"2003 Colloquium on Differential Equations and Applications,
Maracaibo, Venezuela.
Electronic Journal of Differential Equations,
Conference 13, 2005, pp. 49-56.
Title: Friedrichs model operators of absolute type with one singular point
Author: Serguei I. Iakovlev (Univ. Simon Bolivar, Caracas, Venezuela)
Abstract:
Problem of existence of the singular spectrum on the continuous
spectrum emerges in some mathematical aspects of quantum
scattering theory and quantum solid physics.
In the latter field, this phenomenon results from physical
effects such as the Anderson transitions in dielectrics.
In the study of this problem, selfadjoint Friedrichs model
operators play an important part and constitute
quite an apt model of real quantum Hamiltonians.
The Friedrichs model and the Schrodinger operator are
related via the integral Fourier transformation.
Similarly, the relationship between the Friedrichs model and
the one dimensional discrete Schrodinger operator on
$\mathbb{Z}$ is established with the help of the Fourier series.
We consider a family of selfadjoint operators of the Friedrichs model.
These absolute type operators have one singular point $t=0$ of
positive order. We find conditions that guarantee the absence of
point spectrum and the singular continuous spectrum for such operators
near the origin. These conditions are actually necessary and sufficient.
They depend on the finiteness of the rank of a perturbation operator
and on the order of singularity. The sharpness of these conditions
is confirmed by counterexamples.
Published May 30, 2005.
Math Subject Classifications: 47B06, 47B25.
Key Words: Analytic functions; eigenvalues; Friedrichs model;
linear system; modulus of continuity; selfadjoint operators;
singular point.