FIRSTORDER LINEAR BOUNDARY VALUE PROBLEMS
W N EVERITT
w.n.everitt@bham.ac.uk
School of Mathematics and Statistics
University of Birmingham
Edgbaston
Birmingham B15 2TT, UK
ANTHIPPI POULKOU
apoulkou@cc.uoa.gr
Department of Mathematics
University of Athens
Panepistemiopolis
Athens 157 84, Greece
1. ABSTRACT
This lecture reports on joint work with Anthippi Poulkou, Department of
Mathematics, University of Athens.
The general Lagrange symmetric firstorder differential equation with
Lebesgue
integrable coefficients, on the open interval of the real line
has the form, defining the differential expression
for all
where
is the complex spectral parameter. Here the
coefficients satisfy the conditions
The rightdefinite spectral analysis for this differential equation
takes
place in the Hilbert function space
with norm and
innerproduct
A necessary and sufficient condition to ensure that the differential
expression generates a maximal operator in
with
equal deficiency indices whose selfadjoint restrictions
have
discrete spectra, is
With this condition satisfied the GKN boundary condition method can be
applied
to give symmetric boundary value problems with the following properties:
Theorem 1
Let
be a selfadjoint restriction of the maximal operator generated
by
then
has the following spectral properties
 The spectrum of in
is
simple and discrete.
 The spectrum is unbounded above and below on
and so may be denoted by, here
with
 There exists a positive number with
such that
for all
 There exists an entire
function
generated by the boundary value
problem,
with the properties
2. KRAMER ANALYTIC KERNELS
The boundary value problems discussed in Section 1 generate Kramer
analytic
kernels in the Hilbert space
Acknowledgement The authors are indebted to the
Professors
Michael Plum and Hubert Kalf for technical help in the preparation of
the
manuscript and for correcting errors in the first draft of the paper.

 1
 W.N. Everitt and L. Markus. `The GlazmanKreinNaimark
theorem
for ordinary differential operators.' Operator Theory: Advances
and
Applications 98 (1997), 118130.
 2
 W.N. Everitt, G. NasriRoudsari and J. Rehberg. `A note
on the
analytic form of the Kramer sampling theorem.' Results in
Mathematics.
34 (1998), 310319.
 3
 W.N. Everitt and Anthippi Poulkou. `Kramer analytic
kernels and
firstorder boundary value problems.' Jour. Computational Appl.
Math.
(To appear.)
Eastham Meeting at Gregynog, 2627 July 2002Gregynog Abstracts