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We consider self-adjoint realizations of a second-order elliptic differentialexpression 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 formulaewith refined remainder estimates for the singular values of the resolventdifference between the standard self-adjoint realizations and the operatorswith a $\delta$ and $\delta^\prime$-interaction, respectively. Our techniquemakes use of general pseudodifferential methods, classical results on spectralasymptotics of $\psi$do's on closed manifolds and Krein-type resolventformulae.

Spectral asymptotics for resolvent differences of elliptic operators with $\delta$ and $\delta^\prime$-interactions on hypersurfaces