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Reconstruction of the potential of the Sturm-Liouville equation from three spectra of boundary value problems. (English) Zbl 0948.34015

The author considers the following Sturm-Liouville-like problem:

${y}^{\text{'}\text{'}}+\left({\lambda }^{2}-q-i\lambda p\right)y=0\phantom{\rule{1.em}{0ex}}\text{on}\phantom{\rule{4.pt}{0ex}}\left(0,a\right),\phantom{\rule{1.em}{0ex}}y\left(0\right)=y\left(a\right)=0·$

Here, $\lambda$ is the spectral parameter, $q\in {L}^{2}\left(\left(0,a\right),ℝ\right)$ such that the operator $A$ defined by $Ay=-{y}^{\text{'}\text{'}}+qy$ on $\left\{y\in {W}^{2,2}\left(\left(0,a\right),ℝ\right)$; $y\left(0\right)=y\left(a\right)=0\right\}$ is strictly positive, and $p$ is a positive constant $c$ on $\left(0,b\right)$ and 0 on $\left(b,a\right)$ for some $b\in \left(0,a\right)$.

Two associated problems are introduced: one consists of the above differential equation restricted to $\left(0,b\right)$ and the boundary condition $y\left(0\right)=y\left(b\right)=0$, while the other of the equation ${y}^{\text{'}\text{'}}+\left({\lambda }^{2}-q\right)y=0$ on $\left(b,a\right)$ and the condition $y\left(b\right)=y\left(a\right)=0$.

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.

##### MSC:
 34B24 Sturm-Liouville theory 34A55 Inverse problems of ODE 34L40 Particular ordinary differential operators 34K29 Inverse problems in theory of functional-differential equations
##### References:
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