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On perturbation of embedded eigenvalues. (English) Zbl 0865.35093
Hörmander, Lars (ed.) et al., Partial differential equations and mathematical physics. The Danish-Swedish analysis seminar, Copenhagen, Denmark, Lund, Sweden, March 17-19, May 19-21, 1995. Proceedings. Boston, MA: Birkhäuser. Prog. Nonlinear Differ. Equ. Appl. 21, 1-14 (1996).
The paper deals with results extending the following theorem about instability of embedded eigenvalues which is due to Y. Colin de Verdière: Given any open set $$\Omega\subset\subset M$$ with $$M$$ a non-compact hyperbolic surface having a finite area with metric $$g$$, there exists a metric $$h$$ conformal to $$g$$ on $$M$$, $$h=g$$ on $$M\backslash\Omega$$, $$h$$ arbitrarily close to $$g$$ in $$\Omega$$, such that $$\Delta_h$$ (the Laplacian in the metric $$h$$) has no eigenvalues embedded in the continuous spectrum.
Here, it is shown that such results hold for a general class of elliptic operators on Riemannian manifolds. The main assumption is that the operators possess resonances (i.e., certain poles of the resolvent operator function, suitably modified) and a part of the paper is devoted to the presentation of a perturbation theory for resonances.
For the entire collection see [Zbl 0836.00030].
Reviewer: A.Kufner (Praha)

##### MSC:
 35P05 General topics in linear spectral theory for PDEs 58J50 Spectral problems; spectral geometry; scattering theory on manifolds 58J05 Elliptic equations on manifolds, general theory