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Reconstruction of the potential of the Sturm-Liouville equation from three spectra of boundary value problems. (English. Russian original) Zbl 0948.34015
The author considers the following Sturm-Liouville-like problem: $$y''+ (\lambda^2- q-i\lambda p)y= 0\quad\text{on }(0,a),\quad y(0)= y(a)= 0.$$ Here, $\lambda$ is the spectral parameter, $q\in L^2((0,a), \bbfR)$ such that the operator $A$ defined by $Ay= -y''+ qy$ on $\{y\in W^{2,2}((0, a),\bbfR)$; $y(0)= y(a)= 0\}$ is strictly positive, and $p$ is a positive constant $c$ on $(0,b)$ and $0$ on $(b,a)$ for some $b\in (0,a)$. Two associated problems are introduced: one consists of the above differential equation restricted to $(0,b)$ and the boundary condition $y(0)= y(b)= 0$, while the other of the equation $y''+ (\lambda^2- q)y= 0$ on $(b,a)$ and the condition $y(b)= y(a)= 0$. The author gives a procedure for reconstructing, from the spectra of these three problems, the constants $a$, $b$, $c$ and the function $q$ and hence proves the following result: the constants $a$, $b$, $c$ and the function $q$ in the original problem are uniquely determined by the corresponding three spectra.

34B24Sturm-Liouville theory
34A55Inverse problems of ODE
34L40Particular ordinary differential operators
34K29Inverse problems in theory of functional-differential equations
Full Text: DOI
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