This is a question pertaining to a (formerly?) open question from Barry Simon's Schrödinger Semigroups. In Theorem C.5.2 (page 504) of that publication, the existence of a specific function $F(x,y;\lambda) = dE(x,y;\lambda)/d\rho$ is proven, which is
"an integral kernel of the [Radon-Nikodym] derivative of the spectral projection"
It is easy to see that $F(x,x;\lambda)\ge 0$ for almost every $\lambda$. On page 510, Simon states the question:
Open question. The following is somewhat related to unique continuation [...]. Is $dE(x,x;\lambda)/d\rho$ a strictly positive function for a.e. $\lambda$?
My question is exactly the same: Is this still open? Where would I find out more about this specific function, or about the solutions of "minor open questions" in general?