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 $R^d$, 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 Schrödinger 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.