Inverse problem for Sturm Liouville operators with a transmission and parameter dependent boundary conditions

Transcription

1 CJMS. 6()(17), Caspian Journal of Mathematical Sciences (CJMS) University of Mazaran, Iran ISSN: Inverse problem for Sturm Liouville operators with a transmission parameter dependent boundary conditions Mohammad Shahriari 1 Aliasghar Jodayree Akbarfam 1 Department of Mathematics, Faculty of Science, University of Maragheh, P. O. Box , Maragheh - Iran. Faculty of mathematical sciences, University of Tabriz, P. O. Box 51664, Tabriz - Iran. Abstract. In this manuscript, we consider the inverse problem for non self-adjoint Sturm Liouville operator D + q with eigenparameter dependent boundary discontinuity conditions inside a finite closed interval. We prove by defining a new Hilbert space using spectral data of a kind, the potential function can be uniquely determined by a set of value of eigenfunctions at an interior point part of two sets of eigenvalues. Keywords: Inverse Sturm Liouville equation, Non self adjoint operator, Jump condition, Hilbert space. Mathematics subject classification: 34B4; Secondary 34L1, 34L, 31A5. 1. Introduction We consider the boundary value problem L = L(q(x), h, H, a) ly := y + qy = λy, x [, a) (a, π] (1.1) 1 Corresponding author: shahriari@tabrizu.ac.ir Received: November 13 Revised: 6 December 16 Accepted: 19 December 16 17

2 18 M. Shahriari, A. Jodayree Akbarfam with the jump conditions U(y) : = y () hy() =, (1.) V (y) : = Hy (π) + λy(π) =, (1.3) U 1 (y) := y(a + ) a 1 y(a ) =, U (y) := y (a + ) a y (a ) a 3 y(a ) =, (1.4) where q L [, π], h, H R, < H <. a (, π), a 1, a, a 3 are real, a 1, a have the same sign. The method of separation of variables for solving PDEs with discontinuous boundary conditions naturally led to ODE with discontinuities inside of the interval which often appear in mathematics. Inverse spectral problem consists in recovering operators from their spectral characteristics. For example the mathematical formulation of a large variety of technical physical problem led to inverse problems such as identifying the density of the vibrating string from data collected from the sets of frequencies of oscillations of the vibrating string with barrier. The inverse spectral Sturm Liouville problem can be regarded as three aspects, e.g., existence, uniqueness reconstruction of the coefficients given specific properties of eigenvalues eigenfunctions. Here we want to look at the question of uniqueness for the above problem using two sets of spectra, or one spectrum plus part of a set of value of eigenfunctions at an interior point. The applications of boundary value problems with discontinuity conditions inside the interval are connected with discontinuous material properties. Inverse problems with a discontinuity condition inside the interval play an important role in mathematics, mechanics, radio electronics, geophysics, other fields of science technology. As a rule, such problems are related to discontinuous non smooth properties of a medium (e.g., see [1] [14] [18]). There are various formulations of the inverse problems the corresponding uniqueness theorems. Ambarzumian [1] considered the questions of uniqueness in a special case. He considered equation (1.1) with the boundary conditions the equation y () = = y (π) (1.5) y + λy = with the same end conditions. He showed that if the two systems have the same spectrum, {λ n } = {n }, then q(x) is identically zero. Note that Ambarzumian s result [1] is an exception from the rule. In general, the specification of the spectrum does not uniquely determine the potential function. The idea of using two sequences of eigenvalues seems

3 Inverse problem for Sturm Liouville operators to have originated in Borg s paper [] on inverse spectral theory. showed that if q(x) is symmetric, i.e., q(x) = q(π x), then the spectrum of equation (1.1) corresponding to the end condition (1.5) or to the end condition y() = = y(π) determines q(x) uniquely. We refer to the somewhat complementary surveys in [6] [11], [17] [19] for further aspects of this field. For general background on inverse Sturm Liouville problems we refer (e.g.) to the monographs [4], [17], [19], [6]. More recently, Yong Guo [3] have investigated inverse spectra problems for a differential pencil have determined a differential pencil from interior spectral data. In fact in this work, we formulate a new inverse spectral problem for discontinuous parameter dependent boundary conditions. One can find the similar works for continuous discontinuous conditions in [14], [15] [] [5]. Remark 1.1. The same result can be obtained by the same method in the more general case of the parameter dependent boundary conditions y () hy() =, λ(y (π) + H 1 y(π)) H y(π) H 3 y (π) =.. Asymptotic form of solutions eigenvalues In this section, we introduce the special inner product in the Hilbert space (L (, a) L (a, π)) C define a linear operator A in it such that the considered problem (1.1) (1.4) can be interpreted as the eigenvalue problem of A. So, we define a new Hilbert space inner product on H := (L (, a) L (a, π)) C by F, G H = a 1 ( f(x) where F = a π He fḡ + 1 fḡ + 1 a a H a f 1ḡ 1, ( ) g(x) H ḡ, ḡ g 1 are the conjugate 1 ), G = f 1 of g, g 1 respectively. We define R 1 (u) := u(π) R 1 (u) := Hu (π). We now define the operator by ( AF = ) lf R 1 (f) A : H H where F = ( f(x) R 1 (f) ) D(A),

4 11 M. Shahriari, A. Jodayree Akbarfam with domain ( ) f(x) D(A) = F = f 1 f(x), f (x) AC[, a) (a, π], f(a ± ), f (a ± ) is defined, lf L [(, a) (a, π)], U(f) = U 1 (f) = U (f) =, f 1 = R 1 (f). Thus, we can change the boundary value problem (1.1) (1.4) of the form ( ) u(x) AU = λu U := D(A), R 1 (u) in the Hilbert space H. It is easy that verified the eigenvalues of the operator A coincide with those of the problem (1.1) (1.4). Remark.1. We note that the operator L is non self adjoint in the L [, π], where the operator A defined in the new Hilbert space H is self adjoint. The proof of this result is the same as [16]. Suppose that the functions φ(x, λ) ψ(x, λ) be the solutions of (1.1) under the initial conditions: φ(, λ) = 1, φ (, λ) = h, (.1) ψ(π, λ) = H, ψ (π, λ) = λ. (.) By attaching a subscript 1 or to the functions φ ψ, we mean to refer to the first subinterval [, a) or to the second subinterval (a, π]. By φ 1n (x) we mean φ(x, λ n ) for x [, a) by φ n (x) we mean φ(x, λ n ) for x (a, π). Therefore we see that { φ1n (x), x < a, φ n (x) := φ(x, λ n ) = φ n (x), x > a. It is easy to see that equation (1.1) under the initial conditions (.1) or (.) has a unique solution φ 1 (x, λ) or ψ (x, λ), which is an entire function of λ C for each fixed point x [, a) or x (a, π]. From the linear differential equations we obtain that the Wronskians 1 (λ) := W (φ 1 (x, λ), ψ 1 (x, λ)) (λ) := W (φ (x, λ), ψ (x, λ)), are independent on x [, a) (a, π] respectively. Lemma.. The equality (λ) = a 1 a 1 (λ) holds for each λ C. Corollary.3. The zeros of (λ) = (λ) = a 1 a 1 (λ) are coincide, eigenvalues of the problem (1.1) (1.4) coincide with the zeros of the function (λ). Corollary.4. By self adjointness of A Corollary.3., all eigenvalues of the problem (1.1) (1.4) are real simple.

5 Inverse problem for Sturm Liouville operators Theorem.5. Let λ = ρ, τ := Imρ. For equation (1.1) with boundary conditions (1.3) jump conditions (1.4) as λ, the following asymptotic formulas hold. cos ρx + ( h + 1 φ(x; λ) = x q(t)dt) sin ρx ρ sin ρx (b 1 cos ρx + b cos ρ(a x)) + f 1 (x) sin ρ(a x) +f (x) ρ + O ( ) + O exp( τ x) ρ, x < a, ( ) exp( τ x) ρ ρ x > a,, ρ sin ρx + ( ( ) h + 1 x q(t)dt) cos ρx + O exp( τ x) ρ x < a, φ (x; λ) = ρ[( b 1 sin ρx + b sin ρ(a x))] + f 1 (x)) cos ρx x > a, f (x) cos ρ(a x) + O, where ( exp( τ x) ρ b 1 := a 1 + a, b := a 1 a ( f 1 (x) = b 1 h + 1 ( f (x) = b h 1 x x Then the characteristic function is (λ) ) q(t)dt + a 3, q(t)dt + a ) q(t)dt + a 3. = ρ (b 1 cos ρπ + b cos ρ(a π)) + ρ[(f 1 (π) Hb 1 ) sin ρπ +(f (π) + Hb ) sin ρ(a π)] + O(exp( τ π)). (.3) Proof. The arguments for obtaining the asymptotic formulas is similar to that of [18]. Note that by changing x to π x one can obtain the asymptotic form of ψ(x, λ) ψ (x, λ). We consider the boundary value problem L = L(; ; H; a) such that with the boundary conditions the jump conditions l y := y = ρ y, (.4) U (y) := y () =, V (y) = Hy (π) + λy(π) =, y(a + ) = a 1 y(a ), y (a + ) = a y (a ). (.5) Let φ (x; ρ) be the following form { cos ρx, x < a, φ (x; ρ) = b 1 cos ρx + b cos ρ(a x), x > a. (.6)

6 11 M. Shahriari, A. Jodayree Akbarfam One can see that φ (x, ρ) is the solution of (.4) under the initial conditions φ (, ρ) = 1 φ (, ρ) = the jump conditions (.5). Let (ρ) be a characteristic function of problem L. By using (.6) it is easy to see that the characteristic function related to L is (ρ) = ρ[ρ(b 1 cos ρπ+b cos ρ(a π))+h( b 1 sin ρπ+b sin ρ(a π))]. (.7) The roots ρ n of this equation are eigenvalues of problem L. We now prove the main results of this section. Lemma.6. The eigenvalues of problem L are ρ n = n 1 + η n, where sup n η n < M for sufficiently large ρ, η n (, 1) for n N. Proof. For the operator L we can define the operator A similar than the operator A but different boundary jump conditions. From Remark.1 we find that the operator A is self-adjoint. Indeed the zeros of the entire function (ρ) are simple coincide with the eigenvalues of L for which its eigenvalues are real. We restrict the domain of (ρ) to real line. From the fact that a 1 a have the same sign, we see that b 1 > b so for sufficiently large ρ the sign of (ρ) depend on the first term of (ρ). By substituting the points n 1 n instead of ρ we conclude that (n 1) (n) < for n N. According to continuity differentiability of (ρ) there is a point, say η n, in the interval (, 1) such that (n 1 + η n ) =. Now we show that there exists exactly one zero in (n 1, n). Suppose that a = p q π, where p q is a rational number in the interval (, 1). By rewriting (.7) in the following form (ρ) ρ = b 1 i (exp(iρπ) + exp( iρπ)) + b [ ( ( )) ( ( ))] p p exp iρπ i q 1 + exp iρπ q 1 + H [ b 1 (exp(iρπ) exp( iρπ)) (.8) ρ i + b [ ( ( )) ( ( ))]] p p exp iρπ i q 1 exp iρπ q 1 substituting w := exp(i ρπ q ) in (.8), we see that for sufficiently large ρ, (ρ) is a polynomial of degree q in term of w. Since (ρ) is periodic ρ ρ function with period T = q, there are q zeros on the interval (, q) so for each interval (n 1, n) there is exactly one zero. By applying the same method of [1] or as a consequence of Valiron s theorem ([5, Thm. 13.4]) we then get that ρ n = n 1 + η n where sup η n M.

7 Inverse problem for Sturm Liouville operators Theorem.7. The corresponding eigenvalues {λ n } of the boundary value problem L admit the following asymptotic form as n : ( ) 1 λn = n 1 + η n + O, n where η n is defined in Lemma.6. Proof. From (.3) (.7), we see that (ρ) = (ρ) + O (ρ exp( τ π)). For sufficiently large value of ρ, the functions (ρ) (ρ) have the same number of zeros counting multiplicities according to Rouche s theorem. So if ρ n ρ n are eigenvalues of L L respectively, then we have ρ n = ρ n + ε n where lim n ε n =. Since numbers ρ n are zeros of the characteristic function (ρ), therefore we have = (ρ n ) = (ρ n ) + O(ρ n ) = (ρ n + ε n ) + O(ρ n + ε n ) = (ρ n) + ε n (ρ n) + O(ρ n). From (.7), by a simple calculation, we have therefore ε n = O( 1 n ). (ρ n) = O(ρ n ) 3. Main results In this section, together with L we consider the boundary value problem L = L( q(x); h; H; a) of the same form but a different coefficient q a = p q π where p q is a rational number in the interval (, 1). One can find the same results in [], [4] [5]. Theorem 3.1. If λ n = λ n, W (y n, ỹ n ) (a ) = for any n N a π, then q(x) = q(x) a.e. on [, a). Remark 3.. Let φ(x, λ) φ(x, λ) are solutions of L L with the initial conditions φ() = 1, φ () = h φ() = 1, φ () = h respectively. For x a the following representation holds (see [4], [19]) φ(x, λ) = cos(ρx) + x k(x, t) cos(ρt)dt, x φ(x, λ) = cos(ρx) + k(x, t) cos(ρt)dt,

8 114 M. Shahriari, A. Jodayree Akbarfam where k(x, t) k(x, t) are continuous function not depending on λ. Hence φ(x, λ) φ(x, λ) = 1 [ x ] 1 + cos(ρx) + v(x, t) cos(ρt)dt (3.1) where v(x, t) is a continuous function which does not depend on λ. Fix δ > define G δ = {ρ : ρ ρ n δ}. Then (see[18]) for some constant C 1 >. (λ) C 1 ρ exp( τ π), ρ G δ, (3.) Proof of Theorem 3.1. Let φ + qφ = λφ, (3.3) φ + q φ = λ φ. (3.4) If we multiply (3.3) by φ (3.4) by φ subtract them, after integrating on [, a ] we obtain G(λ) = a [ q(x) q(x)]φ(x, λ) φ(x, λ)dx = [ φ (x, λ)φ(x, λ) φ(x, λ)φ (x, λ)] a. (3.5) The functions φ φ are satisfying in the initial condition (1.); from this fact W ( φ n, φ n ) a = it follows that G(λ n ) =, n N. Next we shall show that G(λ) = on the whole λ-plane. We know that From (3.1) we obtain cos(ρx) exp(x τ ). φ(x, λ) φ(x, λ) C exp(x τ ), (3.6) for some constant C. From (3.5) (3.6) we see that the entire function G(λ) satisfies G(λ) C exp(a τ ) (3.7) for some positive constant C λ large enough. Define ( ) q G(λ) ϕ(λ) := ( ) p. (λ) The definition of (λ), G(λ), a π implies ϕ(λ) is an entire function. It follows from (3.) (3.7) that ( ) 1 ϕ(λ) = O λ 4p

9 Inverse problem for Sturm Liouville operators for sufficiently large λ. Thus, ϕ(λ) is bounded so Liouville s theorem implies ϕ(λ) =, which is equivalent to Let G(λ) = for all λ. (3.8) Q(x) = q(x) q(x) From (3.1), (3.5) (3.8), we obtain, on the whole λ-plane, a Q(x)[1 + cos(ρx)]dx + a which can be rewritten as a a [ Q(x)dx + cos(ρs) Q(s) + [ x Q(x) a s ] v(x, s) cos(ρs)ds dx =, ] Q(x)v(x, s)dx ds =. (3.9) Letting λ for real λ in (3.9), by the Riemann Lebesque lemma we see that a a [ cos(ρs) Q(s) + Q(x)dx = a s ] Q(x)v(x, s)dx ds = From the completeness of the function cos(ρs) on the interval [, a], we have Q(s) + a s Q(x)v(x, s)dx =, < s < a But this equation is a homogeneous Volterra integral equation has only the zero solution. Thus Q(x) = on < x < a, that is, q(x) = q(x) almost everywhere on [, a). Remark 3.3. If y z satisfy the jump conditions (1.4) W (y, z) (a ) = then it is easy to verify that W (y, z) (a+) =. Theorem 3.4. Let a ( π, π) be a jump point. Let λ n = λ n, W (y n, ỹ n ) (a ) =, for each n N. Then q(x) = q(x) almost everywhere on (a, π]. Proof. We consider the supplementary problem L by changing x by π x. Let t = π x then from Eq. (1.1) we get y + q(π t)y = λy.

10 116 M. Shahriari, A. Jodayree Akbarfam Define q 1 (t) = q(π t) then the above equation has the following form ly := y + q 1 (t)y = λy, < t < π U(y) := Hy () λy() =, V (y) := y (π) + hy(π) =, by discontinuous conditions y(π a + ) = a 1 1 y(π a ), y (π a + ) = a 1 y (π a ) + a 3 a 1 a y(π a ). For this part we obtain (λ) by the similar form of (.3). Let ψ(t, λ) ψ(t, λ) are solutions of L L with the initial conditions ψ() = H, ψ () = λ ψ() = H, ψ () = λ respectively. For t a the following representation holds (see [4], [19]) [ t ψ(t, λ) = ρ sin(ρt) + k(t, s) sin(ρs)ds ], [ ψ(t, λ) = ρ sin(ρt) + t ] k(t, s) sin(ρs)ds, where k(t, s) k(t, s) are continuous function not depending on λ. Hence ψ(t, λ) ψ(t, [ t ] λ) = ρ 1 cos(ρt) + v(t, s) cos(ρs)ds where v(t, s) is a continuous function which does not depend on λ. From (3.) we have (λ) C1 ρ exp( τ π), ρ G δ, (3.1) for some constant C 1 >. Define Ĝ(λ) = 1 π a ρ [ q(t) q(t)]ψ(t, λ) ψ(t, λ)dt = 1 ρ [ ψ (t, λ)ψ(t, λ) ψ(t, λ)ψ (t, λ)] π a. Thus the initial condition (.) W ( ψ n, ψ n ) π a = implies Ĝ(λ n ) =, n N. Using asymptotic form of ψ(t, λ) ψ(t, λ) there is some constant C such that Ĝ(λ) C exp((π a) τ ). (3.11) Define (Ĝ(λ) ) q ϕ(λ) := ( ) (q p). (λ) By applying the same method of proof of Theorem 3.1 using (3.1) (3.11) we deduce that ϕ(λ) is entire bounded so Ĝ(λ) =. Therefore, q(x) = q(x) on (a, π].

11 Inverse problem for Sturm Liouville operators Let l(n) be a subsequence of natural numbers such that l(n) = n σ (1 + ϵ n), < σ 1, ϵ n let µ n µ n be the eigenvalues of problems (3.3), (3) (3.4), (3) with the jump conditions (1.4), respectively, where H H 1. H 1 y (π) + λy(π) = Theorem 3.5. Let a ( π a, π] be a jump point σ > π 1. Let λ n = λ n, µ l(n) = µ l(n) W (y n, ỹ n ) (a ) =, for each n N. Then q(x) = q(x) almost everywhere on [, a) (a, π]. Proof. Let y n (x, λ n ) ỹ n (x, λ n ) be the eigenfunctions of L(q(x); h; H; a) L( q(x); h; H; a) corresponding to the eigenvalues λ n, respectively. From the fact that the eigenfunctions y n (x, λ n ) ỹ n (x, λ n ) have the same boundary condition at point π Theorem 3.4 we conclude that q(x) = q(x) almost everywhere on (a, π], we obtain ỹ n (x, λ n ) = a n y n (x, λ n ), x (a, π], n N, (3.1) where a n is constant. From (3.5), (3.1), (1.) (1.4) assumptions we get G(λ n ) =, G(µ l(n) ) =. Next, we prove G(λ) =, for all λ C. From (3.1) (3.5) we see that the entire function G(λ) is a function of exponential type G(λ) M exp(ar sin θ ) (3.13) where M is a positive number Imλ = r sin θ. Define the indicator of function G(λ) by h(θ) = lim sup λ + ln G(r exp(iθ)). (3.14) r By applying (3.13) (3.14) using Imλ = r sin θ for which θ = arg λ, we obtain h(θ) = a sin θ. (3.15) Let n(r) be the number of zeros of G(λ) in the disk λ r. From Lemma.6 Theorem.7 we see that there are 1 + r[1 + o(1)] of λ n 1 + rσ[1 + o(1)] of µ l(n) located inside the disc of radius r (for sufficiently large r). Therefore Hence n(r) = + r[1 + σ + o(1)]. n(r) lim = (σ + 1). n r

12 118 M. Shahriari, A. Jodayree Akbarfam Using the condition σ > a π 1 from (3.15), we get n(r) lim (σ + 1) > 4a n r π 1 π π h(θ)dθ. (3.16) According to [], for any entire function G(λ) of exponential type, not identically zero, we see that the following inequality holds lim inf n n(r) r 1 π π h(θ)dθ. (3.17) Also from the inequalities (3.16) (3.17) the relation (3.8) holds. By applying the same method of the proof of Theorem 3.1, we obtain q(x) = q(x) almost everywhere on [, a). Let m(n) r(n) be subsequences of natural numbers such that m(n) = n σ 1 (1 + ϵ 1n ), < σ 1 1, ϵ 1n r(n) = n σ (1 + ϵ n ), < σ 1, ϵ n. Corollary 3.6. Let a (, π ] be a jump point fix σ 1 > a π. Let λ m(n) = λ m(n) for each n N W (y m(n), ỹ m(n) ) (a ) =. Then q(x) = q(x) almost everywhere on [, a). Proof. Using the similar proof of Theorems we obtained easily the result of this corollary. Corollary 3.7. Let a ( π a, π) be a jump point, fix σ > π 1 σ > a π. If for each n N λ n = λ n, µ l(n) = µ l(n), W (y r(n), ỹ r(n) ) (a ) =, then q(x) = q(x) almost everywhere on [, π]. Proof. Using the similar proof of Theorems 3.1, 3.5 Corollary 3.6 we obtained easily the result of this corollary. References [1] V. A. Ambartsumyan, Über eine frage der eigenwerttheorie, Z. Phys., 53 (199) [] G. Borg, Eine umkehrung der Sturm Liouvilleschen eigenwertaufgabe, Acta Math., 78 (1945) [3] Chuan-Fu Yang, Yong-Xia Guo, Determination of a differential pencil from interior spectral data, J. Math. Anal. Appl., 375 (11) [4] B. M. Levitan, Inverse Sturm Liouville problems, (VNU Science Press), [5] B. Ya. Levin, Lectures on entire functions, Transl. Math. Monographs, 15, Amer. Math. Soc., Providence, RI, 1996.

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