Atlas: Eigenvalues of some block operator matrix by Matthias Langer

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18th International Conference on Operator Theory
June 27 - July 1, 2000
University of the West
Timisoara, Romania

Organizers
Dumitru Gaspar, Traian Ceausu, Aurelian Craciunescu, Aurelian Gheondea, Radu-Nicolae Gologan, Ciprian Pop, Dan Popovici, Nicolae Suciu, Alexandru Terescenco, Dan Timotin, Flavius Turcu

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Eigenvalues of some block operator matrix
by
Matthias Langer
Technical University of Vienna

We consider eigenvalues of the block operator matrix

~

A

= æ
ç
è

-\fracd²dx²+p
v

v
u
ö
÷
ø ,

in the space L²(0, 1)\oplusL²(0, 1) with Dirichlet boundary conditions for the first component, where p, u, and v are real functions.

The eigenvalues of this operator which are not embedded in the essential spectrum can be characterized by a min-max principle, which was introduced by P.A. Binding, D. Eschwé, and H. Langer, using the Schur complement of the operator matrix. For the enumeration of the eigenvalues there is a shift involved. This shift is related to the number of those values embedded in the essential spectrum where a vector exists which fulfills the eigenvalue equation but which does not fulfill the boundary condition at the right endpoint of the interval [0, 1].

Date received: June 2, 2000