On Perturbation of Embedded Eigenvalues

Abstract

It is well known that in a proper setup eigenvalues which belong to the discrete spectrum are stable. This property is the basis of the perturbation theory for such eigenvalues. On the other hand, the behavior of eigenvalues which are embedded in the continuous spectrum is completely different. Such eigenvalues may be very unstable under perturbations. A striking example of instability of embedded eigenvalues was given by Colin de Verdière [4]. Consider the Laplace operator Δ _g on a non-compact hyperbolic surface M having a finite area, g the metric. It is well known that the self-adjoint realization of Δ _g (denoted by the same symbol) has a continuous spectrum which is the half-line: (−∞, − 1/4]. The discrete spectrum of Δg is a finite set. However, there are many interesting examples where Δ _g has infinitely many eigenvalues embedded in the continuous spectrum. That these examples are exceptional in some sense is shown by the following theorem due to Colin de Verdière.