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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

##### MSC:
 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
##### Keywords:
spectrum of a differential operator; eigenvalues