= Long-time resonance kernel
{title2=$\sin(tx)/x\to\pi\delta(x)$}
As $t\to+\infty$, the displayed limit holds in <distribution theory>. Indeed $\sin(tx)/x=\tfrac12\int_{-t}^{t}e^{iux}\,du$ 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.
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