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

We prove compactness of a restricted set of real-valued, compactly supported potentials V for which the corresponding Schrödinger operators HV have the same resonances, including multiplicities. More specifically, let BR(0) be the ball of radius R > 0 about the origin in Rd, for d = 1, 3. Let IR(V0) be the set of real-valued potentials in C∞ 0 (BR(0);R) so that the corresponding Schrödinger operators have the same resonances, including multiplicities, as HV0 , for a fixed, but arbitrary, potential V0 ∈ C∞ 0 (BR(0);R). We prove that the set IR(V0) is a compact subset of C∞ 0 (BR(0)) in the C∞-topology. An extension to Sobolev spaces of less regular potentials is discussed.