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Description

In joint work with R.\ Wolf, 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 $d=1$ or $d=3$. Let $\mathcal{I}_R (V_0)$ be the set of real-valued potentials in $C_0^\infty( B_R(0))$ so that the corresponding Schr\"odinger operators have the same resonances, including multiplicities, as $H_{V_0}$. We prove that the iso-resonant set $\mathcal{I}_R (V_0)$ is a compact subset of $C_0^\infty (B_R(0))$ in the $C^\infty$-topology.