Supersymmetric quantum mechanics based on higher excited states

The formalism and the techniques of the supersymmetric (SUSY) quantum mechanics is generalized to the cases where the superpotential is generated/defined by higher excited eigenstates. The generalization is technically almost straightforward but physically quite non-trivial since it yields an infinity of new classes of SUSY-partner potentials, whose spectra are exactly identical except for the lowest (m + 1) states, if the superpotential is defined in terms of the (m + 1) eigenfunction, with m = 0 reserved for the ground state. It is shown that in case of the infinite one-dimensional (ID) potential well nothing new emerges (the partner potential is still of Poschl-Teller type I, for all m), whilst in case of the 1D harmonic oscillator we get a new class of infinitely many partner potentials: for each m the partner potential is expressed as the sum of the quadratic harmonic potential plus rational function, defined as the derivative of the ratio of two consecutive Hermite polynomials. These partner potentials of course have m singularities exactly at the locations of the nodes of the generating (m + 1) wavefunction. The SUSY formalism applies everywhere between the singularities. A systematic application of the formalism to other potentials with known spectra would yield an infinitely rich class of 'solvable' potentials, in terms of their partner potentials. If the potentials are shape invariant they can be solved at least partially and new types of analytically obtainable spectra are expected.