Critical collapse under steep expulsive potentials

Investigate the interplay between critical collapse induced by cubic self-attraction in the one-dimensional Gross–Pitaevskii equation or quintic self-attraction in the two-dimensional Gross–Pitaevskii equation and a strong expulsive potential proportional to r^{2\gamma}.

Background

The paper considers one-dimensional and two-dimensional linear and nonlinear Schrödinger equations with a strongly expulsive potential proportional to r{2\gamma}. For \gamma>1, the linear problem counter-intuitively supports a full continuous spectrum of normalizable, effectively localized bound states. The authors identify the effect of nonlinear self-attraction as an unresolved issue, specifically because cubic attraction in one dimension and quintic attraction in two dimensions can induce critical collapse, whose interaction with the steep expulsive potential may substantially alter localization, stability, or collapse behavior.

References

There remain open questions concerning this setting, especially as concerns effects of the nonlinearity. In particular, it will be interesting to investigate the interplay of the critical collapse, induced by the cubic or quintic self-attraction in the 1D or 2D cases, respectively, with the strong expulsive potential.

— Quantum-mechanical wave functions in singular potentials: linear and nonlinear states  (2608.20282 - Sakaguchi et al., 20 Aug 2026) in Section Discussion and conclusion

There remain open questions concerning this setting, especially as concerns effects of the nonlinearity. In particular, it will be interesting to investigate the interplay of the critical collapse, induced by the cubic or quintic self-attraction in the 1D or 2D cases, respectively, with the strong expulsive potential. Another issue is a possibility of the splitting instability (alias the azimuthal modulational instability) of vortex bound states in the 2D system with self-attraction. It is also relevant to consider the action of the strong expulsive potential on QDs in the 1D and 2D settings.

— Quantum-mechanical wave functions in singular potentials: linear and nonlinear states  (2608.20282 - Sakaguchi et al., 20 Aug 2026) in Section Discussion and conclusion