Existence of resonances for nontrivial compactly supported potentials on R^n × M

Ascertain whether the meromorphically continued resolvent R_V(z) of P_V = -Δ_X + V on X = R^n × M (n ≥ 3 odd, M compact, V ∈ C_c^∞(R^n × M, R)) has at least one pole on \hat{\mathcal{Z}} when M ≠ S^1; in particular, prove the existence of any resonant pole for such V.

Background

In the one-dimensional cylindrical case (X = R × S^1), existence and distribution of resonances have been established. In contrast, for higher-dimensional product spaces with general compact M, the authors cannot currently demonstrate the occurrence of any resonant pole for smooth compactly supported nonzero potentials.

Establishing even a single resonance would be a first step toward broader lower bound or asymptotic results for resonance counting in this product setting.