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Sturm-Liouville difference equations and banded matrices

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*Werner Kratz*

**Address.** Abteilung Angewandte Analysis, Universitat Ulm, D - 89069
Ulm, Germany
**E-mail:** kratz@mathematik.uni-ulm.de

**Abstract.** In this paper we consider {\it discrete} Sturm-Liouville
eigenvalue problems of the form $$ L(y)_k := \sum^n_{\mu =0} (-\Delta)^\mu
\{r_\mu(k)\Delta^\mu y_{k+1-\mu}\} = \lambda \rho(k) y_{k+1} $$ $$ for
\;\; 0\le k \le N-n \;\; with \;\; y_{1-n}= \cdots = y_0 = y_{N+2-n}= \cdots
= y_{N+1} = 0, $$ where $N$ and $n$ are integers with $ 1 \le n \le N$
and with the assumptions that $r_n(k) \not=0,\, \rho(k)>0$ for all $k.$
These problems correspond to eigenvalue problems for symmetric, banded
matrices ${\cal A} \in \IR^{(N+1-n)\times(N+1-n)}$ with band-width $2n+1.$
We present the following results: - a formula for the chracteristic polynomial
of ${\cal A},$ which yields a {\it recursion} for its calculation - an
{\it oscillation theorem}, which generalizes Sturm's well-known theorem
on Sturmian chains, and - an inversion formula, which shows that {\it every}
symmetric, banded matrix corresponds uniquely to a Sturm-Liouville eigenvalue
problem of the above form.

**AMSclassification.** 39A10, 39A12, 65F15, 15A18

**Keywords.** Sturm-Liouville equations, banded matrices, eigenvalue
problems; Sturmian chains.