Rational deformations of conformal mechanics

We study deformations of the quantum conformal mechanics of de Alfaro-Fubini-Furlan with a rational additional potential term generated by applying the generalized Darboux-Crum-Krein-Adler transformations to the quantum harmonic oscillator and by using the method of dual schemes and mirror diagrams. In this way we obtain infinite families of isospectral and nonisospectral deformations of the conformal mechanics model with special values of the coupling constant $g=m(m+1)$, $m\in \mathbb{N}$, in the inverse square potential term, and for each completely isospectral or gapped deformation given by a mirror diagram, we identify the sets of the spectrum-generating ladder operators that encode and coherently reflect its fine spectral structure. Each pair of these operators generates a nonlinear deformation of the conformal symmetry, and their complete sets pave the way for investigation of the associated superconformal symmetry deformations.

Figure 1

The mirror diagram for the dual schemes in (3.7).

Figure 2

Potential (d4) of the system (5.3). The energy levels of the corresponding physical states annihilated by ladder operators ${\mathcal{B}}^{-}$, ${\mathcal{B}}^{+}$, ${\mathcal{A}}^{-}$, ${\mathcal{A}}^{+}$, and ${\mathcal{C}}^{-}$ are indicated from left to right.

Figure 3

Left panel: The numbers on the left correspond to the indices of the physical eigenstates ${\psi }_{2l+1}$ of the half-harmonic oscillator that are mapped “horizontally” by operator ${\mathbb{A}}_{(+)}^{-}$ into eigenstates ${\mathrm{\Psi }}_n$ of the system (5.3). Lines show the action of the ladder operators coherently with their structure (5.12), (5.10), and (5.14). The marked set of the states 0,1,2,3,5,8 on the right corresponds to six eigenstates of $L_{(+)}$ annihilated by ${\mathcal{C}}^{-}$. Right panel: Horizontal lines correspond to the energy levels of $L_{(+)}$. Upward and downward arrows represent the action of the rising and lowering ladder operators, respectively. Following appropriate paths, any eigenstate can be transformed into any other eigenstate by applying subsequently the corresponding ladder operators.