  Let $\mathfrak l:= \mathfrak q(n)\times\mathfrak q(n)$, where $\mathfrak
q(n)$ denotes the queer Lie superalgebra. The associative superalgebra $V$ of
type $Q(n)$ has a left and right action of $\mathfrak q(n)$, and hence is
equipped with a canonical $\mathfrak l$-module structure. We consider a
distinguished basis $\{D_\lambda\}$ of the algebra of $\mathfrak l$-invariant
super-polynomial differential operators on $V$, which is indexed by strict
partitions of length at most $n$. We show that the spectrum of the operator
$D_\lambda$, when it acts on the algebra $\mathscr P(V)$ of super-polynomials
on $V$, is given by the factorial Schur $Q$-function of Okounkov and Ivanov.
This constitutes a refinement and a new proof of a result of Nazarov, who
computed the top-degree homogeneous part of the Harish-Chandra image of
$D_\lambda$. As a further application, we show that the radial projections of
the spherical super-polynomials corresponding to the diagonal symmetric pair
$(\mathfrak l,\mathfrak m)$, where $\mathfrak m:=\mathfrak q(n)$, of
irreducible $\mathfrak l$-submodules of $\mathscr P(V)$ are the classical Schur
$Q$-functions.
