On Inverse Problem for Sturm – Liouville Operator with Discontinuous Coefficients

Abstract

В работе доказана единственность восстановления оператора Штурма-Лиувилля c разрывными коэффициентами по спектральным данным и дан алгоритм построения потенциала. Ключевые слова: обратная задача, оператор Штурма-Лиувилля, собственные значения.

On Singular Sturm-Liouville Problem with Impulse

“…Proof of this Theorem 2 can be done by using proof of Theorem in reference [2]. In addition , we obtain the theorem that the statement above, since the eigenfunctions {ϕ(x, k n )} n≥0 are complete and orthogonal in L 2 (0, π), they form an orthogonal basis in L 2 (0, π) and Parseval's equality (3.2) is valid.…”

“…Inverse spectral analysis has been an important research topic in mathematical physics. Inverse problems of spectral analysis involve the reconstruction of a linear operator from its spectral characteristics e.g., see [2,8,14,20]. Later, inverse problems for a regular and singular Sturm-Liouville operator appeared in various versions [ 4,10,12,14,16,17,19,21,22 ].…”

On an inverse spectral problem for Sturm - Liouville operator with discontinuous coefficient

“…It was proved (see [25]), that the solution ϕ(x, λ) of the equation (1.1) with initial conditions ϕ(0, λ) = 1, ϕ ′ (0, λ) = 0 can be represented as…”

On the boundary value problem for the Sturm–Liouville equation with the discontinuous coefficient

“…Note that direct and inverse scattering problems on the half line for the equation of type (1) with various boundary conditions also has been investigated in [29,30] After the half line inverse scattering problem for the equation of type (1) was successfully solved by using the integral representation of the Jost solutions, it is natural to think about the inverse spectral problems in a finite interval. In this aspect, the direct and inverse spectral problem for the equation (1) with Drichlet boundary conditions on the interval .0, / recently has been investigated in [31,32] where the new integral representations for solutions have been also constructed. But in [31], the author has to realize some inconvenient computations of great volume to obtain the final form of the kernel of the integral representation for the solution.…”

“…In this aspect, the direct and inverse spectral problem for the equation (1) with Drichlet boundary conditions on the interval .0, / recently has been investigated in [31,32] where the new integral representations for solutions have been also constructed. But in [31], the author has to realize some inconvenient computations of great volume to obtain the final form of the kernel of the integral representation for the solution. Moreover, the obtained formulas are also inconvenient to investigate the partial derivatives of the kernel which play an important role in solving of the inverse problem.…”