Matching lower bounds for sandwiched continued resolvents

Establish a matching lower bound for the sandwiched continued resolvent associated with the mass-type semiclassical Dirac operator for every choice of compactly supported spatial cutoffs, thereby determining whether the geometric upper bound with exponential dependence on the positive-sheet connection time is sharp in general.

Background

For mass-type semiclassical Dirac operators in a nontrapping positive-energy window, the paper derives an upper bound for a spectrally localized, sandwiched continued resolvent of the form $\big(e^{s(T_\chi+\varepsilon)/h}-1\big)/s$, where $T_\chi$ is the maximal positive-sheet connection time between microlocal interaction regions associated with the spatial cutoff. At logarithmic depth $s=Mh\log(1/h)$, this produces a polynomial loss determined by the classical flight time.

Flow-box analysis and an explicit free Dirac calculation indicate that continuation below the real axis naturally generates an amplification factor $e^{sT/h}$ along classical connections. However, the paper does not establish a corresponding lower bound for every possible choice of sandwiched cutoffs. Determining whether the geometric upper bound is generally sharp, or identifying the additional conditions under which a matching lower bound holds, remains unresolved.