A Sharp Coercive Spectral Functional Inequality

We prove a sharp coercive lower bound for spectral functionals associated with Schrödinger operators on compact Riemannian manifolds.For nonnegative potentials with critical integrability, the trace of a monotone functional of the operator is bounded from below by the corresponding phase–space integral with the exact semiclassical constant.The argument is based on classical tools from spectral theory, convexity, and coherent-state methods, and avoids auxiliary regularization or perturbative techniques.The constant is shown to be optimal by scaling asymptotics.The presentation is fully self-contained and formulated with complete mathematical rigor.