Upper bounds on resonance counts for R_V(z) on the product space R^n × M

Derive upper bounds for the number of poles (resonances) of the meromorphically continued resolvent R_V(z) of the Schrödinger operator P_V = -Δ_X + V on X = R^n × M (with n ≥ 3 odd, M compact, and V ∈ C_c^∞(R^n × M, R)) either in neighborhoods of the real axis or within specified sheets of the Riemann surface \hat{\mathcal{Z}} on which the square-roots \sqrt{z - \sigma_k^2} are defined.

Background

The paper studies potential scattering for P_V on the product space X = R^n × M and develops a Birman–Krein formula adapted to the presence of threshold energies {±σ_k}. Resonances are defined as poles of the meromorphic continuation of the resolvent R_V(z) to a multi-sheeted Riemann surface \hat{\mathcal{Z}} determined by the square roots \sqrt{z - σ_k^2}.

While sharp resonance counting is known in the one-dimensional cylindrical case (n = 1), the geometry of \hat{\mathcal{Z}} in higher dimensions obstructs the standard complex-analytic zero counting methods used in C, leaving even upper bounds on resonance counts near the real axis or on individual sheets unresolved.