We discuss inverse resonance scattering for the Laplacian on a rotationally symmetric manifold $M = (0,\infty) \times Y$ whose rotation radius is constant outside some compact interval. The Laplacian on $M$ is unitarily equivalent to a direct sum of one-dimensional Schrodinger operators with compactly supported potentials on the half-line. We prove o Asymptotics of counting function of resonances at large radius o Inverse problem: The rotation radius is uniquely determined by its eigenvalues and resonances. Moreover, there exists an algorithm to recover the rotation radius from its eigenvalues and resonances. The proof is based on some non-linear real analytic isomorphism between two Hilbert spaces.

中文翻译：

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旋转对称流形上的逆共振散射

我们讨论了旋转对称流形 $M = (0,\infty) \times Y$ 上拉普拉斯算子的逆共振散射，其旋转半径在某个紧凑区间外是恒定的。$M$ 上的拉普拉斯算子整体上等价于在半线上具有紧支持势的一维薛定谔算子的直接和。我们证明了 o 大半径共振计数函数的渐近性 o 逆问题：旋转半径由其特征值和共振唯一确定。此外，存在一种从其特征值和共振中恢复旋转半径的算法。该证明基于两个希尔伯特空间之间的一些非线性实解析同构。