Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 2 / 33A / a / Solution

Lax's equation in the convention used here isLet $\mathcal{L}\psi =\lambda \psi$ with $(\psi ,\psi )=1$. Since $\mathcal{L}$ is self-adjoint and $\mathcal{A}$ is anti-self-adjoint, differentiating $\lambda =(\psi ,\mathcal{L}\psi )$ givesThe first and third terms equal $\lambda ,d(\psi ,\psi )/dt=0$. For the middle term,Thus $\dot{\lambda }=0$, proving the Isospectral Lax equation property: every eigenvalue is independent of $t$.