Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 61 / 2 / b / Solution

For the specified time operator, $N_k{E}_k=3ikq{E}_k$, because the derivatives of $e^{i(kx+k^{3}t)}$ cancel between ${\partial }_t$ and ${\partial }_x^3$. Expanding the derivatives of $E_k{A}_k$ givesFor completeness, the extra derivative terms before simplification are $3ik{A}_{k,xx}-3k^2{A}_{k,x}+3ikq{A}_k$. Substituting ${\phi }_{xx}+q\phi =M_l\phi +il{\phi }_x$ turns their derivative contribution into $-3k{\int }_L{\phi }_{x}d\lambda =-3k{I}_x$.

Applying $N_k$ to the original linear integral equation and using $I_x=-q$ cancels its right side. ThusPart (a) gives $E_k{\int }_L{M}_l\phi /(k+l)d\lambda =i{M}_k\phi$. Therefore the exact requested identity isSince $M\phi =0$, this becomes homogeneous too. The same uniqueness argument gives $N\phi =0$; it is not necessary to assume a separate uniqueness theorem for the differential time equation.