On the structure of the n=4 supersymmetric quantum mechanics in D=2 and D=3

The superfield formulation of two-dimensional N = 4 extended supersymmetric quantum mechanics (SQM) is described. It is shown that the corresponding classical Lagrangian describes the motion in the conformally flat metric with an additional potential term. The Bose and Fermi sectors of two- and three-dimensional N = 4 SQM are analysed. The structure of the quantum Hamiltonians is such that the usual Schrodinger equation in the flat space arises after some transformation, demonstrating the effect of transmutation of the coupling constant and the energy of the initial model in some special cases.