N-extended supersymmetric quantum mechanics

Supersymmetric quantum mechanics is a theory where the Hamiltonian of conventional quantum mechanics forms an algebraic structure: superalgebra, along with other operators: supercharges. This symmetry is responsible for the degeneracy of the spectrum of the resulting partner Hamiltonians. This work extends the number of supersymmetries - number of supercharge operators - to an arbitrary number. The procedure is given in algebraic and matrix formalism. The scheme is related to the backward potential problem of constructing quantum mechanical systems for a wide range of modified potentials. The totally isospectral potentials are a natural result in the extended supersymmetric system. It is shown how to construct families of isospectral potentials for any number of supersymmetries. It is also shown that the supersymmetric Hamiltonians of the many particle system can be extended. The two particle Calogero potential is given for the case of two pairs of supercharges.