Perturbation of essential spectra of exterior elliptic problems

Abstract

For a second-order symmetric strongly elliptic differential operator on an exterior domain in ℝ ⁿ , it is known from the works of Birman and Solomiak that a change in the boundary condition from the Dirichlet condition to an elliptic Neumann or Robin condition leaves the essential spectrum unchanged, in such a way that the spectrum of the difference between the inverses satisfies a Weyl-type asymptotic formula. We show that one can increase, but not diminish, the essential spectrum by imposition of other Neumann-type nonelliptic boundary conditions. The results are extended to 2m-order operators, where it is shown that for any selfadjoint realization defined by an elliptic normal boundary condition (other than the Dirichlet condition), one can augment the essential spectrum at will by adding a suitable operator to the mapping from free Dirichlet data to Neumann data. We also show here an extension of the spectral asymptotics formula for the difference between inverses of elliptic problems. The proofs rely on Kreĭn-type formulae for differences between inverses, and cutoff techniques, combined with results on singular Green operators and their spectral asymptotics.

Keywords:

• exterior domain
• essential spectrum
• singular Green operator
• Schatten class
• Krein formula
• spectrally negligible cutoffs

AMS Subject Classifications::

• 35J40
• 35P20
• 35S15
• 47A10