What question did this study set out to answer?

The aim is to establish a sharp coercive lower bound for spectral functionals related to Schrödinger operators.

February 2, 2026Open Access

A Sharp Coercive Spectral Functional Inequality

Key Points

The aim is to establish a sharp coercive lower bound for spectral functionals related to Schrödinger operators.
Proved a lower bound using spectral theory and convexity methods.
Employed coherent-state methods without auxiliary regularization.
Demonstrated optimality of the constant through scaling asymptotics.
The trace of the monotone functional is bounded from below by the phase-space integral.
The established constant is optimal for nonnegative potentials with critical integrability.

Abstract

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.

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