Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 358 / 3 / c / Solution

Let $p(x)=x^3-x$. Since $A$ is multiplication by $p$, its spectral projection is multiplication by $1_{p^{-1}(S)}$. HenceSplit $[-1,1]$ at the two critical points $\pm 1/\sqrt{3}$. On each resulting interval, $p$ is monotone, so one-dimensional change of variables shows that the measure is absolutely continuous. For almost every $t$,The density vanishes outsideIts inverse-square-root singularities at the two critical values are locally integrable, so they do not create singular spectral measure.