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I am interested in when and how one can trade a non-liner ODE for a linear PDE. To explain what this could look like here is a physics-inspired discussion. Consider a classical mechanical system with ...

Let's consider the following differential equation on $\mathbb{R}$: $$-u''(x)+u(x)-V(x)u(x)=\lambda u(x),$$ where $\lambda<1$ and $V$ is a bounded. We consider only that solution $u(x) \in C^1$ ...