A weak collisionality constrained solution of a steady electron depleted hole

Electron hole treatments normally match collisionless distribution functions associated with spatially varying electrostatic potential wells containing bound or reflecting electrons to unbound or non-reflecting electrons. When the matching is piecewise continuous at the separatrix between the bound and unbound, the second derivative in velocity space normally diverges, indicating collisions should be retained to remove the singular behaviour. The solubility constraint introduced by the collision operator severely restricts the allowed bound and unbound solutions, as illustrated by solving a model problem for a solitary electron depleted hole. As a result, a narrow collisional boundary layer encloses the separatrix between the bound and barely unbound electrons most sensitive to collisions. The procedure employed does not require resolving the collisional boundary layer, yet self-consistently accounts for the discontinuous behaviour to demonstrate an electron hole can last a long time when collisions are very weak.