The quantum harmonic oscillator on the sphere and the hyperbolic plane

Abstract

A nonlinear model of the quantum harmonic oscillator on two-dimensional spaces of constant curvature is exactly solved. This model depends of a parameter $\la$ that is related with the curvature of the space. Firstly the relation with other approaches is discussed and then the classical system is quantized by analyzing the symmetries of the metric (Killing vectors), obtaining a $\la$-dependent invariant measure $d\mu_\la$ and expressing the Hamiltonian as a function of the Noether momenta. In the second part the quantum superintegrability of the Hamiltonian and the multiple separability of the Schr\"odinger equation is studied. Two $\la$-dependent Sturm-Liouville problems, related with two different $\la$-deformations of the Hermite equation, are obtained. This leads to the study of two $\la$-dependent families of orthogonal polynomials both related with the Hermite polynomials. Finally the wave functions $\Psi_{m,n}$ and the energies $E_{m,n}$ of the bound states are exactly obtained in both the sphere $S^2$ and the hyperbolic plane $H^2$.Comment: 35 pages, 7 figure

An update on the classical and quantum harmonic oscillators on the sphere and the hyperbolic plane in polar coordinates

“…As a final point, it is worth observing that the λ-deformed Hermite polynomials, shown to occur in the remaining two coordinate systems wherein the Schrödinger equation (27) is separable [7], can be identified as classical Gegenbauer polynomials too.…”

“…During many years, there has been a continuing interest for some generalizations [1,2,3,4] of a classical nonlinear oscillator [5,6], which was introduced as a one-dimensional analogue of some quantum field theoretical models, and for the corresponding extensions [2,3,7,8,9,10,11] of its quantum version [12,13]. Such a model is indeed an interesting example of a system with nonlinear oscillations with a frequency showing amplitude dependence.…”

“…Later on, the quantum version of this classical model was also exactly solved in the corresponding three coordinate systems, wherein the Schrödinger equation is separable [7,8].…”

On quantized Liénard oscillator and momentum dependent mass

“…In this section we will be specifically concerned with the quantized version of (21) and seek a solution of the corresponding Schrödinger equation having momentum-dependent mass and the potential function given in (22) in contrast to the coordinate-dependent mass situation that has been well studied in the literature [15,16,17,18,19] in the configuration space. In fact taking cue from such investigations we begin this section with a von Roos type of decomposition [20] for the generic Hamiltonian in the momentum space…”

A Super-Integrable Two-Dimensional Non-Linear Oscillator with an Exactly Solvable Quantum Analog

“…We shall show that the deformed nonlinear oscillator is one of such examples. Furthermore, this can be done in different ways [44]. First, one can check that all the measures dµ = ρ(x, y)dx ∧ dy invariant under the vector fields…”