Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 8 / 3 / ii / Solution

Write $P={\sum }_{j\le \alpha }{p}_j(x){\partial }^j$, setting unused coefficients to zero. Split the Adler trace into its fixed leading part and variable parts:Cyclicity from part (i) givesThus this Hamiltonian trace functional on monic differential operators has the required formTo make each coefficient explicit, expand ${\partial }^l{p}_j$ by the Leibniz rule. A term with $k$ derivatives of $p_j$ contributes to the residue exactly when $l-k+j=-1$. Consequently,These are smooth functions depending on $P$, not on the variable coefficients of $L$. In particular $a_0=p_{-1}$ and $a_1=p_{-2}+p_{-1}^{\prime }$. The coefficient $a_0$ must be included: it appears in the displayed sum even though the printed coefficient list starts at $a_1$.

The monic operators form an affine space, not a vector space. Accordingly $l_P$ is affine in the coordinates $u_l$, with constant $c$, and its differential is linear in variations. Its variational derivatives are $\delta l_P/\delta u_l=a_l$. It is the restriction of a linear trace pairing on the full operator space, which explains the terminology without discarding the constant term.