The duality of $\mathfrak{gl}_{m|n}$ and $\mathfrak{gl}_k$ Gaudin models

We establish a duality of the non-periodic Gaudin model associated with superalgebra $\mathfrak{gl}_{m|n}$ and the non-periodic Gaudin model associated with algebra $\mathfrak{gl}_k$. The Hamiltonians of the Gaudin models are given by expansions of a Berezinian of an $(m+n)\times(m+n)$ matrix in the case of $\mathfrak{gl}_{m|n}$ and of a column determinant of a $k\times k$ matrix in the case of $\mathfrak{gl}_k$. We obtain our results by proving Capelli type identities for both cases and comparing the results.