New potentials for conformal mechanics

We find under some mild assumptions that the most general potential of 1-dimensional conformal systems with time independent couplings is expressed as $V=V_0+V_1$, where $V_0$ is a homogeneous function with respect to a homothetic motion in configuration space and $V_1$ is determined from an equation with source a homothetic potential. Such systems admit at most an $SL(2,\bR)$ conformal symmetry which, depending on the couplings, is embedded in Diff(R)$ in three different ways. In one case, $SL(2,\bR)$ is also embedded in Diff(S^1). Examples of such models include those with potential $V=\alpha x^2+\beta x^{-2}$ for arbitrary couplings $\alpha$ and $\beta$, the Calogero models with harmonic oscillator couplings and non-linear models with suitable metrics and potentials. In addition, we give the conditions on the couplings for a class of gauge theories to admit a $SL(2,\bR)$ conformal symmetry. We present examples of such systems with general gauge groups and global symmetries that include the isometries of $AdS_2 x S^3$ and $AdS_2 x S^3 x S^3$ which arise as backgrounds in $AdS_2/CFT_1$.