  We generalize the formalism and the techniques of the supersymmetric (susy)
quantum mechanics to the cases where the superpotential is generated/defined by
higher excited eigenstates. The generalization is technically almost
straightforward but physically quite nontrivial 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)-st eigenfunction, with m=0 reserved for the ground state. It is shown
that in case of the infinite 1-dim potential well nothing new emerges (the
partner potential is still of P\"oschl-Teller type I, for all m), whilst in
case of the 1-dim 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)-st 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.
  PACS numbers: 03.65.-w, 03.65.Ge, 03.65.Sq
