Monotonicity of the below-threshold scattering region in the coupling

Determine whether the set of initial data satisfying E_a(u_0)<m_a(c) and Q(u_0)>0 is monotone with respect to the inverse-square coupling constant a.

Background

The paper proves that the threshold m_a(c) is strictly increasing as the coupling a increases. However, the energy E_a(u_0) also increases with a for fixed data, so threshold monotonicity alone does not determine how the set of data in the global/scattering region changes.

The open question asks whether the combined conditions defining the positive-Pohozaev-sign region below threshold are preserved monotonically as a varies.

References

Is the set of data satisfying eq:below and Q(u_0)>0 monotone in a? As explained in Remark \ref{rmk:monotoneset}, this does not follow from the monotonicity of the level, because the energy also increases with a.

eq:below:

Ea(u0)<ma(c),c:=∥u0∥L2.E_a(u_0)<m_a(c),\qquad c:=\|u_0\|_{L^{2}} .

— Normalized ground states and a mass-constrained scattering threshold for the inhomogeneous NLS with an inverse-square potential  (2610.02933 - Majdoub et al., 2 Oct 2026) in Section “Complements and open problems,” subsection “Open problems,” item (vii)