Conservation Laws for the Schrödinger—Newton Equations

In this Letter a first-order Lagrangian for the Schrödinger—Newton equations is derived by modifying a second-order Lagrangian proposed by Christian [Exactly soluble sector of quantum gravity, Phys. Rev. D56(8) (1997) 4844–4877]. Then Noether’s theorem is applied to the Lie point symmetries determined by Robertshaw and Tod [Lie point symmetries and an approximate solution for the Schrödinger—Newton equations, Nonlinearity19(7) (2006) 1507–1514] in order to find conservation laws of the Schrödinger—Newton equations.