I apologize if this has been asked before but so far I haven't found it anywhere.

Consider the Nonlinear Schrödinger equation with a potential (i.e. Gross- Pitaevskii) in $\mathbb{R}^{d}$

$$i\Psi_{t} = -\nabla^2\Psi + V(x)\Psi + f(|\Psi|^2)\Psi$$

To simplify things, let $f(|\Psi|^2) = |\Psi|^2$, the cubic term.

Consider a Solitary wave solution $\Psi = \Phi_{\lambda}(x)e^{-i\lambda t}$ and then using that the above equation can be viewed as extremazing a functional (the Hamiltonian) we can linearize it around $\Phi_{\lambda}$ to obtain an equation for the mode $u$ with $\vec{\chi} =\begin{pmatrix} Re\, u \\ Im\, u\end{pmatrix}$

$$\vec{\chi}_{t} = \mathcal{L}(\lambda)\vec{\chi}$$ with $$ \mathcal{L}(\lambda) = \begin{pmatrix}0 & L_0(\lambda)\\ -L_1(\lambda) & 0\end{pmatrix}$$ and $L_0(\lambda) = -\nabla^2 + V(x) + \lambda -\phi_{\lambda}^2$ and $L_0(\lambda) = -\nabla^2 + V(x) + \lambda -3\phi_{\lambda}^2$

The main question is how to understand the spectrum $\sigma (\mathcal{L}(\lambda))$ and its associated spectral subspaces in terms of the spectrum $\sigma(L_0(\lambda))$ and $\sigma(L_1(\lambda))$ and their spectral subspaces?

The only trivial thing is the kernel but otherwise the offdiagonal character of $\mathcal{L}(\lambda)$ really confuses me. More specifically:

I) Can we say that the spectral suspace associated to $\sigma_{pp}(\mathcal{L}(\lambda))$ (abusing and using that $\sigma(\mathcal{L}(\lambda)) \subset i\mathbb{R})$ is "spanned'' by the spectral subspaces of $\sigma_{pp}(L_0(\lambda))$ and $\sigma_{pp}(L_1(\lambda))$? Or something analogous for the continuous spectrum?

II)Is there an algorithmic way to construct the spectrum and its subspaces or does it depend strongly on the choice of $f$?

III) I read somewhere that the interpretation of the discrete spectrum of $\mathcal{L}(\lambda)$ corresponds to bound modes and the continuous part to scattering modes as in standard quantum mechanics but It would be helpful if some one could confirm it.

IV) (minor point) If the components of $\vec{\chi}$ are real, then the operators $L_0, L_1$ are in principle acting on Real Hilbert spaces but in general the solutions for eigenvalue problems of these self-adjoint operators are complex, are we simply extending and allowing the components to be complex?

V)Finally if anyone can recommend a source where this issue is discussed at length at least for some example with $V \neq 0$ , I would be grateful.

Thanks for Reading.