General threshold expansion for resolvents

Establish a threshold expansion of the form Disc R_L(ω)=f(ω)P_0+E(ω), with E(ω)=o(f(ω)), for the outgoing resolvent of a general second-order operator L near a threshold frequency.

Background

The paper’s general threshold-map theorem assumes that the jump of the outgoing resolvent has a leading finite-rank expansion, represented by Disc R_L(ω)=f(ω)P_0+E(ω), where the remainder is asymptotically smaller than f(ω). Under this assumption, the authors show that applying a regular first-order observable T suppresses the leading threshold contribution exactly when T_0P_0=0.

The existence of this type of expansion is not established for a general second-order operator in the paper. Instead, the Schwarzschild Regge–Wheeler application relies on a known fixed-radius small-frequency expansion of the Regge–Wheeler Green function. A general proof would provide the analytic foundation needed to apply the threshold-suppression criterion beyond the specific Schwarzschild setting.