Journal of Spectral Theory


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Volume 5, Issue 4, 2015, pp. 697–729
DOI: 10.4171/JST/111

Published online: 2015-12-03

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

Jussi Behrndt[1], Gerd Grubb[2], Matthias Langer[3] and Vladimir Lotoreichik[4]

(1) TU Graz, Austria
(2) Copenhagen University, Denmark
(3) University of Strathclyde, Glasgow, UK
(4) Nuclear Physics Institute, Řež - Prague, Czech Republic

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.

Keywords: Elliptic operator, $\delta$-potential, $\delta^{\prime}$-potential, singular values, spectral asymptotics

Behrndt Jussi, Grubb Gerd, Langer Matthias, Lotoreichik Vladimir: Spectral asymptotics for resolvent differences of elliptic operators with $\delta$ and $\delta^{\prime}$-interactions on hypersurfaces. J. Spectr. Theory 5 (2015), 697-729. doi: 10.4171/JST/111