Explicit formulas for Eigenvalues of Capelli operators for the Lie superalgebra $\mathfrak{o}\mathfrak{s}\mathfrak{p}(1\mathrm{\mid }2n)$

We define a natural basis for the algebra of $\mathfrak{g}\mathfrak{o}\mathfrak{s}\mathfrak{p}(1\mathrm{\mid }2n)$-invariant differential operators on the affine superspace ${\mathbb{C}}^{1\mathrm{\mid }2n}$. We prove that these operators lie in the image of the centre of the enveloping algebra of $\mathfrak{g}\mathfrak{o}\mathfrak{s}\mathfrak{p}(1\mathrm{\mid }2n)$. Using this result, we compute explicit formulas for the eigenvalues of these operators on irreducible summands of $\mathcal{P}({\mathbb{C}}^{1\mathrm{\mid }2n})$. This settles the Capelli eigenvalue problem for orthosymplectic Lie superalgebras in the cases that were not addressed in Sahi-Salmasian-Serganova. Our main technique relies on an explicit calculation of a certain determinant with polynomial entries.