Open Society Foundation - Armenia. The inverse Sturm-Liouville problem with fixed boundary conditions. Chair of Differential Equations YSU, Armenia

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1 Open Society Foundation - Armenia Physical and Mathematical Sciences 24, p. 8 M a t h e m a t i c s The inverse Sturm-Liouville problem with fixed boundary conditions YU. A. ASHRAFYAN, T. N. HARUTYUNYAN Chair of Differential Equations YSU, Armenia It is well known the necessary and sufficient conditions for two sequences {µ n } n= = { λn 2 } n= and {a n} n= to be the spectral data for a certain Sturm- Liouville problem. We add to these conditions else two and they become necessary and sufficient conditions for { λn 2 } n= and {a n} n= to be the spectral data for a Sturm-Liouville problem with in advance given fixed boundary conditions. MSC2: 34B24, 34L2. Keywords: Sturm-Liouville problem, eigenvalues, norming constants.. Introduction and statements of the results. Let us denote by L(q,α,β the Sturm-Liouville boundary value problem ly y + q(xy = µy, x (,, µ C, ( y(cosα + y (sinα =, α (,], (2 y(cosβ + y (sinβ =, β [,, (3 where q is a real-valued, summable on [,] function (we write q LR [,]. By L(q,α,β we also denote the self-adjoint operator, generated by the problem (- (3 (see []. It is known, that under these conditions the spectra of the operator L(q,α,β is discrete and consists of real, simple eigenvalues ( [], which we denote by µ n = µ n (q,α,β = λn 2 (q,α,β, n =,,2,..., emphasizing the dependence of µ n on q, α and β. We assume that eigenvalues enumerated in the increasing order, i.e. µ (q,α,β < µ (q,α,β <... < µ n (q,α,β <... Let ϕ(x, µ,α,q and ψ(x, µ,β,q are the solutions of the equation (, which satisfy the initial conditions ϕ(, µ,α,q = sinα, ϕ (, µ,α,q = cosα, yuriashrafyan@ysu.am hartigr@yahoo.co.uk ψ(, µ,β,q = sinβ, ψ (, µ,β,q = cosβ,

2 2 Open Society Foundation - Armenia, 24, p. 8. correspondingly. The eigenvalues µ n = µ n (q,α,β, n =,,2,..., of L(q,α,β are the solutions of the equation Φ(µ = Φ(µ,α,β de = f ϕ(, µ,αcosβ + ϕ (, µ,αsinβ =, or the equation Ψ(µ = Ψ(µ,α,β de = f ψ(, µ,βcosα + ψ (, µ,βsinα =. According to the well-known 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( = W( and, consequently Ψ(µ,α,β = Φ(µ,α,β. It is easy to see that the functions ϕ n (x = ϕ(x, µ n,α and ψ n (x = ψ(x, µ n,β, n =,,2,..., are the eigenfunctions, corresponding to the eigenvalue µ n. Since all eigenvalues are simple,there exist constants c n = c n (q,α,β, n =,,2,..., such that ϕ(x, µ n = c n ψ(x, µ n. (4 The squares of the L 2 -norm of these eigenfunctions: a n = a n (q,α,β = b n = b n (q,α,β = ϕ n (x 2 dx, n =,,2,..., ψ n (x 2 dx, n =,,2,... are called norming constants. The sequences { λn 2 } n=, {a n} n= and {b n} n= are called spectral data of problem L(q,α,β. In this paper we consider the case α,β (,, i.e. we assume that sinα and cosβ. In this case we consider the solution ϕ(x, µ,α,q := ϕ(x,µ,α,q sinα of the equation ( which has the initial values ϕ(, µ,α,q =, ϕ (, µ,α,q = cotα, and also we consider the solution ψ(x, µ,β,q := ψ(x,µ,β,q sinβ. Of course, the functions ϕ n (x := ϕ(x, µ n,α,q and ψ n (x := ψ(x, µ n,α,q, n =,,2,..., are the eigenfunctions, corresponding to the eigenvalue µ n. It follows from (4 that for norming constants ã n := ϕ n 2 = a n b sin 2 α n := ψ n 2 = b n the following connections sin 2 β b n = b n sin 2 β = a n c 2 n sin 2 β = ãn sin 2 α c 2 n sin 2 (5 β are held. In this paper we will prove the following assertions. T h e o r e m. Let q LR 2 [,] and α,β (,. Then for norming constants ã n and b n the following connections hold: ( ã + 2 n= ã n b + n=( b 2 n = cotα, (6 = cotβ. (7

3 Ashrafyan Yu.A., Harutyunyan T.N. The inverse Sturm-Liouville problem with fixed boundary conditions 3 The inverse problem by spectral function (see [2, 5 9] is in the reconstruction } the problem L(q,α,β by spectra {µ n } n= and norming constants {ã n} n= { b (or n n=. T h e o r e m 2. For real increasing sequence {µ n } n= and positive sequence {ã n } n= to be spectral data for boundary value problem L(q,α,β with q L2 R [,] and in advance fixed α,β (, is necessary and sufficient that the following relations hold: ã 4 2 ( k= λ n = µ n = n + ω n + κ n n, ω = const,{κ n} n= l2, (8 2 ã n = ( 2 + r n, r n = O n 2, (9 3 ( ã + 2 = cotα, ( n= ã n ( µ k µ 2 + k 2 n= ã n n 4 2 [µ µ n ] 2 ( k=,k n µ k µ n 2 k 2 2 = cotβ. 2. The proof of the Theorem. For the solution ϕ is well known the representation (see [2, 3, 9] x ϕ(x,λ,α,q = cosλx + G(x, t cos λtdt, (2 where about the kernel G(x,t we know (in particular that G(x,x = cotα + 2 x ( q(sds. (3 Besides, it is known that G(x,t satisfies to the Gelfand-Levitan integral equation x G(x,t + F(x,t + n= G(x,sF(s,tds =, t x, (4 where the function F(x,t is defined by formula (see for the details [9] ( cosλn xcosλ n t F(x,t = cosnxcosnt ã n a n where a = and a n = 2 for n =,2,... It is easy follows from (3-(5 that ( G(, = F(, = n= ã n a = n ( = ( ã 2 = cotα. (6 ã n Thus, (6 is proved. n= (5

4 4 Open Society Foundation - Armenia, 24, p. 8. Let us now consider the functions (n =,, 2,... p(x, µ n = ϕ( x, µ n,α,q = ϕ( x, µ n. (7 ϕ(, µ n,α,q ϕ(, µ n Since ϕ(x, µ,α,q satisfies to the equation ( and p (x, µ n = ϕ ( x, µ n ϕ(, µ n we can see, that p(x, µ n satisfy to the equation, p (x, µ n = ϕ ( x, µ n, ϕ(, µ n p (x, µ n + q( xp(x, µ n = µ n p(x, µ n and initial conditions p(, µ n =, p (, µ n = ϕ (, µ n = (cotβ = cotβ = cot( β. (8 ϕ(, µ n Also we have p(, µ n = ϕ(, µ n ϕ(, µ n = sinα sin( α = ϕ(, µ n ϕ(, µ n, p (, µ n = ϕ (, µ n ϕ(, µ n = cosα ϕ(, µ n cos( α =. ϕ(, µ n It follows, that p n (x := p(x, µ n satisfy to the boundary condition p n (cos( α + p n(sin( α =, n =,,2,... Let us denote q (x := q( x. Since µ n (q, β, α = µ n (q,α,β (it is easy to prove and is well known, see, for example [6], it follows, that p n (x, n =,,2,..., are the eigenfunctions of problem L(q, β, α, which have the initial conditions (8, i.e. p n (x = ϕ(x, µ n, β,q, n =,,2,... Thus, as in (6, for norming constants â n = p(, µ n 2 must be true the connection ( â + n= ( â n 2 = cot( β = cotβ. (9 On the other hand, for norming constants â n,according to the (4, (5 and (7, we have â n = p 2 ϕ 2 ( x, µ n (x, µ n dx = ϕ 2 dx = (, µ n = ϕ 2 (, µ n ϕ 2 (s, µ n ds = ϕ 2 ϕ 2 (s, µ n ds = (, µ n = a n(q,α,β ϕ 2 (, µ n = ãn sin 2 α c 2 n sin 2 β = b n. Therefore, we can rewrite (9 in the form ( b n=( b 2 = cot( β = cotβ. n Thus, (7 is true and Theorem is proved.

5 Ashrafyan Yu.A., Harutyunyan T.N. The inverse Sturm-Liouville problem with fixed boundary conditions 5 3. The proof of the Theorem 2. For µ n we have proved the following asymptotic formula [] µ n (q,α,β = [n + δ n (α,β] 2 + where δ n is the solution of the equation q(tdt + r n (q,α,β, (2 δ n (α,β = arccos cosα [n + δ n (α,β] 2 sin 2 α + cos 2 α arccos cosβ [n + δ n (α,β] 2 sin 2 β + cos 2 β (2 and r n (q,α,β = o(, when n, uniformly in α,β [,] and q from any bounded subset of LR [,] (we will write q BL R [,]. It follows from (2 (see [] for details, that if sinα,sinβ, (α,β (,, then ( cotβ cotα δ n (α,β = + O n n 2. (22 It follows from (2 that λ n = µ n = n + δ n (α,β + ( [q] 2[n + δ n (α,β] + l n + O n 2, (23 where l n = [n+δ n (α,β] q(xcos2λ n xdx = o ( n and [q] = q(tdt. In case of q LR 2 [,] and α,β (, it follows from (22 and (23 that we can rewrite (23 in the form λ n = n + ω n + ω n n, (24 where ω = const = cotβcotα+ 2 [q] and {ω n } n= l2, i.e. n= ω n 2 <. For norming constants the following asymptotic formulae hold (when n (see [] a n (q,α,β = 2 b n (q,α,β = 2 [ ( + O [ + O n 2 ( n 2 a n ] sin 2 α + ] sin 2 β + cos 2 [ ( α 2[n + δ n (α,β] 2 + O n 2 cos 2 β 2[n + δ n (α,β] 2 [ + O ( n 2 It is follows that for ã n = the relation (9 is held. sin 2 α In [9] there is a proof of such assertion: T h e o r e m 3. For real numbers { λn 2 } n= and {ã n} n= to be the spectral data for a certain boundary value problem L(q,α,β with q LR 2 [,], it is necessary and sufficient that the relations (24 and (9 are held. ] ] (25 (26

6 6 Open Society Foundation - Armenia, 24, p. 8. It is known that the specification of the spectra {µ n (q,α,β} n= uniquely determines the characteristic function Φ(µ (see [2], Lemma 2.2; see also [6], Lemma, and also its derivative Φ(µ µ = Φ(µ (see [2], lemma2.3. In particular, if α,β (, the following formulae are held: µ k µ Φ(µ = sinα sinβ k= k 2 (27 and (if n, i.e. n =,2,... Φ(µ n = n 2 [µ µ k µ n µ n ]sinα sinβ k=,k n k 2. (28 On the other hand, it is easy to prove the relation (see [2], formula (2.6 in Lemma 2.2 and see [6], Lemma a n = c n Φ(µ n. (29 To take into account the connections (5 and (27-(29 we can find formulae for b and b n,n =,2,...: b n = b = ã 2 ( k= ã n n 4 2 [µ µ n ] 2 ( k=,k n µ k µ n k 2 2, (3 µ k µ n k 2 2. (3 So, we can change the second assertion in Theorem by following + n= ( ã 2 ( k= ã n n 4 2 [µ µ n ] 2 ( k=,k n µ k µ n 2 + k 2 µ k µ n 2 k 2 2 = cotβ. Thus we have a real sequence {µ n } n= = { λn 2 }, which has the asymptotic n= representation (24 and a positive sequence {ã n } n=, which has the asymptotic representation (9. Then, according to the Theorem 3, there exist a function q LR 2 [,] and some constants α, β (, such that λn 2, n =,,2,..., are the eigenvalues and ã n, n =,,2,..., are norming constants of the Sturm-Liouville problem L(q, α, β. The function q(x and constants α,β are obtained on the way of solving the inverse problem by Gel fand-levitan method. The algorithm of that method is: at first we define the function F(x,t by formula (5 (note that this function is defined by {λ n } n= and {ã n} n= uniquely. And we will consider the integral equation (4, where G(x, is unknown function. It is proved that provided (9 and (24 the integral equation (4 has a unique solution G(x, t. And with function G(x, t is constructed a function x ϕ(x,λ = cosλx + G(x, t cos λtdt, (32

7 Ashrafyan Yu.A., Harutyunyan T.N. The inverse Sturm-Liouville problem with fixed boundary conditions 7 defined for all λ C. It is proved (see, for example, [9] that ϕ (x,λ 2 + (2 ddx G(x,x ϕ(x,λ 2 = λ 2 ϕ(x,λ 2, (33 almost everywhere on (,, If we state the condition ϕ(,λ 2 =, ϕ (,λ 2 = G(,. G(, = cotα, (34 then the solution (32 of the equation (33 will satisfy the boundary condition (2 ϕ(,λ 2 cosα + ϕ (,λ 2 sinα = for all λ C. Since from ( (4 follows that G(, = F(, and from (5 follows that F(, = n= ã n, then the condition (34 can be represented as a n ( = cotα, n= ã n a n which is put on the sequence {ã n } n=. It is also proved that the relation ϕ n( ϕ n ( = ϕ (,λn 2 is a constant (i.e. does not ϕ(,λ n 2 depend on n, which we will denote by cot β. So the functions ϕ(x,λ n 2,n =,,2,..., are the eigenfunctions of the problem L(q,α, β, where q(x = 2 d dx G(x,x, α is in advance given α and we want β to be equals β. But we know, from the Theorem, that for problem L(q,α, β is true the equality n=( b b = cot β n n and also are true the connections (3 and (3. Thus if we state the condition (, then β = β. Theorem 2 is proved. R E F E R E N C E S. Naimark M.A. Linear Differential Operators // Nauka, Moscow, 969,(in Russian. 2. Marchenko V.A. Concerning the Theory of a D ifferential Operators of the Second Order // Trudy Moskov. Mat. Obshch., pp , 952, (in Russian. 3. Gel fand I.M., Levitan B.M. On the Determination of a Differential Equation from its Spectral Function // Izv. Akad. Nauk. SSSR., Ser Math. 5, pp , Gasimov M., Levitan B.M. Reconstruction of Differential Equation by Two Spectra // Uspekhi Mat. Nauk, 964, (in Russian. 5. Zikov V.V. On Inverse Sturm-Liouville Problem on a Finite Segment // Izv. Akad. Nauk. SSSR., Ser Math. (5 3,pp , Isaacson E.L., Trubowitz E. The Inverse Sturm-Liouville Problem, I. // Com. Pure and Appl. Math., Vol 36, pp , 983.

8 8 Open Society Foundation - Armenia, 24, p Poshel J., Trubowitz E. Inverse Spectral Theory // N.-Y., Academic Press, Levitan B.M. The Inverse Problems // Moskva, Nauka, Yurko V.A. An Introductions to the Theory of Inverse Spectral Problems. // Fizmatlit, Moscow, 27, (in Russian.. Harutyunyan T.N. The Dependence of the Eigenvalues of the Sturm-Liouville Problem on Boundary Conditions // Matematicki Vesnik (Beograd (4 6, pp , 28.. Harutyunyan T.N. Asymptotics of the Norming Constants of the Sturm-Liouville Problem // Proceedings of YSU, 3, pp. 3-, Harutyunyan T.N. Representation of the Norming Constants by Two Spectra // Electronic Journal of Differential Equations, N59, pp. -, 2.