We study the spatial properties of the modes responsible for the Hawking effect in the presence of high frequency dispersion. Near the horizon, the modes are regularized on a small distance which only depends on the surface gravity and the scale of dispersion. The regularization explains why the spectrum is hardly affected by dispersion as long as the background geometry does not significantly vary over this composite length. For relevant frequencies, the regularization differs from the usual WKB resolution of wave singularity near a turning point. The latter only applies when the frequency is so high that the Hawking effect is negligible.
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