FIRST-ORDER 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 first-order differential equation with Lebesgue integrable coefficients, on the open interval $(a,b)$ of the real line $\mathbb{R},$ has the form, defining the differential expression $M[\cdot],$

$$M[y](x):=i\rho(x)y^{\prime}(x)+\tfrac{1}{2}i\rho^{\prime}%% (x)y(x)+q(x)y(x)=\lambda w(x)y(x)\;$$for all$$\;x\in(a,b)$$

where $\lambda\in\mathbb{C}$ is the complex spectral parameter. Here the coefficients $\rho,q,w$ satisfy the conditions

$$%% \begin{array}[c]{ll}%% (i) & \rho,q,w:(a,b)\rightarrow\mat... ...\ (iv) & w(x)>0\;\text{for almost all}\;x\in(a,b). \end{array}$$

The right-definite spectral analysis for this differential equation takes place in the Hilbert function space $L^{2}((a,b);w)$ with norm and inner-product

$$\left\Vert f\right\Vert _{w}^{2}:=\int_{I}w\left\vert f\right\vert ^{2}\;and\;(f,g)_{w}%% :=\int_{a}^{b}w(x)f(x)\overline{g}(x)\,dx.$$

A necessary and sufficient condition to ensure that the differential expression $M[\cdot]$ generates a maximal operator in $L^{2}((a,b);w)$ with equal deficiency indices $d^{\pm}=1$ whose self-adjoint restrictions have discrete spectra, is

$$\int_{a}^{b}\frac{w(x)}{\rho(x)}\,dx<+\infty.$$

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 $T$ be a self-adjoint restriction of the maximal operator generated by $M[\cdot];$ then $T$ has the following spectral properties$:$

$(i)$

The spectrum $\sigma(T)$ of $T$ in $L^{2}((a,b);w)$ is simple and discrete.

$(ii)$

The spectrum $\sigma(T)$ is unbounded above and below on $\mathbb{R}\subset\mathbb{C},$ and so may be denoted by, here $\mathbb{Z}%% =\{\ldots-2,-1,0,1,2,\ldots\},$

$$\sigma(T)=\{\lambda_{n}\in\mathbb{R}:n\in\mathbb{Z\}}%%$$

with

$$\lambda_{n}<\lambda_{n+1}\;$$for all$$\;n\in\mathbb{Z},\;$$and$$%% \;\lim_{n\,\rightarrow\,\pm\infty}\lambda_{n}=\pm\infty.$$

$(iii)$

There exists a positive number $k>0,$ with

$$k=2\pi\left( \int_{a}^{b}\frac{w(x)}{\rho(x)}\,dx\right) ^{-1},$$

such that

$$\lambda_{n+1}-\lambda_{n}=k\;$$for all$$\;n\in\mathbb{Z}.$$

$(iv)$

There exists an entire $(integral)$ function $\varphi :\mathbb{C}\rightarrow\mathbb{C},$ generated by the boundary value problem, with the properties

$$%% \begin{array}[c]{ll}%% (i) & \varphi(\lambda)=0\;\text{if ... ...\lambda_{n})\neq0\;\text{for all}\;n\in\mathbb{Z}. \end{array}$$

2. KRAMER ANALYTIC KERNELS

The boundary value problems discussed in Section 1 generate Kramer analytic kernels in the Hilbert space $L^{2}((a,b);w).$

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.

Bibliography

1

W.N. Everitt and L. Markus. `The Glazman-Krein-Naimark theorem for ordinary differential operators.' Operator Theory: Advances and Applications 98 (1997), 118-130.

2

W.N. Everitt, G. Nasri-Roudsari and J. Rehberg. `A note on the analytic form of the Kramer sampling theorem.' Results in Mathematics. 34 (1998), 310-319.

3

W.N. Everitt and Anthippi Poulkou. `Kramer analytic kernels and first-order boundary value problems.' Jour. Computational Appl. Math. (To appear.)