Assume we have a linear operator P_0: D->H $P_0: D->H$ where D $D$ is the domain and H $H$ some Hilbert space it is acting on. Assume moreover that the spectrum of P_0 $P_0$ is purely essential. If z $z$ is a complex number not in the spectrum of P_0 $P_0$ then the resolvent (P_0 $(P_0 - z)^{-1} z)^{-1}$ exists as a bounded operator H->D, $H->D$, meaning there is a constant (depending on z) $z$) such that |u|\leq C|(P_0-z)u| $|u|\leq C|(P_0-z)u|$ where the norm on then the left hand side is that of the domain D.

Now perturb P_0 $P_0$ by a "potential" V:R^n->C $V:R^n->C$ (complex-valued function on Euclidean n-space) and denote P=P_0+V $P=P_0+V$ where we assume V $V$ has compact support. Assume also that V $V$ generates an eigenvalue z_0 $z_0$ away from the essential spectrum of P_0 $P_0$ (which we may assume is also the essential spectrum of P). $P$). Thus there is a non-zero u $u$ in D $D$ with (P-z_0)u=0. $(P-z_0)u=0$. Let X $X$ be the characteristic function of some set (call it X $X$ as well) which is disjoint from the support of V. $V$. Then |Xu|\leq C|(P_0-z_0)Xu|=C|(P-z_0)Xu|=0$|Xu|\leq C|(P_0-z_0)Xu|=C|(P-z_0)Xu|=0$, i.e. the eigenfunction must be 0 wherever the potential is zero.

Am I forgetting something?

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 then 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?