  The development of supersymmetric (SUSY) quantum mechanics has shown that
some of the insights based on the algebraic properties of ladder operators
related to the quantum mechanical harmonic oscillator carry over to the study
of more general systems. At this level of generality, pairs of eigenfunctions
of so-called partner Hamiltonians are transformed into each other, but the
entire spectrum of any one of them cannot be deduced from this intertwining
relationship in general -- except in special cases. In this paper, we present a
more general structure that provides all eigenvalues for a class of
Hamiltonians that do not factor into a pair of operators satisfying canonical
commutation relations. Instead of a pair of partner Hamiltonians, we consider
two pairs that differ by an overall shift in their spectrum. This is called
coupled supersymmetry. In that case, we also develop coherent states and
present some uncertainty principles which generalize the Heisenberg uncertainty
principle. Coupled SUSY is explicitly realized by an infinite family of
differential operators which admit orthonormal bases of eigenfunctions of
generalized harmonic oscillators.
