Apology in advance for the long question.
I have this family of operators parametrized by a real $k$: $$ {{\mathcal{L}}} v=\left((1-e^{-|x|})v\right)'-k\,{\rm{sgn}}(x)e^{-|x|}v $$ on the space $C_b(\mathbb{R})$ of continuous bounded functions on the whole line. On $L^2(\mathbb{R})$, it would make the analysis slightly different but the same question would arise. The solution of $$ {{\mathcal{L}}} v-\lambda v=0 $$ for $x<0$ is $$ v_{H-}=e^{\lambda x}(e^x-1)^{k-1-\lambda}, $$ and for $x>0$ $$ v_{H+}=e^{\lambda x}(1-e^{-x})^{k-1+\lambda}. $$ We restrict ourselves to $\lambda>0$ real for simplicity. For $k>1$, $v_{H-}$ is bounded and continuous if $0<\lambda< k-1$ and we have the eigenfunction $$ v= \left\{ \begin{array}{ll} v_{H-},&x<0\\ \displaystyle{0},&x>0. \end{array}\right. $$ The resolvent equation $$ {{\mathcal{L}}} v-\lambda v=f $$ for $x<0$ is solved by $$ v_{-}=e^{\lambda x}(e^x-1)^{k-1-\lambda}\int_0^x \frac{f(x')e^{-\lambda x'}}{(e^{x'}-1)^{k-\lambda}}dx', $$ and for $x>0$ by $$ v_{+}=e^{\lambda x}(1-e^{-x})^{k-1+\lambda}\int_\infty^x \frac{f(x')e^{-\lambda x'}}{(1-e^{-x'})^{\lambda+k}}dx'. $$ Still in the case $k>1$ but here for $\lambda>k-1$, the equation $$ v= \left\{ \begin{array}{ll} v_{-},&x<0\\ \displaystyle{v_+},&x>0. \end{array}\right. $$ defines a solution to the resolvent equation in $C_b(\mathbb{R})$. Thus in the case $k>1$, the spectrum on the real positive real line is $0< \lambda\leq k-1$, almost all point spectrum.
If $k<1$ (remember we consider $\lambda>0$ real), $v_{H-}$ is not bounded at the origin and $v_{H+}$ is not bounded at $+\infty$. Thus there is no point spectrum. However, for $\lambda<1-k$, $v_+$ above, while bounded at $+\infty$, is not bounded at $0$ and there is no way to fix that by changing the limits of integration. If you change $\infty$ to $0$ in the integral for example, then $v_+$ is not bounded at $+\infty$. So I have this region of the spectrum $0<\lambda<1-k$ that I don't really understand. Is this just essential spectrum? It just does not "look" as I would expect for essential spectrum. Or maybe I made a mistake in my reasoning. Admittedly, my question is a bit vague. Any insight would be appreciated.