Qualitative spectrum conjecture for the linearized focusing delta NLS

Fix $q<0$ and $p>4$, and consider the linearized operator $\mathcal{L}(\omega)$ for $\omega\in(\tfrac12q^2,\infty)$. Let $csigma_c(\mathcal{L}(\omega))$ denote its continuous spectrum, let $\Omega$ be the minimal unstable frequency, and let the spectral condition be the condition in Definition~. Spectrum conjecture. There exists $\omega_1\in(\tfrac12q^2,\Omega)$ such that $\mathcal{L}(\omega_1)$ has resonances at $\pm i\omega_1$, the spectral condition holds for $\omega\in(\tfrac12q^2,\omega_1)$, and for $\omega>\omega_1$ there is a first eigenvalue pair $\pm\lambda_1(\omega)$ satisfying

$$\pm\lambda_1(\omega)\in\begin{cases} i\mathbb{R}\backslash\sigma_c(\mathcal{L}(\omega))&\text{for }\omega_1<\omega<\Omega,\\ \{0\}&\text{for }\omega=\Omega,\\ \mathbb{R}&\text{for }\omega>\Omega. \end{cases}$$

There exists $\omega_2>\omega_1$ such that $\mathcal{L}(\omega_2)$ has a second pair of resonances at $\pm i\omega_2$; for $\omega>\omega_2$ there is a second eigenvalue pair $\pm\lambda_2(\omega)$, corresponding to two dimensions of the generalized kernel of the limiting free operator $\mathcal{L}^0$, with

$$\pm\lambda_2(\omega)\in i\mathbb{R}\backslash\sigma_c(\mathcal{L}(\omega)),\qquad \pm\lambda_2(\omega)\to0\quad\text{as }\omega\to\infty.$$

For every $\omega\in(\tfrac12q^2,\infty)$, there are no eigenvalues embedded in the continuous spectrum. The conjecture gives the expected evolution of the discrete spectrum across the three frequency regimes; the absence of embedded eigenvalues and the stated spectral transitions are not proved in the paper.