  We consider self-adjoint realizations of a second-order elliptic differential
expression on ${\mathbb R}^n$ with singular interactions of $\delta$ and
$\delta^\prime$-type supported on a compact closed smooth hypersurface in
${\mathbb R}^n$. In our main results we prove spectral asymptotics formulae
with refined remainder estimates for the singular values of the resolvent
difference between the standard self-adjoint realizations and the operators
with a $\delta$ and $\delta^\prime$-interaction, respectively. Our technique
makes use of general pseudodifferential methods, classical results on spectral
asymptotics of $\psi$do's on closed manifolds and Krein-type resolvent
formulae.
