Critical dipoles in one, two, and three dimensions

The Schrödinger equation for a point charge in the field of a stationary electric dipole admits bound states only when the dipole moment exceeds a certain critical value. It is not hard to see why this might be the case, but it is surprisingly difficult to calculate the critical dipole moment. The analogous problem should be simpler in one and two dimensions, but a general theorem forbids critical moments in one dimension, and explicit calculation shows that there is no critical moment in two dimensions.

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