AnalyticQball solutions in a parabolic-type potential

We introduce a piecewise parabolic potential for a complex scalar field and we show that it admits stable Q ball solutions. These solutions can be found analytically, unlike the case of polynomial potentials. We find stable Q ball solutions, for large enough values of the charge, even when the potential has only one minimum. There can also exist Q balls immersed in a supercooled false vacuum.