Coulomb blockade of a nearly-open Majorana island

We consider the ground-state energy and the spectrum of the low-energy excitations of a Majorana island formed of topological superconductors connected by a single-mode junction of arbitrary transmission. Coulomb blockade results in $e$-periodic modulation of the energies with the gate-induced charge. We find the amplitude of modulation as a function of reflection coefficient $\mathcal{R}$. The amplitude scales as $\sqrt{\mathcal{R}}$ in the limit $\mathcal{R}\to 0$. At larger $\mathcal{R}$, the dependence of the amplitude on the Josephson and charging energies is similar to that of a conventional-superconductor Cooper-pair box. The crossover value of $\mathcal{R}$ is small and depends on the ratio of the charging energy to superconducting gap.