We prove compactness of a restricted set of real-valued, compactly supported potentials $V$ for which the corresponding Schr\"odinger operators $H_V$ have the same resonances, including multiplicities. More specifically, let $B_R(0)$ be the ball of radius $R > 0$ about the origin in $R^d$, for $d=1,3$. Let $\mathcal{I}_R (V_0)$ be the set of real-valued potentials in $C_0^\infty( \overline{B}_R(0); R)$ so that the corresponding Schr\"odinger operators have the same resonances, including multiplicities, as $H_{V_0}$. We prove that the set $\mathcal{I}_R (V_0)$ is a compact subset of $C_0^\infty (\overline{B}_R(0))$ in the $C^\infty$-topology. An extension to Sobolev spaces of less regular potentials is discussed.

中文翻译：

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一维和三维薛定谔算子的等共振势的紧致性

我们证明了一组受限制的实值、紧支持势 $V$ 的紧致性，其中相应的 Schr\"odinger 算子 $H_V$ 具有相同的共振，包括多重性。更具体地说，让 $B_R(0)$ 是半径为 $R > 0$ 的球在 $R^d$ 的原点附近，对于 $d=1,3$。令 $\mathcal{I}_R (V_0)$ 是 $C_0 中实值势的集合^\infty( \overline{B}_R(0); R)$ 使得相应的 Schr\"odinger 算子具有与 $H_{V_0}$ 相同的共振，包括多重性。我们证明了集合 $\mathcal{I}_R (V_0)$ 是 $C^\infty$ 拓扑中 $C_0^\infty (\overline{B}_R(0))$ 的紧凑子集。讨论了对不规则势的 Sobolev 空间的扩展。