# Strict positivity on the diagonal of a particular integral kernel: A question from Simon's Schrödinger Semigroups

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?

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