DELTA FUNCTIONS OF OBSERVABLES AND RADON–NIKODYM DERIVATIVES OF SPECTRAL MEASURES

Abstract

In this paper, we study the mapping r ↦ δr(Q), where Q is an observable satisfying mild conditions and δr(Q) is the delta function of Q with r ∈ ℝ. We find that, under certain integrable conditions, the mapping r ↦ δr(Q) is just the Radon–Nikodym derivative of Q 's spectral measure with respect to Lebesgue measure and, moreover, Q can be represented as an integral of the mapping r ↦ rδr(Q) with respect to Lebesgue measure. An example is also given.