Full-line instability of scattering-resonance-induced states

Establish dynamical instability in H^1(R) for scattering-resonance-induced nonlinear bound states associated with symmetric potentials, by showing that the two exterior soliton wells generate a pair of discrete eigenvalues near zero and applying the Hamiltonian NLS/GP instability criteria.

Background

For symmetric potentials, the paper restricts scattering-resonance-induced states to the half-line space H1_0(R_+) (equivalently, the odd subspace of H1(R)) in order to obtain the stated stability theory. The authors observe that the corresponding full-line odd state consists of two oppositely signed free-soliton components, one far to the right and one far to the left of the potential.

Because the linearized operator then contains two exterior wells separated from the central potential, the authors expect a pair of discrete eigenvalues near zero. Proving that this spectral configuration produces dynamical instability in the full-line problem remains unresolved, whereas the half-line restriction removes the additional mode and restores stability in the regimes covered by the paper.

References

We believe therefore that it can be shown that our resonance-induced nonlinear bound states are dynamically \underline{unstable} in $H1(\mathbb R)$, as an application of the instability criteria of Theorem~\ref{thm:W-GSS}(ii).

— Stability and instability analysis of resonance-induced nonlinear bound states  (2610.01875 - Turner et al., 1 Oct 2026) in Remark 4.1, Section 4.1 ("Half-line restriction for scattering resonances")

How these constructions extend to the unstable resonance-induced branches studied here is an open question.

— Stability and instability analysis of resonance-induced nonlinear bound states  (2610.01875 - Turner et al., 1 Oct 2026) in Item 4, Section 11 ("Discussion and Future Directions", subsection "Future Directions")