For context, this question is related to a previous one and other one of mine.
Consider the operator $H$ acting on $L^2(\mathbb{R})$ and of the form $$Hv:=\phi\ast v-f v,$$ where $\phi=e^{-|x|}/2$ and $\ast$ denotes the convolution operation. Since $\phi$ is the Green's function for $1-\partial_x^2$, we have that $$(1-\partial_x^2)(\phi\star v)=v.\;\;\;\;\;(1)$$ Also $f$ is a differentiable function that is bounded and converging to $f_0>0$ as $\xi\to \pm\infty$.
From (1), the eigenvalue problem $Hv=\lambda v$ is equivalent to $$ v''=\frac{1}{\lambda}\left[\lambda-1+(1-\partial_x^2)f\right]\,v.\;\;\;\;(2) $$ I cannot say anything about the point spectrum of $H$. Intuitively/heuristically, I find the essential spectrum by using Fourier analysis on the asymptotic limit of (2), with $v=e^{ikx}$ and find the dispersion relation $$ \lambda=\frac{1}{1+k^2}-f_0. $$ Thus I have the following result: The essential spectrum of $H$ is the interval $[-f_0,1-f_0]$.
Is my result valid? I am asking because I find the proof to be heuristic thus have some doubts. Perhaps somebody can help me with the validity (or invalidity) of one of them?