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A potential of the form $r^{-n}$ is often considered long-range, while one that decays exponentially is considered short-range.

Is this characterization simply relative/conventional, or is there a more fundamental reason?

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I think one of the most important reasons for that distinction is that an exponentially decaying function in space necessarily has a natural length scale: $\exp(-r/\xi)$. So one can sensibly talk about distances that don't really notice the force (i.e $r >> \xi$), whereas power law functions look the same at all scales. – Ruben Verresen Jun 11 at 12:56

Graphing examples of each would rapidly convince you that the power law has a "long tail" whereas the exponential gets small rather quickly. That's the basic reason. But understand that it is possible to pick a distance from the origin and then to pick the constants in the potential such that the exponential will be larger than the power law.

All that said, it should also be noted that in the power law once one gets past $1/r$ potentials, that even the power law potential gets small quickly.

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In scattering theory $r^{-n}$ is a short range potential for n > 1 and a long range potential for n <= 1.

For short range potentials the (unmodified) wave operators exist, for long range potentials not.

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First and foremost, this is a relative/conventional distinction. But secondly, it also divides the ‘long-range’ forces (electromagnetism and gravity) from the short-range interactions (strong and weak force): While the former decay according to $\frac{1}{r}$, the latter are usually described by means of exponentially decaying potentials1.

Apart from this, the further distinction brought up by Paul between $r^{-1}$ and $r^{-n}, n>2$ is interesting insofar as that simple models of interactions in magnets, for example, are usually based on $r^{-2}$ potentials, the same is true for multipole moments of electric charges or currents - which are therefore only ‘visible’ at short ranges.

1:This is sometimes associated with the fact that, in the standard model, the weak and the strong interactions are linked to massive bosons (or gluons), while the electromagnetic force and gravity (if included) are linked to massless bosons (photon, graviton). If you then take into account that massive bosons cannot travel at the speed of light and hence are capable to decaying (something that always occurs exponentially), you can see why the strong and the weak force are often described with exponential potentials

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It's explained here in a lecture from the Clay Mathematics institute:

Because Area grows like $r^2$. So the classical "field lines" in a $1/r$ potential aka $1/r^2$ force get less dense, but integration over the surface at all distances still yields a constant. A short range potential gets damped with distance. The mass of the exchange particle actually shows up as a damping term in the wave equation.

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