Symmetry, Local Linearization, and Gauge Classification of the Doebner-Goldin Equation

For the family of nonlinear Schr\"odinger equations derived by H.-D.~Doebner and G.A.~Goldin (J.Phys.A 27, 1771) we calculate the complete set of Lie symmetries. For various subfamilies we find different finite and infinite dimensional Lie symmetry algebras. Two of the latter lead to a local transformation linearizing the particular subfamily. One type of these transformations leaves the whole family of equations invariant, giving rise to a gauge classification of the family. The Lie symmetry algebras and their corresponding subalgebras are finally characterized by gauge invariant parameters.