The Hamiltonians of the Schrödinger algebra

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Abstract
We give a complete characterization of the one‐dimensional nonconjugate subalgebras of the extended Schrödinger algebra (and of the Schrödinger algebra). This yields a classification into conjugacy classes of all the Hamiltonians admitting the extended Schrödinger algebra as time dependent invariance algebra in the Schrödinger picture. Moreover, we determine the action of the inner automorphism induced by the general element of the extended Schrödinger algebra from which the explicit time dependence of the generators is deduced, whatever Hamiltonian is considered.