Thông tin tài liệu
Hindawi Publishing Corporation Boundary Value Problems Volume 2010, Article ID 171967, 17 pages doi:10.1155/2010/171967 Research Article On an Inverse Scattering Problem for a Discontinuous Sturm-Liouville Equation with a Spectral Parameter in the Boundary Condition Khanlar R. Mamedov Mathematics Department, Science and Letters Faculty, Mersin University, 33343 Mersin, Turkey Correspondence should be addressed to Khanlar R. Mamedov, hanlar@mersin.edu.tr Received 9 April 2010; Accepted 22 May 2010 Academic Editor: Michel C. Chipot Copyright q 2010 Khanlar R. Mamedov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. An inverse scattering problem is considered for a discontinuous Sturm-Liouville equation on the half-line 0, ∞ with a linear spectral parameter in the boundary condition. The scattering data of the problem are defined and a new fundamental equation is derived, which is different from the classical Marchenko equation. With help of this fundamental equation, in terms of the scattering data, the potential is recovered uniquely. 1. Introduction We consider inverse scattering problem for the equation −ψ q x ψ λ 2 ρ x ψ 0 <x<∞ , 1.1 with the boundary condition − α 1 ψ 0 − α 2 ψ 0 λ 2 β 1 ψ 0 − β 2 ψ 0 , 1.2 where λ is a spectral parameter, qx is a real-valued function satisfying the condition ∞ 0 1 x q x dx < ∞, 1.3 ρx is a positive piecewise-constant function with a finite number of points of discontinuity, α i ,β i i 1, 2 are real numbers, and γ α 1 β 2 − α 2 β 1 > 0. 2 Boundary Value Problems The aim of the present paper is to investigate the direct and inverse scattering problem on the half-line 0, ∞ for the boundary value problem 1.1–1.3. In the case ρx ≡ 1, the inverse problem of scattering theory for 1.1 with boundary condition not containing spectral parameter was completely solved by Marchenko 1, 2,Levitan3, 4, Aktosun 5, as well as Aktosun and Weder 6. The discontinuous version was studied by Gasymov 7 and Darwish 8. In these papers, solution of inverse scattering problem on the half-line 0, ∞ by using the transformation operator was reduced to solution of two inverse problems on the intervals 0,a and a, ∞. In the case ρx / 1, the inverse scattering problem was solved by Guse ˘ ınov and Pashaev 9 by using the new nontriangular representation of Jost solution of 1.1. It turns out that in this case the discontinuity of the function ρ x strongly influences the structure of representation of the Jost solution and the fundamental equation of the inverse problem. We note that similar cases do not arise for the system of Dirac equations with discontinuous coefficients in 10. Uniqueness of the solution of the inverse problem and geophysical application of this problem for 1.1 when qx ≡ 0 were given by Tihonov 11 and Alimov 12. Inverse problem for a wave equation with a piecewise-constant coefficient was solved by Lavrent’ev 13. Direct problem of scattering theory for the boundary value problem 1.1–1.3 in the special case was studied in 14. When ρx ≡ 1in1.1 with the spectral parameter appearing in the boundary conditions, the inverse problem on the half-line was considered by Pocheykina-Fedotova 15 according to spectral function, by Yurko 16–18 according to Weyl function, and according to scattering data in 19, 20. This type of boundary condition arises from a varied assortment of physical problems and other applied problems such as the study of heat conduction by Cohen 21 and wave equation by Yurko 16, 17. Spectral analysis of the problem on the half-line was studied by Fulton 22. Also, physical application of the problem with the linear spectral parameter appearing in the boundary conditions on the finite interval was given by Fulton 23 . We recall that inverse spectral problems in finite interval for Sturm-Liouville operators with linear or nonlinear dependence on the spectral parameter in t he boundary conditions were studied by Chernozhukova and Freiling 24, Chugunova 25, Rundell and Sacks 26, Guliyev 27, and other works cited therein. This paper is organized as follows. In Section 2, the scattering data for the boundary value problem 1.1–1.3 are defined. In Section 3, the fundamental equation for the inverse problem is obtained and the continuity of the scattering function is showed. Finally, the uniqueness of solution of the inverse problem is given in Section 4. For simplicity we assume that in 1.1 the function ρx has a discontinuity point: ρ x ⎧ ⎨ ⎩ α 2 , 0 ≤ x<a, 1,x≥ a, 1.4 where 0 <α / 1. The function f 0 x, λ 1 2 1 1 ρ x e iλμ x 1 2 1 − 1 ρ x e iλμ − x , 1.5 is the Jost solution of 1.1 when qx ≡ 0, where μ ± x±x ρxa1 ∓ ρx. Boundary Value Problems 3 It is well known 9 that, for all λ from the closed upper half-plane, 1.1 has a unique Jost solution fx, λ which satisfies the condition lim x →∞ f x, λ e −iλx 1 1.6 and it can be represented in the form f x, λ f 0 x, λ ∞ μ x K x, t e iλt dt, 1.7 where the kernel Kx, t satisfies the inequality ∞ μ x | K x, t | dt ≤ C exp ∞ x t q t dt , 0 <C const, 1.8 and possesses the following properties: dK x, μ x dx − 1 4 ρ x 1 1 ρ x q x , 1.9 d dx K x, μ − x 0 − K x, μ − x − 0 1 4 ρ x 1 − 1 ρ x q x . 1.10 In addition, if qx is differentiable, Kx, t satisfies a.e. the equation ρ x ∂ 2 K ∂t 2 − ∂ 2 K ∂x 2 q x K 0, 0 <x<∞,t>μ x . 1.11 Denote that ϕ λ α 2 β 2 λ 2 f 0,λ − α 1 β 1 λ 2 f 0,λ . 1.12 According to Lemma 2.2 in Section 2, the equation ϕλ0 has only a finite number of simple roots in the half-plane Im λ>0; all these roots lie in the imaginary axis. The behavior of this boundary value problem 1.1–1.3 is expressed as a self-adjoint eigenvalue problem. We will call the function S λ α 2 β 2 λ 2 f 0,λ − α 1 β 1 λ 2 f 0,λ α 2 β 2 λ 2 f 0,λ − α 1 β 1 λ 2 f 0,λ 1.13 the scattering function for the boundary value problem 1.1–1.3, where f0,λ denotes the complex conjugate of f0,λ. 4 Boundary Value Problems We denote by m −2 k the normalized numbers for the boundary problem 1.1–1.3: m −2 k ≡ ∞ 0 ρ x f x, iλ k 2 dx 1 γ β 2 f 0,iλ k − β 1 f 0,iλ k 2 , 1.14 where k 1, 2, ,n. It turns out that the potential qx in the boundary value problem 1.1– 1.3 is uniquely determined by specifying the set of values {Sλ,λ k ,m k }. The set of values is called the scattering data of the boundary value problem 1.1–1.3. The inverse scattering problem for boundary value problem 1.1–1.3 consists in recovering the coefficient qx from the scattering data. The potential qx is constructed by slightly varying the method of Marchenko. Set F 0 x 1 2π ∞ −∞ S 0 λ − S λ e −iλx dλ n k1 m 2 k e −λ k x , F x, y 1 2 1 1 ρ x F 0 y μ x 1 2 1 − 1 ρ x F 0 y μ − x , 1.15 where S 0 λ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ f 0 0,λ f 0 0,λ e −2iλa 1 τe −2iλaα e −2iλaα τ , if β 2 0, f 0 0,λ f 0 0,λ −e −2iλa 1 − τe −2iλaα e −2iλaα − τ , if β 2 / 0, 1.16 and τ α − 1/α 1. We can write out the integral equation F x, y ∞ μ x K x, t F 0 t y dt K x, y 1 − ρ x 1 ρ x K x, 2a − y 0, 1.17 for the unknown function Kx, t. The integral equation is called the fundamental equation of the inverse problem of scattering theory for the boundary problem 1.1–1.3.The fundamental equation is different from the classic equation of Marchenko and we call the equation the modified Marchenko equation. The discontinuity of the function ρx strongly influences the structure of the fundamental equation of the boundary problem 1.1–1.3.By Theorem 4.1 in Section 4, the integral equation has a unique solution for every x ≥ 0. Solving this equation, we find the kernel Kx, y of the special solution 1.7, and hence according to formula 1.10 it is constructed the potential qx. We show that formula 1.7 is valid for 1.1. For this, let us give the algorithm of the proof in 9. For fx, λ let us consider the integral equation f x, λ f 0 x, λ ∞ x Φ x, t, λ q t f t, λ dt, 1.18 Boundary Value Problems 5 where Φ x, t, λ s 0 t, λ c 0 x, λ − s 0 x, λ c 0 t, λ , 1.19 while s 0 x, λ and c 0 x, λ are solutions of 1.1 when qx ≡ 0, satisfying the initial conditions c 0 0,λs 0 0,λ1andc 0 0,λs 0 0,λ0. It is not hard to show that the function Φx, t, λ satisfies the formula Φ x, t, λ σx,t −σx,t K 0 x, t, z e iλz dz, 1.20 where K 0 x, t, z ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1 2α , | z | ≤ σ x, t ,x≤ t ≤ a, 1 4 1 1 α ,t− a − α a − x ≤ | z | ≤ σ x, t ,x≤ a ≤ t, 1 2 , | z | ≤ t − a − α a − x ,x≤ a ≤ t, 1 2 , | z | ≤ σ x, t ,t≥ x ≥ a, σ x, t t x ρ s ds ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ α t − x ,x≤ t ≤ a, α a − x t − a, x ≤ a ≤ t, t − x, a ≤ x ≤ t. 1.21 Substituting the expression 1.7 for fx, λ in the integral equation 1.18 and using formula 1.20 for Φx, t, λ after elementary operations, the following integral equations for the kernel Kx, t are obtained: K x, t 1 4α 1 1 α a αxαa−at/2α q z dz 1 4α 1 − 1 α a αxαaa−t/2α q z dz 1 4 1 1 α ∞ a q z dz − 1 4 1 − 1 α t−αxαaa/2 a q z dz 1 2α mint,ααa−a/α x q z tαz−x t−αz−x K z, s ds dz − 1 4 taαa−αx/2 a q z t−z−a−αaαx tz−a−αaαx K z, s ds dz, 1.22 6 Boundary Value Problems for 0 <x<a, αx − αa a<t<−αx αa a; K x, t 1 4 1 1 α ∞ tαx−αaa/2 q z dz 1 4 1 − 1 α ∞ t−αxαaa/2 q z dz 1 2α a x q z tαz−x t−az−x K z, s ds dz − 1 4 1 − 1 α aαa−αx a q z t−zaαa−αx tz−a−αaαx K z, s ds dz 1 4 1 − 1 α ∞ aαa−αx q z tz−a−αaαx t−zaαa−αx K z, s ds dz, 1.23 for 0 <x<a,t>−αx αa a; K x, t 1 2 ∞ xt/2 q z dz 1 2 ∞ x q z dz tz−x t−z−x K z, s ds, 1.24 for t ≥ x ≥ a. The solvability of these integral equations is obtained through the method of successive approximations. By using integral equations 1.22–1.24 for Kx, t, equalities 1.9, 1.10 are obtained. By substituting the expressions for the functions fx, λ and f x, λ in 1.1, it can be shown that 1.11 holds. 2. The Scattering Data For real λ / 0, the functions fx, λ and fx, λ form a fundamental system of solutions of 1.1 and their Wronskian is computed as W{fx, λ, fx, λ} 2iλ. Here the Wronskian is defined as W{f, g} f g − fg . Let ωx, λ be the solution of 1.1 satisfying the initial condition ω 0,λ α 2 β 2 λ 2 ,ω 0,λ α 1 β 1 λ 2 . 2.1 The following assertion is valid. Lemma 2.1. The identity 2iλω x, λ α 2 β 2 λ 2 f 0,λ − α 1 β 1 λ 2 f 0,λ f x, λ − S λ f x, λ 2.2 holds for all real λ / 0,where S λ α 2 β 2 λ 2 f 0,λ − α 1 β 1 λ 2 f 0,λ α 2 β 2 λ 2 f 0,λ − α 1 β 1 λ 2 f 0,λ 2.3 Boundary Value Problems 7 with S λ S −λ S −λ −1 . 2.4 The function Sλ is called the scattering function of the boundary value problem 1.1– 1.3. Lemma 2.2. The function ϕλ may have only a finite number of zeros in the half-plane Im λ>0. Moreover, all these zeros are simple and lie in the imaginary axis. Proof. Since ϕλ / 0 for all real λ / 0, the point λ 0 is the possible real zero of the function ϕλ. Using the analyticity of the function ϕλ in upper half-plane and the properties of solution 1.7 are obtained that zeros of ϕλ form at most countable and bounded set having 0 as the only possible limit point. Now let us show that all zeros of the function ϕλ lie on the imaginary axis. Suppose that μ 1 and μ 2 are arbitrary zeros of the function ϕλ. We consider the following relations: −f x, μ 1 q x f x, μ 1 μ 2 1 ρ x f x, μ 1 , − f x, μ 2 q x f x, μ 2 μ 2 2 ρ x f x, μ 2 . 2.5 Multiplying the first of these relations by fx, μ 2 and the second by fx, μ 1 , subtracting the second resulting relation from the first, and integrating the resulting difference from zero to infinity, we obtain μ 2 1 − μ 2 2 ∞ 0 ρ x f x, μ 1 f x, μ 2 dx − W f x, μ 1 , f x, μ 2 x0 0. 2.6 On the other hand, according to the definition of the function ϕλ, the following relation holds: ϕ μ j α 2 β 2 μ 2 j f 0,μ j − α 1 β 1 μ 2 j f 0,μ j 0,j 1, 2. 2.7 Therefore, f x, μ j 1 γ β 2 f 0,μ j − β 1 f 0,μ j ω x, μ j ,j 1, 2. 2.8 This formula yields W f x, μ 1 , f x, μ 2 x0 1 γ β 2 f 0,μ 1 − β 1 f 0,μ 1 × β 2 f 0,μ 2 − β 1 f 0,μ 2 μ 2 2 − μ 2 1 . 2.9 8 Boundary Value Problems Thus, using 2.6 and 2.9 we have μ 2 1 − μ 2 2 ∞ 0 ρ x f x, μ 1 f x, μ 2 dx 1 γ β 2 f 0,μ 1 − β 1 f 0,μ 1 × β 2 f 0,μ 2 − β 1 f 0,μ 2 0. 2.10 Here ρx > 0, γ>0. In particular, the choice μ 2 μ 1 at 2.10 implies that μ 2 1 − μ 1 2 0, or μ 1 iλ 1 , where λ 1 ≥ 0. Therefore, zeros of the function ϕλ can lie only on the imaginary axis. Now, let us now prove that function ϕλ has zeros in finite numbers. This is obvious if ϕ0 / 0, because, under this assumption, the set of zeros cannot have limit points. In the general case, since we can give an estimate for the distance between the neighboring zeros of the function ϕλ, it follows that the number of zeros is finite see 2, page 186. Let m −2 k ≡ ∞ 0 ρ x f x, iλ k 2 dx 1 γ β 2 f 0,iλ k − β 1 f 0,iλ k 2 1 2iμ k γ ϕ iλ k β 2 f 0,iλ k − β 1 f 0,iλ k ,k 1, 2, ,n. 2.11 These numbers are called the normalized numbers for the boundary problem 1.1–1.3. The collections {Sλ, −∞ <λ<∞; λ k ; m k k 1, 2, ,n} are called the scattering data of the boundary value problem 1.1–1.3. The inverse scattering problem consists in recovering the coefficient qx from the scattering data. 3. Fundamental Equation or Modified Marchenko Equation From 1.9, 1.10, it is clear that in order to determine qx it is sufficient to know Kx, t.To derive the fundamental equation for the kernel Kx, t of the solution 1.7, we use equality 2.2, which was obtained in Lemma 2.1. Substituting expression 1.7 for fx, λ into this equality, we get 2iλω x, λ ϕ λ − f 0 x, λ S 0 λ f 0 x, λ ∞ μ x K x, t e −iλt dt S 0 λ − S λ f 0 x, λ ∞ μ x K x, t S 0 λ − S λ e iλt dt − S 0 λ ∞ μ x K x, t e −iλt dt. 3.1 Boundary Value Problems 9 Multiplying both sides of relation 3.1 by 1/2πe iλy and integrating over λ from −∞ to ∞, for y>μ x at the right-hand side we get K x, y 1 2π ∞ −∞ S 0 λ − S λ f 0 x, λ e iλy dλ ∞ μ x K x, t 1 2π ∞ −∞ S 0 λ − S λ e iλty dλ dt − ∞ μ x K x, t 1 2π ∞ −∞ S 0 λ e iλty dλ dt. 3.2 Now we will compute the integral 1/2π ∞ −∞ S 0 λe iλty dλ. By elementary transforms we obtain S 0 λ e −2iλa 1 − τ 2 e 2iλaα 1 τe 2iλaα τe −2iλa e −2iλa1−α 1 − τ 2 ∞ k0 −1 k τ k e 2iλaαk τe −2iλa , 3.3 where β 2 0. Thus we have 1 2π ∞ −∞ S 0 λ e iλty dλ 1 − τ 2 ∞ k0 −1 k τ k δ t y − 2a 1 − α 2aαk τδ t y − 2a , 3.4 where δt is the Dirac delta function. For β 2 / 0, similarly we get 1 2π ∞ −∞ S 0 λ e iλty dλ τ 2 − 1 ∞ k0 −1 k τ k δ t y − 2a 1 − α 2aαk τδ t y − 2a . 3.5 Consequently, 3.2 can be written as K x, y F S x, y ∞ μ x K x, t F 0S t y dt − τK x, 2a − y − 1 − τ 2 ∞ k0 −1 k τ k K x, 2a 1 − α − 2aαk − y , 3.6 10 Boundary Value Problems where F 0S x ≡ 1 2π ∞ −∞ S 0 λ − S λ e iλx dλ, F S x, y ≡ 1 2 1 1 ρ x F 0S μ x y 1 2 1 − 1 ρ x F 0S μ − x y . 3.7 Let us show that for y>μ x the last expression in the sum equals zero. We note that Kx, z0forz<x. For y>μ x we have 2a 1 − α − 2aαk − y<μ x ,k 0, 1, 2, 3.8 If 0 <x<a,then μ xαx − αa a, and hence 2a 1 − α − 2aαk − y<2a − 2aα k 1 − αx αa − a a − aα − 2aαk − αx < a 1 − α ≤ μ x . 3.9 If x ≥ a, then μ xx, and hence, for this case, the inequality holds. Therefore, for y>μ x3.2 takes the form K x, y F S x, y ∞ μ x K x, t F 0S t y dt 1 − ρ x 1 ρ x K x, 2a − y . 3.10 On the left-hand side of 3.1 with help of Jordan’s lemma and the residue theorem and by taking Lemma 2.2 into account for y>μ x, we obtain − n k1 2iλ k ω x, iλ k ϕ iλ k e −λ k y . 3.11 From the definition of normalized numbers m k k 1, 2, ,n in 2.11 we have − n k1 2iλ k ω x, iλ k e −λ k y ϕ iλ k − n k1 2iλ k e −λ k y f x, iλ k β 2 f 0,iλ k − β 1 f 0,iλ k ϕ iλ k − n k1 m 2 k f x, iλ k e −λ k y − n k1 m 2 k f 0 x, iλ k e −λ k xy ∞ μ x K x, t e −λ k ty dt . 3.12 [...]... operator with a spectral parameter in the boundary condition,” Mathematical Notes, vol 74, no 1-2, pp 136–140, 2003 20 Kh R Mamedov, On the inverse problem for Sturm-Liouville operator with a nonlinear spectral parameter in the boundary condition,” Journal of the Korean Mathematical Society, vol 46, no 6, pp 1243–1254, 2009 21 D S Cohen, An integral transform associated with boundary conditions containing... F0 y μ− x Equation 3.14 is called the fundamental equation of the inverse problem of the scattering theory for the boundary problem 1.1 – 1.3 The fundamental equation is different from the classic equation of Marchenko and we call equation 3.14 the modified Marchenko equation The discontinuity of the function ρ x strongly in uences the structure of the fundamental equation of the boundary problem 1.1... discrete spectra for the radial Schrodinger equation, ” Inverse Problems, vol 22, no 1, pp 89–114, 2006 ¨ 7 M G Gasymov, The direct and inverse problem of spectral analysis for a class of equations with a discontinuous coefficient,” in Non-Classical Methods in Geophysics, M M Lavrent’ev, Ed., pp 37–44, Nauka, Novosibirsk, Russia, 1977 8 A A Darwish, The inverse problem for a singular boundary value problem, ”... containing an eigenvalue parameter, ” SIAM Journal on Applied Mathematics, vol 14, pp 1164–1175, 1966 22 C T Fulton, “Singular eigenvalue problems with eigenvalue parameter contained in the boundary conditions,” Proceedings of the Royal Society of Edinburgh Section A, vol 87, no 1-2, pp 1–34, 1980/81 23 C T Fulton, “Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions,”... Zeland Journal of Mathematics, vol 22, pp 37–66, 1993 9 I M Guse˘nov and R T Pashaev, On an inverse problem for a second-order differential equation, ” ı Uspekhi Matematicheskikh Nauk, vol 57, no 3 345 , pp 147–148, 2002 10 Kh R Mamedov and A Col, On the inverse problem of the scattering theory for a class of systems ¸¨ of Dirac equations with discontinuous coefficient,” European Journal of Pure and Applied... Sturm-Liouville operator with eigenvalue parameter dependent boundary conditions,” in Operator Theory, System Theory and Related Topics (BeerSheva/Rehovot, 1997), vol 123 of Operator Theory Advances and Applications, pp 187–194, Birkh¨ user, a Basel, Switzerland, 2001 26 W Rundell and P Sacks, “Numerical technique for the inverse resonance problem, ” Journal of Computational and Applied Mathematics, vol 170,... Lavrent’ev Jr., An inverse problem for the wave equation with a piecewise-constant coefficient,” Sibirski˘ Matematicheski˘ Zhurnal, vol 33, no 3, pp 101–111, 219, 1992, translation in ı ı Siberian Mathematical Journal, vol 33, no 3, pp 452–461, 1992 14 Kh R Mamedov and N Palamut, On a direct problem of scattering theory for a class of SturmLiouville operator with discontinuous coefficient,” Proceedings... conditions,” Proceedings of the Royal Society of Edinburgh Section A, vol 77, no 3-4, pp 293– 308, 1977 24 A Chernozhukova and G Freiling, A uniqueness theorem for the boundary value problems with non-linear dependence on the spectral parameter in the boundary conditions,” Inverse Problems in Science and Engineering, vol 17, no 6, pp 777–785, 2009 25 M V Chugunova, Inverse spectral problem for the Sturm-Liouville. .. Levitan, On the solution of the inverse problem of quantum scattering theory,” Mathematical Notes, vol 17, no 4, pp 611–624, 1975 4 B M Levitan, Inverse Sturm-Liouville problems, VSP, Zeist, The Netherlands, 1987 5 T Aktosun, “Construction of the half-line potential from the Jost function,” Inverse Problems, vol 20, no 3, pp 859–876, 2004 6 T Aktosun and R Weder, Inverse spectral -scattering problem with. .. fundamental equation 3.14 , it suffices to know the functions F0 x and F x, y In turn, to find the functions F0 x , F x, y , it suffices to know only the 1, 2, , n } Given the scattering data, scattering data {S λ −∞ < λ < ∞ ; λk , mk k 16 we can use formulas 3.15 to fundamental equation 3.14 for the fundamental equation has a K x, y of the special solution constructed the potential q x Boundary Value . cited. An inverse scattering problem is considered for a discontinuous Sturm-Liouville equation on the half-line 0, ∞ with a linear spectral parameter in the boundary condition. The scattering data. An integral transform associated with boundary conditions containing an eigenvalue parameter, ” SIAM Journal on Applied Mathematics, vol. 14, pp. 1164–1175, 1966. 22 C. T. Fulton, “Singular. 1977. 24 A. Chernozhukova and G. Freiling, A uniqueness theorem for the boundary value problems with non-linear dependence on the spectral parameter in the boundary conditions,” Inverse Problems in Science
Ngày đăng: 21/06/2014, 16:20
Xem thêm: Báo cáo sinh học: " Research Article On an Inverse Scattering Problem for a Discontinuous Sturm-Liouville Equation with a Spectral Parameter in the Boundary Condition" potx, Báo cáo sinh học: " Research Article On an Inverse Scattering Problem for a Discontinuous Sturm-Liouville Equation with a Spectral Parameter in the Boundary Condition" potx