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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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