Trace formulae and singular values of resolvent power differences of self-adjoint elliptic operators

Jussi Behrndt, Matthias Langer, Vladimir Lotoreichik

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Abstract

In this note self-adjoint realizations of second order elliptic differential expressions with non-local Robin boundary conditions on a domain Ω⊂Rn with smooth compact boundary are studied.  A Schatten--von Neumann type estimate for the singular values of the difference of the mth powers of the resolvents of two Robin realizations is obtained, and for m>n/2-1 it is shown that the resolvent power difference is a trace class operator. The estimates are slightly stronger than the classical singular value estimates by M.Sh. Birman where one of the Robin realizations is replaced by the Dirichlet operator.  In both cases trace formulae are proved, in which the trace of the resolvent power differences in L2(Ω) is written in terms of the trace of derivatives of Neumann-to-Dirichlet and Robin-to-Neumann maps on the boundary space L2(∂Ω).

Original languageEnglish
Pages (from-to)319-337
Number of pages19
JournalJournal of the London Mathematical Society
Volume88
Issue number2
Early online date21 Apr 2013
DOIs
Publication statusPublished - Oct 2013

Keywords

  • elliptic operator
  • resolvent power differences
  • trace formulae
  • singular values
  • self-adjoint elliptic operators

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