Long-time resonance kernel ($\sin (tx)/x\to \pi \delta (x)$)

As $t\to +\infty$, the displayed limit holds in distribution theory. Indeed $\sin (tx)/x=\frac{1}{2}{\int }_{-t}^t{e}^{iux}du$ and the Fourier representation of the Dirac delta function gives the limit on smooth rapidly decaying test functions. For a regular detuning $\Delta (\mathbf{J})$, the coarea formula reduces its use under an integral over action variables to this one-dimensional identity. The kernel grows like $t$ at exact resonance, so this is not pointwise convergence.