Negative examples for spectral asymptotics in the product setting with n ≥ 3

Identify or construct explicit counterexamples demonstrating the failure of general spectral asymptotic results for the Schrödinger operator P_V = -Δ_X + V on X = R^n × M (n ≥ 3 odd, M compact), or determine rigorously whether such negative examples exist.

Background

The paper compares the product setting (n ≥ 3) with the n = 1 cylindrical case, noting that spectral asymptotics akin to those available in simpler geometries may be out of reach due to trapping mechanisms induced by the compact factor M and the presence of thresholds.

Despite this expectation, no explicit negative examples are currently known that would conclusively preclude spectral asymptotics in broad generality for the product case.