JGSP 6 (2006) 109–117
FREDHOLM ANALYTIC OPERATOR FAMILIES AND
PERTURBATION OF RESONANCES
STANISLAV A. STEPIN
Communicated by Martin Schlichenmaier
Abstract. The purpose of this contribution is to display an outlook of a certain
approach that enables one to study various spectral perturbation phenomena such
as perturbation of eigenvalues and resonances (scattering poles) from the general
viewpoint. Some applications of this elaborated technique are presented as well.
1. Introduction
Let us consider a Hilbert space H and an operator function K(λ, α) which is
analytic in two variables and takes its values in the class of compact operators.
The variable λ stands for the “spectral parameter”, while the second variable α is
treated as a “perturbation parameter”. For a fixed α = α0 the point λ = λ0 is
called singular for the operator family K(λ, α0 ) if ker(K(λ0 , α0 ) − I) 6= {0}.
The problem in question is to study the analytic properties of the singular points
λ(α) as functions of the parameter α.
Below the notation K(λ) = K(λ, α0 ) will be used. Let n = dim ker(K(λ0 ) − I)
(0)
(0)
and {ϕ1 , . . . , ϕn } be a basis of the subspace ker(K(λ0 ) − I). It may hap(0)
pens that for a given eigenvector ϕj there exists a chain of adjoint vectors
(1)
(mj −1)
{ϕj , . . . , ϕj
} of maximal length (mj − 1) such that
(1)
(0)
(1)
K(λ0 )ϕj + K 0 (λ0 )ϕj = ϕj
1
(1)
(2)
(2)
(0)
K(λ0 )ϕj + K 0 (λ0 )ϕj + K 00 (λ0 )ϕj = ϕj
2
.....................
1
(m −1)
(m −1)
(0)
K(λ0 )ϕj j
+ ... +
K (mj −1) (λ0 )ϕj = ϕj j .
(mj − 1)!
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