The duality of ${\mathfrak{g}\mathfrak{l}}_{m\mathrm{\mid }n}$ and ${\mathfrak{g}\mathfrak{l}}_k$ Gaudin models

We establish a duality of the non-periodic Gaudin model associated with superalgebra ${\mathfrak{g}\mathfrak{l}}_{m\mathrm{\mid }n}$ and the non-periodic Gaudin model associated with algebra ${\mathfrak{g}\mathfrak{l}}_k$. 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 ${\mathfrak{g}\mathfrak{l}}_{m\mathrm{\mid }n}$ and of a column determinant of a $k\times k$ matrix in the case of ${\mathfrak{g}\mathfrak{l}}_k$. We obtain our results by proving Capelli type identities for both cases and comparing the results.