Consider the elliptic PDE:
$$-\Delta u= f(x) u. $$
Assume that $f,u$ are defined in some reasonable bounded domain $\Omega \subset \mathbb{R}^n$ and impose the boundary condition $u=0$ on $\partial \Omega.$
Suppose first $f\equiv \lambda \in \mathbb{R}.$ Then is a fact that there are infinitely many, discrete choices of $\lambda$ such that this equation holds.
I want to know more generally about the structure of the set of $f$ which admit solutions to the above equation.
In particular, is the set of $f$ such that the above equation holds "exceptional" in any sense?
E.g. could the following statement be true? "for any $f \in C^0(\Omega)$ which admits a solution to the above equation, there exists $\epsilon>0$ such that if $\|g-f\|_{C^0} < \epsilon$ and $g$ also admits a solution to the equation then $f \equiv g.$"
(N.B. I am actually interested in the case where $f,u$ are defined on a closed manifold and $\Delta$ is the Laplace-Beltrami operator, but I have asked the question in the Euclidean setting since I assume this is more familiar to most people.)
Robert Lewis has the right line. $fu$ is a multiplication operator (and is self-adjoint). If $f$ is bounded, then $\Delta+f$ is bounded and self-adjoint so it has a spectral decomposition. The decomposition varies continuously with $f$, no? That is, the only way for it to be the same is if $f$ and $g$ differ on a set of measure zero?