Spectral asymptotics for resolvent differences of elliptic operators with δ and δ'-interactions on hypersurfaces

Behrndt, Jussi and Grubb, Gerd and Langer, Matthias and Lotoreichik, Vladimir (2015) Spectral asymptotics for resolvent differences of elliptic operators with δ and δ'-interactions on hypersurfaces. Journal of Spectral Theory, 5 (4). pp. 697-729. ISSN 1664-0403 (https://doi.org/10.4171/JST/111)

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Abstract

We consider self-adjoint realizations of a second-order elliptic differential expression on R n with singular interactions of δ and δ'-type supported on a compact closed smooth hypersurface in 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 δ and δ'-interaction, respectively. Our technique makes use of general pseudodifferential methods, classical results on spectral asymptotics of ψdo's on closed manifolds and Krein-type resolvent formulae.

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