The Dixmier trace is an important example of non-normal trace. It's domain is the weak trace class. It also plays the role of the integral in the framework of Connes’ noncommutative geometry. During a conference at Fudan University in Shanghai in 2017 Connes claimed that we should be able to understand the Dixmier trace from purely spectral data without any reference to ultrafilters or generalized limits as in previous constructions. In this talk, we shall present an affirmative answer to Connes’ claim. We will also explain some relationship with the work of Birman-Solomyak on Weyl laws for compact operators from the 70s. Birman-Solomyak’s work was motivated by the spectral analysis of Schroedinger operator on domains or manifolds. Thus, this highlight a neat link between noncommutative geometry and semiclassical analysis, which are two different facets of quantum theory.