Step-potential equilibrium in a ring condensate ($N_L/N_R=(2{\mu }_0-V_0)/(2{\mu }_0+V_0)$)

For equal ring halves with potentials $V_0$ and zero, the zero-current Thomas–Fermi bulk solution has $\mu ={\mu }_0+V_0/2$ and populations $N_{L,R}=N(1\mp V_0/2{\mu }_0)/2$ while both halves are occupied. For $V_0>2{\mu }_0$ the raised half is depleted and $\mu =2{\mu }_0$ in this approximation.