All I know about this is from what I just Googled. A "cusp" in the electron wavefunction in an atom appears at the nucleus' surface. We would expect to find a discontinuity here as the charge density has a discontinuity at this point. (This is a first order approximation. The nuclear charge density smears out a bit due to the Uncertainty Principle and the electron wavefunction changes rapidly there, but without an actual discontinuity.)
That's all I can tell you without a more in depth search.
Is this a curiosity question or are you working on something?
-Dan
<Wiktionary:Cusp>
Note that the figurative usage and the geometric usage are actually closely connected.
Making a small change near a cusp point can produce a large change in the outcome.
An analogy might be the (American) continental divide where a drop of water falling on the west side will end up in the Pacific, while a drop of water falling just a very short distance to the east will end up in the Atlantic.
Thus one might suspect that a cusp electron is one where a tiny "push" (which would be practically negligible for a "normal" electron) would send the cusp electron along dramatically different future paths.
Note that the figurative usage and the geometric usage are actually closely connected.
Making a small change near a cusp point can produce a large change in the outcome.
An analogy might be the (American) continental divide where a drop of water falling on the west side will end up in the Pacific, while a drop of water falling just a very short distance to the east will end up in the Atlantic.
Thus one might suspect that a cusp electron is one where a tiny "push" (which would be practically negligible for a "normal" electron) would send the cusp electron along dramatically different future paths.