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 Sfb 288 Differential Geometry and Quantum Physics

Abstract for Sfb Preprint No. 598

A remark on the inverse Sturm-Liouville problem with mixed boundary conditions

E. Korotyaev

Consider the operator $Tu=-u''+q(x)u,\ u(0)=u'(1)=0$ acting on the Hilbert space $L_C^2(0,1)$, where the real potential $q\in H=\{q\in L^2(0,1), \int_0^1 q(x)dx=0\}$. Let $\m_n(q),n\ge 1$ be the eigenvalues of $T$ and $\n_n(q)$ be the so-called norming constants. We show that the mapping $\P: H\to M\ts \ell_1^2$ given by $q\to \P(q)=\{\m_n(q), \n_{n+1}(q)\}_1^\iy$ is a real analytic isomorphism between $H$ and $\P(H)$, where $M=\{\m=\{\m_n\}_1^\iy,\ \ \m_1<\m_2<.., \m_n=\pi^2(n-{1\/2})^2+c_n,\ \ c=\{c_n\}_0^\iy\!\in\ell^2\}$. In fact, we do not use the norming constant $\n_1$.

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Copyright © 1999 Sfb 288, Mathematics 8-5, Strasse des 17 Juni 136, TU-Berlin, 10623 Berlin
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