Global compensated-to-essential connection coefficient

Construct a finite canonical domain chain and corresponding global connection operator for the strong-drive Weber reduction of the driven single-mode Kerr cavity, and prove that the compensated-to-essential connection coefficient is proportional to the exact continued-fraction spectral function \(\Xi(E)\), so that its zeros coincide with the physical Bargmann eigenvalues and yield the stated strong-drive quantization and energy asymptotics.

Background

The strong-drive analysis reduces the coalescing-turning-point region to a parabolic-cylinder (Weber) comparison problem. Local asymptotics identify a branch whose gauge factor cancels at the irregular origin and a second branch with an essential singularity. Physical quantization requires a global continuation from the compensated branch through the turning-point region and back to the origin, with compatible sheets, canonical normalizations, and controlled errors.

The paper has an exact scalar spectral condition $\Xi(E)=0$ from the Bargmann coefficient recurrence, but it does not establish that the conjectured Weber connection coefficient has the same zero set. Proving this connection would justify the conditional next-order strong-drive energy coefficient and convert the local WKB construction into a global spectral proof.