On the Structure of the $N=4$ Supersymmetric Quantum Mechanics in $D=2$ and $D=3$

Abstract

The superfield formulation of two - dimensional $N=4$ Extended Supersymmetric Quantum Mechanics (SQM) is described. It is shown that corresponding classical Lagrangian describes the motion in the conformally flat metric with additional potential term. The Bose and Fermi sectors of two- and three-dimensional $N=4$ SQM are analyzed. The structure of the quantum Hamiltonians is such, that the usual Shr\"{o}dinger equation in the flat space arises after some unitary transformation, demonstrating the effect of transmutation of the coupling constant and the energy of the initial model in some special cases.