ, converges uniformly by λ = it = t, ]. Thus, we can pass to the limit t in the relation Φ t, π, 0 sin iπt it = c π µ k π, 0 µ k π, 0 + t + t and according to.4 and uniform convergence of the right hand side, we obtain = c k+ π µ k π,0. Thus, we obtain the formula.8. Lemma. is proved. The proof of the lemma.3 coincides with simple differentiation with respect to µ at the point µ = µ n α, β in With this aim we represent the infinite product a k in the form a k = n a k k=n+ n a k = a n a k k=n+ a k = a n a k.

9 EJDE-00/59 REPRESENTATION OF THE NORMING CONSTANTS 9 For example, Φµ, α, 0 = µ nπ, 0 µ sin α µ k π, 0 µ. Now differentiating with respect to µ and taking the result at µ = µ n α, 0 we obtain Φµ n α, 0, α, 0 = where sin α = sin α µ k α, 0 µ µ=µnα,0 + µ nα, 0 µ Pµ µ=µα,0 µ k α, 0 µ n α, 0, Pµ = sin α µ k α, 0 µ. The process is analogous in other cases. Lemma.3 is proved. Proof of lemma.4. Since all the eigenvalues of Lq, α, β are simple, there exist the constants c n = c n q, α, β, n = 0,,,..., such that By direct computation, from.4 we obtain m α,β,ε µ ϕ n x = c n ψ n x. 3.7 = [ ψ 0, µ, β ψ0, µ, β ψ0, µ, β ψ 0, µ, β] sinα ε [ψ0, µ, β cos ε + ψ 0, µ, β sin ε]. 3.8 On the other hand, by standard methods, we obtain π 0 ψ x, µ, β dx = ψ 0, µ, β ψ0, µ, β ψ0, µ, β ψ 0, µ, β. 3.9 Since for µ = µ n q, α, β we have 3.7, using.4, from 3.8 and 3.9 we obtain m α,β,ε µ = ψ 0 nxdx sinα ε µ=µnα,β [ψ n 0 cos ε + ψ n0 sin ε] ϕ 0 nxdx sinα ε = c n c n[ϕ n 0 cos ε + ϕ n0 sin ε] = a nq, α, β sinα ε. We prove.7 in a similar way. Lemma.4 is proved. References [] Freiling, G.; Yurko, V.; Inverse Sturm-Liouville problems and their application, 00, Nova Science Publ., Inc. [] Gasymov, M. G.; Levitan,B. M.; Determination of differential equation by two of its spectra, Usp. Mat. Nauk 9, no. 964, 3 63; English transl. in Russian Math. Surveys 9964, 64. [3] Harutyunyan, T. N.; The dependence of the eigenvalues on the boundary conditions, Matematicki Vesnik, 008, no. 4, pp

10 0 T. N. HARUTYUNYAN EJDE-00/59 [4] Harutyunyan, T. N.; Hovsepyan, M. S.; On the solutions of the Sturm-Liouville equation. Mathem. in Higher School, Yerevan, vol. I, N 3 005, pp in Russian. [5] Isaacson, E. L.; Trubowitz, E.; The inverse Sturm-Liouville problem, I. Com. Pure and Appl. Math., vol. 36, pp , 983. [6] Levitan, B. M.; Sargsyan, I. S.; Sturm-Liouville and Dirac operators in Russian. Nauka, Moskwa, 988. [7] Levitan, B. M.; On the determination of Sturm-Liouville equation by two spectra, Izvestia AN SSSR, ser. Math., 964, v. 8, N, p [8] Marchenko, V. A.; The Sturm-Liouville operators and their applications in Russian. Naukova Dumka, Kiev, 977. [9] Pöschel, J.; Trubowitz, E.; Inverse Spectral Theory. Acad. Press, 987. [0] Zikov, V. V.; On the inverse Sturm-Liouville problems on a finite segment, Izv. Akad. Nauk SSSR, Ser. Mat., vol. 3, N 5 967, pp in Russian. Tigran N. Harutyunyan Department of Mathematics and Mechanics of Yerevan State University, A. Manukian str., 005, Yerevan, Armenia address: hartigr@yahoo.co.uk