Workshop
ON MODERN MATHEMATICAL
ANALYSIS AND APPLICATIONS
*******
Topic 1:
Spectral problems nonlinear in the spectral
parameter and applications

Prof. Reinhard Franz Ernst Norbert Mennicken,
Department of Mathematics, University of Regensburg, Germany

From 14.08.2009 to 19.08.2009
at the Lecture Room 416 (4-th oor), House T1,
334 Nguyen Trai str., Hanoi.

Spectral problems nonlinear in the spectral parameter frequently occur in applications such as in physics and engineering.

Often they appear with the investigation of boundary value problems for systems of ordinary or partial differential equations of mixed order.

Examples of such problems are the linear Navier-Stokes equation in uid dynamics, the system of equations for the force operator in magnetohydrodynamics (MHD), the system of equations for rotating compressible uid masses in astrophysics, and the system of equations
concerning the movements of a coil spring.

In my (introductory) lecture I will talk about joint work from a series of papers with V. Adamjan (Odessa), M. Faierman (Johannesburg, now Sydney), H. Langer (Vienna), M. Moller (Johannesburg), S. Naboko (St. Petersburg), A. Shkalikov (Moscow), and C. Tretter (Bern).

For motivation I will start my lecture with a series of examples. I will speak about the essential and discrete spectrum of these examples, further on limit points of the discrete spectrum, and on spectral decomposition and basis properties. The spectral problems for systems of differential equations of mixed order are considered as spectral problem for 2x2 block operator matrices in product of Hilbert spaces. By eliminating one variable one obtains an equation in one of the Hilbert spaces which is nonlinear in the spectral parameter. In case of MHD this is the so-called Hain-Lust equation. It is a modi cation of the transfer function in system theory.

One of the main results is the characterization of the essential spectral for the block operator in terms of the essential spectrum of the operator function nonlinear in the spectral parameter. Another main result is the characterization of spectral subspaces of the 2x2 block
operator in terms of its entries which is related to basis properties in the underlying Hilbert
spaces.

For further investigations and an extensive list of references I would like to mention the
recently published monograph of (my former student ) Christiane Tretter, University of Bern,
Switzerland, on
"Spectral Theory of Block Operator Matrices and Applications",
Imperial College Press, 2008.

 ( Download file PDF)