The following problem keeps bothering me:
Let $H:=-\Delta+V$ be a Schrödinger-Operator in $\mathbb{R}^n$, where $V$ is a Kato-Potential of type $K_n$, which especially yields that $H$ is e.s.a. on $C_0^\infty$ and the resolvent operator is an Integral operator (see e.g. Simon- Schrödinger Semigroups). Furthermore, let $V$ be $C^\infty$ everywhere but on a finite set of points, say $N$, where $V$ is discontinuous.
Now let $\varphi$ be a test function, i.e. smooth and vanishing together with all derivatives on $N$. (In my case, $\varphi$ is furthermore a Schwartz function, but I don't think this plays a role in this question.) Furthermore, let $\lambda$ be in the resolvent set of $H$ and $\psi=R_\lambda(H)\varphi$, which is equivalent to $\psi$ being a strong $\mathcal{L}^2$-solution to the PDE $$ (-\Delta+V-\lambda)\psi=\varphi $$ By elliptic regularity, I instantly get that $\psi$ is $C^\infty$ on $\mathbb{R}^n-N$.
I want to know, how $\psi$ behaves on $N$. My Intuition tells me, that $\psi$ has to vanish together with all derivatives on $N$, just as $\varphi$ does. However, I am not able to provide a rigorous proof for this. I am quite sure that I am overlooking something trivial, this has to be solvable by standard methods. Does anyone have an idea?