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On the inverse Sturm-Liouville problem for spatially symmetric operators. I. (English) Zbl 0547.34017
For $$p\in C^ 1[0,1]$$, $$h\in {\mathcal R}$$ and $$H\in {\mathcal R}$$, let $$A=A_{p,h,H}$$ be the Sturm-Liouville operator $$-d^ 2/dx^ 2+p(x)$$ with $$(-d/dx+h)\cdot|_{x=0}=(d/dx+H)\cdot|_{x=1}=0$$. It is said to be symmetric if $$p\in C^ 1_ s[0,1]\equiv\{q| q(1-x)=q(x)$$ (0$$\leq x\leq 1)\}$$ and $$h=H$$. For the symmetric operator $$A=A_{p,h,h}$$, $$\{\lambda_ n=\lambda_ n(p,h)| n=0,1,...\}$$ denotes the totality of its eigenvalues. In 1946, G. Borg proved that for two symmetric operators $$A_{p,h,h}$$ and $$A_{q,j,j}$$, the relation (1) $$\lambda_ n(q,j)=\lambda_ n(p,h) (n=1,2,...)$$, $$j=h$$ implies $$q\equiv p$$. In this article, we show two related theorems. First, the relation (2) $$\lambda_ n(q,j)=\lambda_ n(p,h) (n=0,1,2,...)$$ is equivalent to $$q\equiv p$$, $$j=h$$. Second, for $$n_ 1\geq 1$$, the relation (3) $$\lambda_ n(q,j)=\lambda_ n(p,h) (n\neq n_ 1)$$, $$j=h$$ holds if and only if either $$q=p$$ or $$q=p-2d^ 2/dx^ 2 \log W,$$ where $$W=\phi^*_{n_ 1}{}'\phi_{n_ 1}-\phi^*_{n_ 1}\phi_{n-1}{}'$$. Here $$\phi_ n=\phi_ n(x) (n=0,1,2,...)$$ and $$\phi^*_ n=\phi^*_ n(x) (n=0,1,2,...)$$ denote n-th eigenfunctions of $$A_{p,h,h}$$ and $$A^*_ p$$, respectively, where $$A^*_ p$$ is the Sturm-Liouville operator $$-d^ 2/dx^ 2+p(x)$$ with Dirichlet condition $$\cdot|_{x=0}=\cdot|_{x=1}=0$$.

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)
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References:
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