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$.