Compactness of iso-resonant potentials for Schr\"odinger operators in dimensions one and three

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.