M. Robnik, SUPERSYMMETRIC QUANTUM-MECHANICS BASED ON HIGHER EXCITED-STATES, Journal of physics. A, mathematical and general, 30(4), 1997, pp. 1287-1294
The formalism and the techniques of the supersymmetric (SUSY) quantum
mechanics is generalized to the cases where the superpotential is gene
rated/defined by higher excited eigenstates. The generalization is tec
hnically almost straightforward but physically quite non-trivial since
it yields an infinity of new classes of SUSY-partner potentials, whos
e 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 o
f 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 inf
initely many partner potentials: for each m the partner potential is e
xpressed as the sum of the quadratic harmonic potential plus rational
function, defined as the derivative of the ratio of two consecutive He
rmite polynomials. These partner potentials of course have m singulari
ties exactly at the locations of the nodes of the generating (m + 1) w
avefunction. The SUSY formalism applies everywhere between the singula
rities. 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 ar
e shape invariant they can be solved at least partially and new types
of analytically obtainable spectra are expected.