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Half-inverse problem for diffusion operators on the finite interval. (English) Zbl 1116.34010
The following uniqueness theorem is proved: Theorem. Let $\{\lambda_n\}$ be a spectrum of the problems $$-y''+ [q(x)+ 2\lambda p(x)] y=\lambda^2 y,\qquad x\in [0,\pi],$$ $$y'(0)- hy(0)= 0,\quad y'(\pi,\lambda)+ Hy(\pi,\lambda)= 0,$$ and $$-y''+ [\widetilde q(x)+ 2\lambda p(x)]y= \lambda^2 y,\qquad x\in [0,\pi],$$ $$y'(0)- hy(0)= 0,\quad y'(\pi,\lambda)+ Hy(\pi,\lambda)= 0,$$ where $h$, $H$ are finite, $p\in W^{m+1}_2[0,\pi]$, $q,\widetilde q\in W^m_2[0,\pi]$, $m\ge 1$ are real-valued. If $q(x)=\widetilde q(x)$ on $[{\pi\over 2},\pi]$, then $q(x)=\widetilde q(x)$ almost everywhere on $[0,\pi]$.

34A55Inverse problems of ODE
34B24Sturm-Liouville theory
Full Text: DOI
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