Widom Conjecture for Spectral Projectors of momentum-restricted Schr"odinger Operators

Release time:2025-12-13

Journal:Submitted

Abstract:We investigate the Widom conjecture for the spectral projection operator \(\Pi = \ind_{\{H_{\Gamma,\hbar} < \mu\}}\) of momentum-restricted Schr\"odinger operators \(H_{\Gamma,\hbar}\coloneq \text{Op}_{\hbar}(\ind_{\Gamma}(\xi))\text{Op}_{\hbar}( p(x,\xi))\text{Op}_{\hbar}(\ind_{\Gamma}(\xi))\) in $\mathbb R^n$. Using semiclassical and operator-theoretic techniques, we establish the Widom conjecture for momentum-restricted Schr\"odinger operators, obtaining precise asymptotic expansions of the corresponding trace and entropy functionals. We further derive limiting formulas for the entanglement entropy in the semiclassical regime and identify the extremal configurations among compact convex domains. In particular, we show that the limiting entropy constant admits a variational characterization over convex bodies and that the extremal values are uniquely attained by two congruent spheres, up to translation. These results provide a geometric and functional-analytic perspective on Widom-type asymptotics, linking spectral theory, convex geometry, and entropy in the semiclassical analysis of Schr\"odinger operators.

Co-author:Pan Ma, Daoyuan Zheng

Indexed by:Journal paper

Discipline:Natural Science

First-Level Discipline:Mathematics

Document Type:J

Translation or Not:no