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On determination of Sturm-Liouville operator with discontinuity conditions with respect to spectral data. (English) Zbl 1368.34027

From the introduction: Let us consider the equation \[ -y''+ q(x)y=\lambda y,\quad 0<x<\pi,\tag{1} \] with discontinuity conditions at a point \(a\in (0,\pi)\) \[ y(a+0)=\alpha y(a-0),\quad y'(a+ 0)= \alpha^{-1}y'(a-0),\tag{2} \] and boundary conditions \[ y(0)= y(\pi)= 0.\tag{3} \] Here \(\lambda\) is a complex parameter \(q(x)\), \(\alpha\) are real; \(q(x)\in L_2(0,\pi)\), \(\alpha\neq 1\), \(\alpha>0\).
Let \(S(x,\lambda)\) be the solution of (1) with discontinuity conditions (2) and initial conditions \(S(0,\lambda)=0\), \(s'(0,\lambda)=1\).
Denote by \(\lambda_n\) the eigenvalues and by \(\alpha_n\) the normalized numbers of (1)–(2): \[ \alpha_n= \int^\pi_0 S^2(x,\lambda_n)\,dx. \] The numbers \(\{\lambda_n,\alpha_n\}\) are said to be spectral data of (1)–(3).
We are interested in the following inverse problem: determine the function \(q(x)\) in (1) with respect to spectral data \(\{\lambda_n,\alpha_n\}\) of (1)–(3).

MSC:

34A55 Inverse problems involving ordinary differential equations
34B09 Boundary eigenvalue problems for ordinary differential equations
34B24 Sturm-Liouville theory
34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators
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