Symmetry of time-dependent Schrödinger equations. II. Exact solutions for the equation {∂x x+2i∂t−2g2(t)x2−2g1(t)x −2g0(t)}Ψ(x, t) = 0

The maximal kinematical algebra of the Schrödinger equation {∂xx+2i∂t−2g2(t)x2−2g1(t)x−2 g0 (t)}Ψ(x, t) = 0 is known to be the Schrödinger algebra 𝓈1. The kinematical symmetries are realized as first-order differential operators in the space and time variables. A subalgebra 𝒢 of 𝓈1 is chosen and from 𝒢 and its invariants a complete set of commuting observables are constructed. The solution space of the Schrödinger equation is identified with the appropriate irreducible representation space of 𝒢. The wave functions, simultaneous eigenvectors of the compatible observables, are computed as explicit functions of space and time. The properties of a system with a potential V(x, t) = g2(t)x2+g1(t)x+g0(t) are discussed.