On the Schrodinger-Newton equation and its symmetries: a geometric view

The Schrodinger-Newton (SN) equation is recast on purely geometrical grounds, namely in terms of Bargmann structures over (n + 1)-dimensional Newton-Cartan (NC) spacetimes. Its maximal group of invariance, which we call the SN group, is determined as the group of conformal Bargmann auto-morphisms that preserve the coupled Schrodinger and NC gravitational field equations. Canonical unitary representations of the SN group are worked out, helping us recover, in particular, a very specific occurrence of dilations with dynamical exponent z=(n+2)/3.