Low-energy boundedness for oscillating short-range potentials

Determine whether the modified wave operators associated with the oscillating potentials in Examples 26662124 and 26662236, for decay exponents bc in the range [1,2), are bounded on L^p(R) in the low-energy regime.

Background

The paper establishes Lp-boundedness of the modified wave operators in the middle and high energy regimes for the oscillating potentials presented in Examples 26662124 and 26662236 when bce[1,2). However, the low-energy behavior is not resolved. The authors explain that their construction of generalized eigenfunctions, particularly the Liouville-transform reduction used in the cited prior work, is not suitable for these oscillatory models at low energy. Resolving the problem would require sufficiently uniform bounds on the Jost functions and their energy derivatives, possibly through an alternative construction of the Jost functions.

References

Though modified wave operators are bounded in the middle and high energy regimes for V in Examples \ref{26662124} and \ref{26662236} with \mu \in [1, 2), the low energy regime is completely unknown.

— Modified wave operators are unbounded on $L^p$ for $p\neq 2$  (2609.29729 - Hoshiya et al., 24 Sep 2026) in Section 1, subsection "Further observations," underlined paragraph "Oscillating short range potentials"