On perturbation of embedded eigenvalues. (English) Zbl 0865.35093

Agmon, Shmuel

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

Keywords:

continuous spectrum; elliptic operators on Riemannian manifolds; perturbation theory for resonances