Compactness of iso-resonant potentials for Schrödinger operators in dimensions one and three

Abstract:

We prove compactness of a restricted set of real-valued, compactly supported potentials $V$ for which the corresponding Schrödinger 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 $\mathbb {R}^d$, for $d=1,3$. Let $\mathcal {I}_R (V_0)$ be the set of real-valued potentials in $C_0^\infty ( \overline {B}_R(0); \mathbb {R})$ so that the corresponding Schrödinger operators have the same resonances, including multiplicities, as $H_{V_0}$, for a fixed, but arbitrary, potential $V_0 \in C_0^\infty ( \overline {B}_R(0); \mathbb {R})$. 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.