Logarithmic boundary conditions and AKO

In Allen–Kay–Ottewill AKO1996, their scattering length $R$ and pole $q$ map to our $L$ and ${\kappa }_L$:

Their equations (3.17), (4.3), and Appendix A give the $\nu =0$ Krein coefficient and principal-value integral. Equation (4.8) agrees with the leading term of (eq:logPV).

Their equations (4.13)–(4.15) are the one-real-field $\nu =0$ versions of our diagonal stress integrands. The complex tensor here is twice those expressions. Their equation (4.17) is the same radial conservation identity.

AKO fit idealized exterior formulas to a smooth core. The literal zero-thickness logarithmic extension has an unstable band and no stationary finite-moment state by Theorem thm:nogo. The principal-value value is therefore not silently called its ground-state expectation.