Spectral condition conjecture for attractive delta NLS solitary waves

Fix $q<0$ and $p>4$. Let $\sigma=-1$ denote the focusing case and $\sigma=1$ the defocusing case, and let $\Omega=\Omega(q,p)$ be the minimal unstable frequency. The spectral condition is the condition given in Definition~. Spectral condition conjecture. In the focusing case, there exists $\omega_1=\omega_1(q,p)\in(\tfrac12q^2,\Omega)$ such that the spectral condition holds for $\omega\in(\tfrac12q^2,\omega_1)$. In the defocusing case, the spectral condition holds for all $\omega\in(0,\tfrac12q^2)$. This conjecture describes the frequency ranges in which the spectral hypothesis needed for the asymptotic stability theorem is expected to hold; the paper combines analytic and numerical evidence but does not establish the full claim.