Conformal generation of an exotic rotationally invariant harmonic oscillator

An exotic rotationally invariant harmonic oscillator (ERIHO) is constructed by applying a nonunitary isotropic conformal bridge transformation (CBT) to a free planar particle. It is described by the isotropic harmonic oscillator Hamiltonian supplemented by a Zeeman type term with a real coupling constant $g$. The model reveals the Euclidean ($|g|<1$) and Minkowskian ($|g|>1$) phases separated by the phases $g=+1$ and $g=-1$ of the Landau problem in the symmetric gauge with opposite orientation of the magnetic field. A hidden symmetry emerges in the system at rational values of $g$. Its generators, together with the Hamiltonian and angular momentum produce nonlinearly deformed $\mathfrak{u}(2)$ and $\mathfrak{g}\mathfrak{l}(2,\mathbb{R})$ algebras in the cases of $0<|g|<1$ and $\infty >|g|>1$, which transmute one into another under the inversion $g\to -1/g$. Similarly, the true, $\mathfrak{u}(2)$, and extended conformal, $\mathfrak{g}\mathfrak{l}(2,\mathbb{R})$, symmetries of the isotropic Euclidean oscillator ($g=0$) interchange their roles in the isotropic Minkowskian oscillator ($|g|=\infty$), while two copies of the $\mathfrak{g}\mathfrak{l}(2,\mathbb{R})$ algebra of analogous symmetries mutually transmute in Landau phases. We show that the ERIHO system is transformed by a peculiar unitary transformation into the anisotropic harmonic oscillator generated, in turn, by anisotropic CBT. The relationship between the ERIHO and the subcritical phases of the harmonically extended Landau problem, as well as with a plane isotropic harmonic oscillator in a uniformly rotating reference frame, is established.

Figure 1

Trajectories for some rational values of $g$. In cases (b), (e), and (h), $p_{\phi }=0$ and trajectories pass through the origin. In cases (a), (d) and (g), $p_{\phi }<0$, while cases (c), (f) and (i) correspond to $p_{\phi }>0$.

Figure 2

Trajectories for some rational values of $g$ and $R_1/R_2$. “Dual” figures (a) and (d), see below, correspond to a general case $R_1|{\ell }_1|=R_2|{\ell }_2|$ of the trajectories with cusps, in which velocity turns into zero. In dual cases (b)-(e) and (c)-(f) the indicated equality is violated, and corresponding trajectories are smooth.

Figure 3

Trajectories of the anisotropic harmonic oscillator with commensurable frequencies.