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Uniqueness of reconstruction of binomial differential operators from two spectra. (English) Zbl 0659.34028
Two theorems are presented. Let \({}^ i\lambda_{kj}\) \((i,j=1,2;k=1,2,...)\) be eigenvalues of the problems \[ y^{(2n)}+q_ i(t)y=\lambda y,\quad 0\leq t\leq \pi,\quad n\geq 2, \] \[ (1)\quad y(0)=y'(0)=...=y^{(2n-2)}(0)=y^{(j-1)}(\pi)=0, \] where \(q_ i\) are sectionally analytic functions on \(<0,\pi >\). If \({}^ 1\lambda_{kj}=^ 2\lambda_{kj}\) for \(j=1,2\) and \(k=1,2,...\), then \(q_ 1=q_ 2\). let \({}^ i\mu_{kj}\) \((i,j=1,2\); \(k=1,2,...)\) be eigenvalues of the boundary value problems for equations \(y^{(2n)}+\sum^{2n-2}_{\nu =0}q_{ir}(t)y^{(\nu)}=\lambda y\) with boundary conditions (1), where \(q_{i\nu}\) \((i=1,2;\nu =0,1,....,2n-2)\) are sectionally analytic function on \(<0,\pi >\). Let \(\rho\) be a positive integer, \(0\leq \rho \leq 2n-2\). If \({}^ 1\mu_{kj}=^ 2\mu_{kj}\) for \(j=1,2\), \(k=1,2,..\). and \(q_{1\nu}=q_{2\nu}\) for all \(\nu\),\(\nu\neq \rho\), then \(q_{1\rho}=q_{2\rho}\).
Reviewer: S.Stanek

34L99 Ordinary differential operators
47E05 General theory of ordinary differential operators (should also be assigned at least one other classification number in Section 47-XX)
34A30 Linear ordinary differential equations and systems, general