Bernoulli Numbers and Multi-brane Solutions in Cubic String Field Theory

In a previous paper [arXiv:1901.01681], we presented an analytic construction of multi-brane solutions in cubic open string field theory (CSFT) for any integer brane number. Our $(N+1)$-brane solution is given in the pure-gauge form $\Psi=U Q_\textrm{B}U^{-1}$ in terms of a unitary string field $U$ which is specified by $[N/2]$ independent real parameters $\alpha_k$. We saw that, for various sample values of $N$ $(=2, 3, 4, 5,\cdots)$, $\alpha_k$ can be consistently determined by two requirements: The energy density from the action should reproduce that of $(N+1)$-branes, and the EOM of the solution against the solution itself should hold. In this paper, we complete our construction by determining $\alpha_k$ satisfying the two requirements for a generic $N$. We find that each $\alpha_k$ is given in a closed form by using the Bernoulli numbers. We also present some supplementary results on our solution; the energy density of the solutions determined from its gravitational coupling, and the unitary string field $U$ as an exponential function.