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We define a natural basis for the algebra of $\frak{gosp}(1|2n)$-invariantdifferential operators on the affine superspace $\mathbb{C}^{1|2n}$. We provethat these operators lie in the image of the centre of the enveloping algebraof $\frak{gosp}(1|2n)$. Using this result, we compute explicit formulas for theeigenvalues of these operators on irreducible summands of$\mathcal{P}(\mathbb{C}^{1|2n})$. This settles the Capelli eigenvalue problemfor orthosymplectic Lie superalgebras in the cases that were not addressed inSahi-Salmasian-Serganova. Our main technique relies on an explicit calculationof a certain determinant with polynomial entries.

Explicit formulas for Eigenvalues of Capelli operators for the Lie superalgebra $\frak{osp}(1|2n)$