Assume we have a linear operator $P_0: D->H$ where $D$ is the domain and $H$ some Hilbert space it is acting on. Assume moreover that the spectrum of $P_0$ is purely essential. If $z$ is a complex number not in the spectrum of $P_0$ then the resolvent $(P_0 - z)^{-1}$ exists as a bounded operator $H->D$, meaning there is a constant (depending on $z$) such that $|u|\leq C|(P_0-z)u|$ where the norm on the left hand side is that of the domain D.
Now perturb $P_0$ by a "potential" $V:R^n->C$ (complex-valued function on Euclidean n-space) and denote $P=P_0+V$ where we assume $V$ has compact support. Assume also that $V$ generates an eigenvalue $z_0$ away from the essential spectrum of $P_0$ (which we may assume is also the essential spectrum of $P$). Thus there is a non-zero $u$ in $D$ with $(P-z_0)u=0$. Let $X$ be the characteristic function of some set (call it $X$ as well) which is disjoint from the support of $V$. Then $|Xu|\leq C|(P_0-z_0)Xu|=C|(P-z_0)Xu|=0$, i.e. the eigenfunction must be 0 wherever the potential is zero.
Am I forgetting something?
