Title: ABC of ladder operators for rationally extended quantum harmonic oscillator systems,
Abstract: The problem of construction of ladder operators for rationally extended
quantum harmonic oscillator (REQHO) systems of a general form is investigated
in the light of existence of different schemes of the Darboux-Crum-Krein-Adler
transformations by which such systems can be generated from the quantum
harmonic oscillator. Any REQHO system is characterized by the number of
separated states in its spectrum, the number of `valence bands' in which the
separated states are organized, and by the total number of the missing energy
levels and their position. All these peculiarities of a REQHO system are shown
to be detected and reflected by a trinity $(\mathcal{A}^\pm$,
$\mathcal{B}^\pm$, $\mathcal{C}^\pm$) of the basic (primary) lowering and
raising ladder operators related between themselves by certain algebraic
identities with coefficients polynomially-dependent on the Hamiltonian. We show
that all the secondary, higher-order ladder operators are obtainable by a
composition of the basic ladder operators of the trinity which form the set of
the spectrum-generating operators. Each trinity, in turn, can be constructed
from the intertwining operators of the two complementary minimal schemes of the
Darboux-Crum-Krein-Adler transformations.