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A uniqueness theorem for inverse eigenparameter dependent Sturm-Liouville problems. (English) Zbl 0890.34022
This paper deals with the inverse spectral problem for a regular Sturm-Liouville problem with boundary conditions depending on the eigenvalue parameter. More exactly, a uniqueness theorem is proved for second order boundary eigenvalue problems $$-y''+ qy= \lambda y\quad \text{on } [0,1],$$ $$\cos \alpha y(0)+ \sin\alpha y'(0)=0, \quad (a \lambda +b) y(1)= (c\lambda +d)y'(1).$$ It is assumed that the potential $q$ is integrable, $\alpha \in (0,\pi)$, $ad- bc>0$ and $c\ne 0$. It is shown that two problems of the above form for which the eigenvalues and suitable norming constants coincide must have equal potentials. This result is an extension of a well-known theorem of Gel’fand and Levitan for Sturm-Liouville problems with standard boundary conditions to the interesting case of boundary conditions containing the eigenvalue parameter linearly.

34B24Sturm-Liouville theory
34A55Inverse problems of ODE
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