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Given a bounded open set $\Omega\subseteq{\mathbb{R}}^n$, we consider theeigenvalue problem of a nonlinear mixed local/nonlocal operator with vanishingconditions in the complement of $\Omega$.  We prove that the second eigenvalue $\lambda_2(\Omega)$ is always strictlylarger than the first eigenvalue $\lambda_1(B)$ of a ball $B$ with volume halfof 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 eigenvalueproblem does not exist, but a minimizing sequence is given by the union of twodisjoint balls of half volume whose mutual distance tends to infinity.

A Hong-Krahn-Szeg\"{o} inequality for mixed local and nonlocal operators