Minoration de la resolvante dans le cas captif

TL;DR

This paper establishes an optimal universal lower bound on the truncated resolvent for semiclassical Schrödinger operators near trapping energies, confirming the optimality of known upper bounds for hyperbolic trapping.

Contribution

It provides the first optimal universal lower bound on the truncated resolvent in the presence of trapping, extending understanding of semiclassical Schrödinger operators.

Findings

Proves an optimal universal lower bound near trapping energies.

Shows known upper bounds for hyperbolic trapping are optimal.

Uses propagation of coherent states for Ehrenfest times in the proof.

Abstract

In this note, we prove an optimal universal lower bound on the truncated resolvent for semiclassical Schroedinger operators near a trapping energy. In particular, this shows that known upper bounds for hyperbolic trapping are optimal. The proof rely on an idea of X. P. Wang, and on propagation of coherent states for Ehrenfest times.