Oscillators, 3-dimensional harmonic algebras

In 32 the authors consider the solution of the two-dimensional time-independent Schrodinger equation by partial discretization. The discretized problem is treated as a problem of the numerical solution of system of ordinary differential equations and has been solved numerically by symplectic methods. The problem is then transformed into an algebraic eigenvalue problem involving real, symmetric, large sparse matrices. As numerical illustrations the authors have found the eigenvalues of the two-dimensional harmonic oscillator... 

As is well known, the standard 3-dimensional harmonic oscillator is a manifestation of the standard u 3) D so(3) symmetry. It is instructive to see how a g-deformed version of the 3-dimensional harmonic oscillator is related to the u,(3) algebra and its so,(3) subalgebra. The construction of the Hamiltonian of the g-deformed, 3-dimensional harmonic oscillator is a non-trivial problem, because one has first to construct the square of the g-deformed angular momentum operator. 

The algebraic version of an n-dimensional harmonic oscillator is a completely standard exercise in modern quantum mechanics textbooks [12]. Here we simply recall those aspects of some import to our pursuit. First, we recall that one has to replace usual and p, space coordinates... 

This expression includes terms up to two-body interactions. The algebraic Hamiltonian (2.27) of the ( -dimensional) harmonic oscillator is, of course, a special case of Eq. (2.36). One then observes that it is possible to arrange the Hamiltonian (2.36), in the framework of a dynamical algebra, by explicitly introducing the bilinear products... 

At this point one can introduce an algebraic bookkeeping of states for the three-dimensional (truncated) harmonic oscillator by taking into account the degeneracy chain of transformation groups given by... 

To summarize, we have seen how the simple Lie algebra U(2) embeds within itself the analytical solutions of a potentially infinite class of one-dimensional quantum problems. Furthermore, the harmonic, Morse, and Poschl-Teller oscillators are solved instantaneously in terms of the trivial expansions (2.61) and (2.64). However, the most important aspect of this study rests in the construction of a zeroth-order Hamiltonian which... 

To begin with, we recall that in certain cases, the algebraic model has been already put in a one-to-one correspondence with a specific potential function for the usual space coordinates. We have already studied dynamic symmetries providing exact solutions for the one-, two-, and three-dimensional truncated harmonic oscillators, the Morse and Poschl-Teller potential functions. When we consider more complicated algebraic expansions in terms of Casimir operators, or when we deal with coupled... 

The space of the 3-dimensional q-deformed harmonic oscillator consists of the completely S3nnmetric irreducible representations of the quantum algebra u (3) [12-14]. In this space a deformed angular momentum algebra, sOg(3), can be defined [10]. The Hamiltonian of the 3D -HO is defined so that it satisfies the following requirements ... 

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