J. Behrndt, G. Grubb, M. Langer and V. Lotoreichik,
Spectral asymptotics for resolvent differences of elliptic operators with δ
and δ'-interactions on hypersurfaces,
J. Spectr. Theory 5 (2015), 697–729
Abstract:
We consider self-adjoint realizations of a second-order elliptic
differential expression on ℝn with singular
interactions of δ and δ'-type
supported on a compact closed smooth hypersurface in ℝn.
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.