Resonant rigidity for Schr\"odinger operators in even dimensions

This paper studies the resonances of Schr\"odinger operators with bounded, compactly supported, real-valued potentials on d-dimensional Euclidean space, where d is even. If the potential V is non-trivial and d is not 4 then the meromorphic continuation of the resolvent of the Schr\"odinger operator has infinitely many poles, with a quantitative lower bound on their density. A somewhat weaker statement holds if d =4. We prove several inverse-type results. If the meromorphic continuations of the resolvents of two Schr\"odinger operators $-\Delta +V_1$ and $-\Delta +V_2$ have the same poles, with both potentials bounded, compactly supported and real-valued, if k is a natural number and if $V_1\in H^k({\mathbb R}^d; {\mathbb R})$, then $V_2\in H^k$ as well. Moreover, we prove that certain sets of isoresonant potentials are compact. We also show that the poles of the resolvent for a smooth potential determine the heat coefficients and that the (resolvent) resonance sets of two bounded, real-valued potentials with compact support cannot differ by a nonzero finite number of elements away from $0$.