$\newcommand{\div}{\operatorname{div}}$Suppose I have an elliptic operator $\mathcal{L} u = -\div (A \nabla u) $ on some open set $\Omega \subseteq \mathbb{R}^d$ where here $A$ is uniformly elliptic on $\Omega$. Assume that I can prove the existence of the Green's function for $ \mathcal{L}_v u = -\div (A \nabla u) + vu$ on $\Omega$ for any bounded, nonnegative $v$ on $\Omega$.
Is there any simple way to go from this to proving the existence of the Green's function $ \mathcal{L}_v u = -\div (A \nabla u) + vu$ on $\Omega$ for any $v$ real and bounded on $\Omega$? (that is, removing the nonnegative assumption).
The obvious point here being that for $E > 0$ large enough, $v + E$ is bounded and nonnegative on $\Omega$ and so we would know the existence of the Green's function for $ \mathcal{L}_{v, E} u = -\div (A \nabla u) + (v+E)u$ on $\Omega$. Can we use this knowledge to somehow get the Green's function for $\mathcal{L}_v?$
Even just the Schrödinger /fundamental solution case ($\Omega = \mathbb{R}^d$ and $A = I_{d \times d}$ so $\div (A \nabla u) = \Delta u$ ) would be very helpful.
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