  Given a bounded open set $\Omega\subseteq{\mathbb{R}}^n$, we consider the
eigenvalue problem of a nonlinear mixed local/nonlocal operator with vanishing
conditions in the complement of $\Omega$.
  We prove that the second eigenvalue $\lambda_2(\Omega)$ is always strictly
larger than the first eigenvalue $\lambda_1(B)$ of a ball $B$ with volume half
of that of $\Omega$.
  This bound is proven to be sharp, by comparing to the limit case in which
$\Omega$ consists of two equal balls far from each other. More precisely,
differently from the local case, an optimal shape for the second eigenvalue
problem does not exist, but a minimizing sequence is given by the union of two
disjoint balls of half volume whose mutual distance tends to infinity.
