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Complete description of the classical spectrum for p=2 in the p-adic Jaynes–Cummings model

Determine the full classical spectrum F(S^2_2 × Q_2^2) of the p-adic Jaynes–Cummings model F=(J,H): S^2_p × (Q_p)^2 → (Q_p)^2 with J(u,v,z)=(u^2+v^2)/2+z and H(u,v,x,y)=(ux+vy)/2, in the case p=2. Provide necessary and sufficient conditions on (j,h) ∈ (Q_2)^2 for membership in F(S^2_2 × Q_2^2) and give a complete, explicit characterization of the image.

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Background

The paper fully characterizes the image and fibers of the p-adic Jaynes–Cummings model for odd primes, but for p=2 only partial necessary and sufficient conditions are provided. The authors note structural complications arising from the compactness of the 2-adic sphere S2_2, unlike S2_p for p≠2.

A complete description of the image F(S2_2 × Q_22) would unify the spectrum analysis across all primes and resolve the current gaps for p=2, where only inclusions and partial conditions are available.

References

In part (3b) of the previous theorem, we use “contains” because we do not have a complete description of the image of the system for p=2. Deciding whether some points are in the image seems more complicated than for other primes, partly because S2_2 is compact while S2_p is not compact for any other p.

— The $p$-adic Jaynes-Cummings model in symplectic geometry (2406.18415 - Crespo et al., 26 Jun 2024) in Remark after Theorem “Classical spectrum and critical points of p-adic Jaynes-Cummings model”, Section 2 (Main results)