In this article a study was made of the conditions under which a Hamiltonian which is an element of the complex $ \left\{ h (1) \oplus h(1) \right\} \uplus u(2) $ Lie algebra admits ladder operators which are also elements of this algebra. The algebra eigenstates of the lowering operator constructed in this way are computed and from them both the energy spectrum and the energy eigenstates of this Hamiltonian are generated in the usual way with the help of the corresponding raising operator. Thus, several families of generalized Hamiltonian systems are found, which, under a suitable similarity transformation, reduce to a basic set of systems, among which we find the 1:1, 2:1, 1:2,
su(2) and some other non-commensurate and commensurate anisotropic 2D quantum oscillator systems. Explicit expressions for the normalized eigenstates of the Hamiltonian and its associated lowering operator are given, which show the classical structure of two-mode separable and non-separable generalized coherent and squeezed states. Finally, based on all the above results, a proposal for new ladder operators for the
p:q coprime commensurate anisotropic quantum oscillator is made, which leads us to a class of Chen
SU(2) coherent states.
Recently a
f-deformed Fock space which is spanned by
∣n>λ has been introduced. These bases are indeed the eigen-states of a deformed non-Hermitian Hamiltonian. In this contribution, we will use a rather new non-orthogonal basis vectors for the construction of coherent and squeezed states, which in special case lead to the earlier known states. For this purpose, we first generalize the previously introduced Fock space spanned by
∣n>λ bases, to a new one, spanned by an extended two-parameters bases
∣n>λ1,λ2. These bases are now the eigen-states of a non-Hermitian Hamiltonian
Hλ1,λ2=aλ1,λ2†a+1/2, where
aλ1,λ2†=a†+λ1a+λ2 and
a are respectively, the deformed creation and ordinary bosonic annihilation operators. The bases
∣n>λ1,λ2 are non-orthogonal (squeezed states), but normalizable. Then, we deduce the new representations of coherent and squeezed states, in our two-parameters Fock space. Finally, we discuss the quantum statistical properties, as well as the non-classical properties of the obtained states, numerically.