Albrecht Böttcher, Hermann Brunner, Arieh Iserles, Syvert P. Nørsett : On the Singular Values and Eigenvalues of the Fox-Li and Related Operators

Author(s):

Albrecht Böttcher
Hermann Brunner
Arieh Iserles
Syvert P. Nørsett

Title:

On the Singular Values and Eigenvalues of the Fox-Li and Related Operators

Electronic source:

application/pdf

Preprint series:

Technische Universität Chemnitz, Fakultät für Mathematik (Germany). Preprint 4, 2010

Mathematics Subject Classification:

47B35	[ Toeplitz operators, Hankel operators, Wiener-Hopf operators ]
45C05	[ Eigenvalue problems ]
47B06	[ Riesz operators; eigenvalue distributions; approximation numbers, $s$-numbers, Kolmogorov numbers, entropy numbers, etc. of operators ]
65R20	[ Integral equations ]
78A60	[ Lasers, masers, optical bistability, nonlinear optics ]

Abstract:

The Fox-Li operator is a convolution operator over a finite in- terval with a special highly oscillatory kernel. It plays an important role in laser engineering. However, the mathematical analysis of its spectrum is still rather incomplete. In the present paper we show how standard Wiener-Hopf theory can be used to obtain insight into the behaviour of the singular values of the Fox-Li operator. In addition, several approximations to the spectrum of the Fox-Li operator are discussed and results on the singular values and eigenvalues of certain related operators are derived.

Keywords:

Fox-Li operator, Wiener-Hopf operator, oscillatory kernel, eigenvalue, singular value

Language:

English

Publication time:

03/2010