Creation and Annihilation Operators in QM

Suppose a self-adjoint operator F on a Hilbert space has discrete
eigenvalues m_1 , m_2, m_3, etc. It is sometimes possible to find a
"creation operator" C such that if \phi is an eigenfunction of F for
the eigenvalue m_n, then C\phi is an eigenfunction with eigenvalue
m_{n+1}. Similarly an annihilation operator A does the opposite. Thus
if \phi has eigenvalue m_n then A\phi has eigenvalue m_{n-1}. The
reason for the terminology is that one thinks of C and A as creating or
annihilating quanta. Two classical examples of this in QM:

1) If L_z is the z-component of angular momentum (eigenvalues are hm
for m=0,1,2) then one can show that L_x + iL_y and L_x - iL_y are
creation and annihilation operators respectively.

2) For the harmonic oscillator with potential kx^2 one can show that C
= cx + i dp and A = cx - dp are creation and annihilation operators for
the Hamiltonian (x is position, p is momentum, and c and d are suitable
constants. This is the most elegant way to mathematically derive the
spectrum of energy eigenvalues.

Question: Is there any theorem along the following lines. If an
operator F has discrete spectrum and satisfies suitable hypotheses,
then there exist creation and annihilation operators for F? I am no
expert in functional analysis, but it seems to me that such a theorem
should be possible.

.

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