FIRST-ORDER LINEAR BOUNDARY VALUE PROBLEMS
W N EVERITT
School of Mathematics and Statistics
University of Birmingham
Birmingham B15 2TT, UK
Department of Mathematics
University of Athens
Athens 157 84, Greece
This lecture reports on joint work with Anthippi Poulkou, Department of
Mathematics, University of Athens.
The general Lagrange symmetric first-order differential equation with
integrable coefficients, on the open interval of the real line
has the form, defining the differential expression
is the complex spectral parameter. Here the
coefficients satisfy the conditions
The right-definite spectral analysis for this differential equation
place in the Hilbert function space
with norm and
A necessary and sufficient condition to ensure that the differential
expression generates a maximal operator in
equal deficiency indices whose self-adjoint restrictions
discrete spectra, is
With this condition satisfied the GKN boundary condition method can be
to give symmetric boundary value problems with the following properties:
be a self-adjoint restriction of the maximal operator generated
has the following spectral properties
- The spectrum of in
simple and discrete.
- The spectrum is unbounded above and below on
and so may be denoted by, here
- There exists a positive number with
- There exists an entire
generated by the boundary value
with the properties
2. KRAMER ANALYTIC KERNELS
The boundary value problems discussed in Section 1 generate Kramer
kernels in the Hilbert space
Acknowledgement The authors are indebted to the
Michael Plum and Hubert Kalf for technical help in the preparation of
manuscript and for correcting errors in the first draft of the paper.
- W.N. Everitt and L. Markus. `The Glazman-Krein-Naimark
for ordinary differential operators.' Operator Theory: Advances
Applications 98 (1997), 118-130.
- W.N. Everitt, G. Nasri-Roudsari and J. Rehberg. `A note
analytic form of the Kramer sampling theorem.' Results in
34 (1998), 310-319.
- W.N. Everitt and Anthippi Poulkou. `Kramer analytic
first-order boundary value problems.' Jour. Computational Appl.
Eastham Meeting at Gregynog, 26-27 July 2002