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Bounds for eigenvalue counting and scattering matrix in the product setting beyond bounded Euclidean domains

Develop upper and lower bounds for eigenvalue counting and for the scattering matrix associated with the Schrödinger operator P_V = -Δ_X + V on X = R^n × M (n ≥ 3 odd, M compact) for generic manifolds M, extending beyond the case when M is a bounded Euclidean domain.

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Background

The Birman–Krein formula derived in the paper connects spectral measures with the normalized scattering matrix on Rn × M and treats threshold contributions. For M a bounded Euclidean domain, the authors establish an upper bound for the scattering phase using commutator and heat kernel methods.

For generic compact manifolds M, however, no general upper or lower bounds are currently known for eigenvalue counting or for quantitative behavior of the scattering matrix in this product geometry with thresholds.

References

In fact, except the case that M is a bounded Euclidean domain which is presented in this paper, the author does not know any upper bound or lower bound results for eigenvalues counting or the scattering matrix in the setting \mathbb{R}n \times M for generic manifold M.

— The Birman-Krein Trace Formula and Scattering Phase on Product space (2509.06372 - Zhang, 8 Sep 2025) in Introduction, Further possible result (bullet 2)