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

Abstract

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.