On $\mathbb{R}^3$, we consider the operator \begin{equation} \mathcal{H}= \left( \begin{matrix} -\Delta +1 -2 \phi^2 & -\phi^2 \\ \phi^2 & \Delta -1 +2 \phi^2 \end{matrix} \right) , D(\mathcal{H})=H^2 \times H^2 (\mathbb{R}^3)\subset L^2 \times L^2(\mathbb{R}^3), \label{linearized oeprator: general case} \end{equation} where $\phi$ is the ground state of $-\Delta u + u= u^3$, and $\phi$ is strictly positive, radial, decreasing, smooth, and exponentially-decaying. Moreover, we can see that $\sigma_{ess}(\mathcal{H})= (-\infty, 1] \cup [1, \infty)$. Thus we are curious about the properties of discrete spectum of $\mathcal H$, expecially on the imaginary axis, which is eaiser to talk about than what on $[-1,1] \setminus \{ 0 \}$. As for question above, in W. Schlag: Stable manifolds for an orbitally unstable NLS, Schlag claimed in Proposition 16 that there are only finite eigenvalues of $\mathcal H$ on the imaginary axis. More precisely, there are only finite $\sigma \in \mathbb{R} \setminus \{0\} $ such that \begin{equation} \mathcal H f= i \sigma f, \quad f\in D(\mathcal {H})=H^2 \times H^2 \subset L^2 \times L^2. \end{equation}
and he just said that it followed from Fredholm's alternative immediately, but I do not know how to check it. So can someone give me some tips about it?
